BC Calculus 12

Edvantage Math

BC Calculus 12

2026

BC Calculus 12 Development Team

Authors

Bruce McAskill, BSc, BEd, MEd, PhD Deanna Catto, BSc, BEd Mathew Geddes, BSc, BEd, MMT Steve Bates, BEd, MEd

COPIES OF THIS BOOK MAY BE OBTAINED BY CONTACTING: Edvantage Interactive E-MAIL: info@edvantageinteractive.com TOLL-FREE CALL: 866.422.7310

BC Calculus 12: Student Edition Copyright © 2022, Edvantage Interactive 2026 Printing

All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, or stored in a database or retrieval system, without the prior written permission of Edvantage Interactive.

ISBN: 978-1-77430-307-8 Editorial Development Lead: Lionel Sandner Editor: Ryan Chowdhry Care has been taken to trace ownership of copyright material contained in this text. The publishers will gladly accept any information that will enable them to rectify any reference or credit in subsequent printings.

The Big Ideas at the beginning of each chapter are quoted from AP Calculus: Course and Exam Description published by the College Board, New York, NY. Advanced Placement, AP and College Board are registered trademarks of the College Board.

ii

Unit 1: Behavior of Functions

1

Chapter 1: Characteristics of Functions 1.1.

Characteristics of Polynomial Functions ……………………………………………………………………………………….

1.2.

Characteristics of Exponential and Logarithmetic Functions ………………….…………………………...……….

1.3.

Characteristics of Rational Functions …………………………….…………………………………………………………….

1.4.

Characteristics of Periodic Functions ………………………..….…….……………………..……………….……………….

1.5.

Chapter Review………………………………………………………………………………………………………….…………..…….

2 3 16 23 33 42

Chapter 2: Analyzing Functions Using Limits 2.1.

Rates of Change and Limits – Determing Limits Graphically …….……………………………………….….………

2.2.

Exploring Limits ……………….…………………………………………………………………………………………….…………….

2.3.

Asymptotic Behavior ……………………………………….……………………………………………………………….………….

2.4.

End Behavior of Functions ….…………………………………………………………………………………..………….….……

2.5.

Function Continuity …………………………………………………………………..………………………………………..………

2.6.

Intermediate Value Theorem ………….…………………………………………………………………………….…….……..

2.7.

Chapter Review……………………………………………………………………………………………………..……………….…….

49 50 57 71 82 90 104 110

Unit 1 Limits and Discontinuity………………………………………………………………………….…………………………..…………

114

Unit 2: Differential Calculus

115

Chapter 3: The Derivative 3.1.

Slope of a Function ……………………………………………………………………………………………………………………..

3.2.

Definition of the Derivative…………………………………………………………………………………………….…………….

3.3.

Differentiability of a Function…………………………………………………………………………………..………….….……

3.4.

Graphing the Derivative of a Function …………………………………………………………………………….…….……..

3.5.

Chapter Review……………………………………………………………………………………………………..……………….…….

116 117 128 139 149 161

Chapter 4: Differentiation Techniques 4.1.

Derivatives of Polynomial Functions …………………………………………………………………………………….………

4.2.

Higher Order Derivatives……………………………………………………………………………………………………...…….

4.3.

Velocity and Other Rates of Change………………………………………………………………………………………………

4.4.

Derivatives of Trigonometric Functions ………………………………………………………………………………………..

4.5.

Introducing the Chain Rule ………………………………………………………………………………………………….……….

4.6.

Chapter Review…………………………………………………………………………………………………………………………….

167 168 179 189 206 216 223

Chapter 5: Advanced Differentiation Techniques 5.1.

The Chain Rule …………………………………………………………………………………………………….……………………….

5.2.

Composite Functions and Function Notation ……………………………………………………………………………….

5.3.

Implicit Differentiation………………………………………………………………………………………………………………….

5.4.

Derivatives of Exponential and Logarithmic Functions …………………………………………………………………

5.5.

Logarithmic Differentiation………………………………………………………………………………….……………………….

5.6.

Chapter Review…………………………………………………………………………………………………………………………….

229 230 237 245 255 267 277

Unit 2 Project: Roller Coaster Design………………………………………………………………………….…………………..…………

282

Unit 3: Applications of Derivatives

285

Chapter 6: Analyzing Functions Using Derivatives 6.1.

The First Derivative Test …………………………………………………………………………………………..………………….

6.2.

Modelling and Optimization …………………………………………………………………………………….…………………

6.3.

The Second Derivative Test ………………………………………………………………………………………….………………

6.4.

Curve Sketching …………………………………………………………………………………………………………………..………

6.5.

The Mean Value Theorem ……………………………………………………………………………………………………………

6.6.

Chapter Review ……………………………………………………………………………………………………………………………

286 287 296 305 315 324 330

Chapter 7: Solving Problems Using Derivatives 7.1.

Related Rates Involving Shape and Space ………………………………………………………………………….…………

7.2.

Related Rates Involving Motion ……………………………………………………………………………………………………

7.3.

Linearization and Newtons Method of Approximation.………………..…….………………………………...……..

7.4.

Chapter Review ……………………………………………………………………………………………………………………………

335 336 344 352 360

Unit 3 Project: Shrinking Lollipop .…………………………………………………………………………………………………………….

365

Unit 4: Integral and the Fundamental Theorem of Calculus

367

Chapter 8: Antidifferentiation 8.1.

The Antiderivative – Working Backwards ………………………………………………………..…………..………………

8.2.

Antiderivative of Trigonometric Functions ……………………………………………………………..……………………

8.3.

Antidifferentiation Using Substitution.…………………………………………………………………………………………

8.4.

Advanced Antidifferentiation Techniques ……………………………………………………………………………………

8.5.

Antidifferentiation Involving Exponential and Logarithmic Functions …………………………………………..

8.6.

Integration by Parts ……………………………………………………………… …………………………………………………….

8.7.

Chapter Review ……………………………………………………………………………………………………………………………

368 369 374 378 385 390 396 405

Chapter 9: Solving Differential Equations 9.1.

Solving Differential Equations Analytically ……………………………………………………………………………………

9.2.

Solving Differential Equations Graphically ……………………………………………………………………………………

9.3.

Approximating Area Under a Curve Using Riemann Sums ……………………………………..…………………….

9.4.

Calculating Area Under Functions Graphically …………………………………………..…………………………………

9.5.

Chapter Review ……………………………………………………………………………………………………………………………

409 410 419 427 438 448

Unit 4 Project: Newton’s Law of Cooling .………………………………………………………………………………………………….

456

Unit 5: Applications of Integration

461

Chapter 10: Integral and the Fundamental Theorem of Calculus 10.1.

Integrating Using the Fundamental Theorem …………………………………………………….……..…………………

10.2.

Derivative of a Definite Integral ……………………………………………………………………………………………………

10.3.

The Average Value of a Function ……………………………………………………………………………….…………………

10.4.

The Integral as Net Change ………………………………………………………………………………………….………………

10.5.

Chapter Review ……………………………………………………………………………………………………………………………

462 463 475 485 492 501

Chapter 11: Area and Volume 11.1.

Area Between Two Curves ……………………………………………………………………………………………………………

11.2.

Volumes of Solids …………………………………………………………………………………………………………………………

11.3.

Finding Volume Using Cross-Sectional Area ..………………………………………………………….……………………

11.4.

Chapter Review …………………………………………………………………………………………………………………………...

505 506 516 523 531

Unit 5 Project: Solid of Revolution ..…………………………………………………..……………………………………………….…….

535

Unit 1: Behavior of Functions This unit focuses on the following Big Idea from the British Columbia provincial Calculus 12 curriculum: 

The concept of a limit is foundational to calculus

By the end of this unit, you should be able to:  Express limits symbolically using correct notation  Interpret limits expressed symbolically  Estimate limits of functions  Determine limits of functions  Deduce and interpret behavior of functions using limits  Analyze functions for intervals of continuity or points of discontinuity  Determine the applicability of important calculus theorems using continuity

By the end of this unit, you should know the meaning of these key terms:     

Constant multiple rule Continuity at a point Continuous function Difference rule End behavior models (left-end and right-end)

             

Extended function Horizontal asymptote Infinite discontinuity Intermediate Value Theorem (for continuous functions) Jump discontinuity Left-hand limit Limit One-sided limit Product rule Quotient rule Right-hand limit Sum rule Two-sided limit Vertical asymptote

When the Mars Rover lands it’s important to know its velocity at all times.

1

Chapter 1: Characteristics of Functions Many events that occur in everyday life involve two related quantities. For example, to operate a vending machine, you insert money and make a selection. The machine gives you the selected item and any change due. Once your selection is made, the amount of change you receive depends on the amount of money you put into the machine.

Money is the INPUT

Change is the OUTPUT

The vending machine is an example of a function where there is a rule that determines the OUTPUT (change) depending on the INPUT (money inserted). This chapter will review the characteristics of several important functions that you will be using throughout this course. These functions include:

2



polynomial



exponential



logarithmic



rational



periodic

1.1 Characteristics of Polynomial Functions

Which of the following functions are examples of polynomials? Explain. a)

b)

c)

d)

For each polynomial, identify the degree, leading coefficient and constant.

Polynomial Function

Degree

Leading Coefficient

Constant Term

Is it possible to determine the yintercept of the above polynomials? If so, how?

3

Definition: A polynomial function has the form where:     

n is a whole number x is a variable the coefficients an to a0 are real numbers the degree of the polynomial function is n, the exponent of the greatest power of x the leading coefficient is an, the coefficient of the greatest power of x, the constant term is a0.

Polynomial Graphs – A Summary

4



If the graph of a polynomial has two raising arms, then the degree of the polynomial is and the leading coefficient is . This means the left end behaviour is and the right end behaviour is .



If the graph of a polynomial has two drooping arms, then the degree of the polynomial is and the leading coefficient is is . This means the end behaviour is .



If the graph of a polynomial has a right arm up and a left arm down, then the degree of the polynomial is and the leading coefficient is . This means the end behaviour is .



If the graph of a polynomial has the left arm up and the right arm down, then the degree of the polynomial is and the leading coefficient is . This means the end behaviour is .

Example 1: Sketching a Polynomial Function a) Sketch the polynomial . b) Does the function have a minimum or maximum value? c) Explain your choice.

How to Do It

What to Think About What do the degree and leading coefficient tell you about the shape of the graph?

What does the constant term tell you about the graph?

5

Your Turn a) Sketch the polynomial

.

b) Does the function have an overall (global or absolute) minimum or maximum value?

c) Explain your choice.

6

Example 2: Determine the Equation of a Polynomial Function What is the equation of the polynomial, P(x), that has 3 identical x-intercepts at x = 2 (multiplicity of 3), 1 at x = 1 (multiplicity of 1) and a y-intercept at 16?

How to Do It

What to Think About What is the minimum degree of the polynomial function?

How does the degree of the polynomial relate to the shape of the graph?

What is the x-value when the yintercept is 16?

7

Your Turn Determine the equation of the polynomial function shown below.

8

Example 3: Write an Absolute Value Equation as a Piecewise Function What is the piecewise equation of the absolute function ?

How to Do It

What to Think About What is the equation of the line to the right of x = 2?

What is the equation of the line to the left of x = 2?

Why are the two functions opposite signs?

Your Turn What is the piecewise equation of the absolute function

?

9

Practice 1. For each polynomial function, state the degree. If the function is not a polynomial, explain why. a.

b.

c.

d.

2. State whether the polynomial function is odd or even degree. Then, state whether the function has a global maximum value, a global minimum value, or neither. a.

b.

c.

d.

10

3. State the number of real x-intercepts, domain, and range for each polynomial function. a.

b.

c.

11

4. Determine the equation for each polynomial function in question 3 a)

b)

c)

12

5. State the possible number of x-intercepts and the value of the y-intercept for each polynomial function. a.

b.

c.

d.

6. Identify the following characteristics for each polynomial function: • the type of function and whether it is of even or odd degree • the end behaviour of the graph of the function • the number of possible x-intercepts • whether the function will have a maximum or minimum value • the y-intercept a.

b.

13

7. Given the polynomial graphing.

14

, determine the following without

a.

Describe the end behaviour of the graph of the function.

b.

Determine the possible number of x-intercepts of the function.

c.

Determine the y-intercept of the function.

d.

Now, use graphing technology to create a sketch of the graph.

8.

Write the following functions in piecewise notation and graph the absolute value function.

a.

b.

9.

For

a. Algebraically determine the vertex of

b. Algebraically determine the x and y-intercepts of

c. Identify the domain and range of

d. Write

in piecewise notation.

15

1.2 Characteristics of Exponential and Logarithmic Functions On the same grid graph and its inverse. (Reminder: To graph an inverse function switch the x and y-values.)

Complete the table below:

Domain Range x-intercept y-intercept Equation of any Asymptotes

16

A logarithmic function is the inverse of an exponential function.

argument of the logarithm

value of the logarithm

base of the logarithm

    

The inside of the logarithm, in this case x, is called the argument of the logarithm.

The argument can never be negative.

The base, b > 0, b ≠ 1 When the base is e (Euler’s Number  2.71828…), then it is called the natural logarithm and is denoted as .

When the base is 10, then it called a common logarithm and is denoted by .

Did You Know

Euler’s number is also know as Napier’s constant because John Napier, a Scottish mathematian, published a book 113 years before Euler used the symbol e in his manuscript Meditatio en experimenta explosione tormentorum nuper instituta. Napier described the approximate motion of someone covering a distance d whose speed at each instant is equal to the remaining distance to be covered (in appropriate units).

He divided the time into short intervals of length λ, and assumed that the speed was constant within each short interval. He tabulated the corresponding values of distance and time obtained in this way. He coined a name for their relationship out of the Greek words logos (ratio) and arithmos (number). He used a latinized version of his word: logarithm.

17

Example 1: Convert Between and Exponential and Logarithmic Forms How to Do It

What to Think About What is the base of the logarithm?

What is the argument of the logarithm?

What does the logarithm represent?

What does the result in question e suggest?

Your Turn What is the logarithmic equivalent of

18

Laws of Logarithms Name

Law

Product Quotient

Power

Description The logarithm of a product of numbers is the sum of the logarithms of the numbers.

The logarithm of a quotient of numbers is the difference of the logarithms of the numbers.

The logarithm of a power of a number is the exponent times the logarithm of the number.

Example 2: Rewrite The Expression using a Single Logarithm. State the Restrictions on the Variable.

How to Do It

What to Think About Which logarithm rules can you use?

How can you prove the two answers are equivalent?

Why is there a restriction?

Your Turn Rewrite the expression

using a single logarithm. State the

restrictions on the variable.

19

Example 3: Solve the Following Equation using Two Methods How to Do It

What to Think About What must happen before exponential rules can be used?

Why can you equate the exponents?

Why was function?

applied to the

Could a different base have been used?

Your Turn Solve each equation algebraically to the nearest hundredth. a)

20

b)

Practice 1. Use the definition of a logarithm to evaluate each expression. a)

b)

c)

d)

c)

d)

4. Write each expression in terms of the individual logarithms of x, y, and z. a)

2. Express in logarithmic form. a) b) b)

c)

d)

3. Determine the value of x. a)

5. Use the laws of logarithms to simplify and evaluate each expression. a)

b) b)

21

6. Evaluate each of the following. a) If , determine the value of

.

8. If , write an algebraic expression in terms of k for each of the following. a)

b) b) Determine the value of and

7. Simplify. a)

b)

22

if

.

9. Write each expression as a single logarithm in simplest form. State any restrictions on the variable. a)

b)

1.3 Characteristics of Rational Functions Warm-Up: a) Graph

b) Use the graph of f(x) to graph

, the

reciprocal function.

c) Summarize the characteristics of the function

using a

table.

Characteristic Non-permissible Value Behaviour near non-permissible value.

End behaviour Domain Range Equation of Vertical Asymptote Equation of Horizontal Asymptote.

What equations could you use to graph

using

transformations?

23

Example 1: Sketch a Rational Function For How to Do It

: a) identify the characteristics; and, b) sketch the graph of the function.

What to Think About What is the parent function that you should start with and what does this look like?

How is the non-permissible value related to the domain and equation of the vertical asymptote?

How is the end behavior related to the range and horizontal asymptote?

24

What values should be labelled on the sketch?

Your Turn Identify the characteristics of

and use them to sketch a graph of the function.

Non-permissible Value Behaviour near non-permissible value.

x-intercept

y-intercept

25

Domain

Range

Equation of Vertical Asymptote Equation of Horizontal Asymptote.

26

Definition: Discontinuities If a rational function has non-permissible values, then the graph of the rational function will have some kind of discontinuity in the graph.

There are two types of discontinuities found in rational functions: Infinite Discontinuity – leading to a vertical asymptote on the graph.

Point Discontinuity – leading to a hole on the graph.

Example 2: Identify Types of Discontinuities For each function identify the indicated characteristics and use this information to determine the type of discontinuity. a) How to Do It

b) What to Think About Does the rational function simplify?

Should the non-permissible values be included in the simplified expression?

What do the graphs look like?

27

Your Turn a) Sketch the graph of the function

.

b) Identify the type(s) of discontinuity and where it occurs.

28

Practice 1. Sketch and graph each function. Identify the domain and range, intercepts, and asymptotes. a)

b)

29

c)

2. Analyse the function

. Then, complete the table.

Characteristic Non-permissible value Behaviour near non-permissible value End behaviour Domain Range Equation of vertical asymptote Equation of horizontal asymptote

30

3. Explain the behaviour at each non-permissible value in the graph of the rational function

.

4. Explain how the equation of the rational function

can be analysed

to determine whether the graph of the function has an asymptote or a point of discontinuity.

31

5. Complete the table for the given rational function.

Characteristic Non-permissible value(s) Feature exhibited at each non-permissible value Behaviour near each non-permissible value Domain Range 6. Analyse each function and predict the location of any vertical asymptotes, points of discontinuity, and intercepts. Then, graph the function to verify your predictions. a)

b)

32

1.4 Characteristics of Periodic Functions Warm Up The point P(12, 9) is on the terminal arm of an angle in standard position. a) Sketch a diagram to represent the angle in standard position.

b) Determine the length of the terminal arm, r.

c) Determine the exact value of the primary trigonometric ratios and their inverses.

d) Which trigonometric function would we have to use to calculate the measure of the reference angle? Why?

e) Calculate the measure of R, in radians.

f) Use your reference angle to calculate the measure of .

33

Example 1: Graph the Cosine Function How to Do It

What to Think About What special triangle is being used?

Will the graph look different if the angle is in degrees rather than radians?

Can the amplitude be greater than 1?

How many x-intercepts are there?

34

Your Turn Graph the sine function a) Complete the table of values for the function x

0



2

y b) Use the values above to graph the function

c) Complete the following characteristics.

Maximum Value Minimum Value Amplitude Period y-intercept x-intercept(s) Domain Range

35

Example 2: Graph the Tangent Function How to Do It

What to Think About What special triangle is being used?

Will the graph look different if the angle is in degrees rather than radians?

Can the amplitude be greater than 1?

How many x-intercepts are there?

How many vertical asymptotes are there?

36

Your Turn Complete the characteristic chart for

, then sketch the graph over

.

Max/Min Value Amplitude Period y-intercept x-intercept(s) Asymptote(s)

37

Summary

Sinusoidal functions of the form or are transformations of the parent functions and respectively. The parameters and the corresponding transformations include: a – vertical stretch by a factor of changes the amplitude to . If , the function is also reflected through the horizontal midline of the function. b – horizontal stretch by a factor of

changes the period to

radians. If

, the function is also reflected in the y-axis. c – horizontal translation, also called the phase shift, of c units to the left if c > 0 or right if c < 0. d – vertical translation, also called vertical displacement, of d units up if d > 0, or down if d < 0.

period = 6 amplitude = 2

, amplitude = 2, phase shift = 2, vertical displacement = 1 Sinusoidal functions are periodic functions. Tangent is a periodic function but is not a sinusoidal function. On the unit circle, you can express the coordinates of the point P on the terminal arms of angle as (x, y) or The slope of the terminal arm is represented by the tangent function,

. Therefore,

you can use the tangent function to model the slope of a line from a fixed point to a moving object as the object moves through a range of angles.

38

Practice 1. Graph each pair of functions on the same grid. For each, clearly plot the key points. a)

and

b)

and

c)

d)

and

and

39

2. For each function, determine the phase shift and vertical displacement with respect to .

a)

b)

3. Determine the period and range for each function. a)

b)

40

4. Given the following characteristics, write the equation of the sine function for each in the form . a) phase shift of

, period of

, vertical displacement of 5, and amplitude of 3

b) period of

and vertical displacement of 2

5. Graph

for

then:

a) State the coordinates of the x-intercepts.

b) State the equations of the asymptotes.

41

1.5 Chapter Review 1. Complete the chart for each polynomial function.

Polynomial Function

Degree

Type

Leading Coefficient

Constant Term

a) b) c) 2. For each of the following,  determine whether the graph represents on odd-degree or an even-degree polynomial function  determine whether the leading coefficient of the corresponding function is positive or negative  state the number of x-intercepts  state the domain and range a) b)

42

3. For each exponential function, state the domain, range, y-intercept, horizontal asymptote, and whether a graph of the function would be increasing or decreasing. Verify your answers by using technology to graph the functions. a)

b)

c)

4. Solve each of the following equations algebraically. Use graphing technology to checkyour answer. a)

b)

5. Write each expression in logarithmic form. a) b)

c)

c)

6. Use the laws of logarithms to evaluate each of the following. a)

b)

c)

43

7. Sketch a graph of each function. Label the important parts of the graph. a)

b)

8. Solve each rational equation algebraically. a)

44

b)

9. Graph at least two cycles of

. What are the amplitude and period in

radians? y

0

θ

10. Graph at least two cycles of in radians?

. What are the amplitude and period

y

0

θ

45

11.Without graphing, determine the amplitude and period, in radians, of each function. a)

b)

c)

d)

46

12. Determine the amplitude, period, phase shift, and vertical displacement with respect to or for each function. a)

b)

13. Graph at least two cycles of

.

47

14. Graph

15. Solve

48

over the domain

.

graphically over the domain

.

Chapter 2: Analyzing Functions Using Limits Students often wonder “What is Calculus?” Calculus uses the logic of mathematics to analyze situations involving continuous change by breaking them down into the smallest components (differential calculus) and then rebuilding them (integral calculus). As such, Calculus is considered the science and the art of change. It provides a mechanism to understand the world around us, make predictions, and interpret behavior. One type of behavior involves a limit. What is a limit and how is it useful? The mathematical answer is that a limit is the y-value that a function,

, approaches as the value of x

approaches some number. A few “real life” examples can be found in: 

Physics - pressure at a point is calculated as the average pressure (force per unit area) applied to an area that is shrinking to zero (i.e., shrinking to a point). 

Ecology - sustainable population (or carrying capacity) is often extrapolated by determining, over an extended period of time (i.e., infinite), the upper bound on some population that can be sustained by a given ecosystem.



Chemistry - if you drop an ice cube into a glass of warm water and measure the temperature against time, the temperature will eventually approach the room temperature where the glass is stored. Determining the final temperature is a limit as time approaches infinity.

A limit describes the behavior of a function at a particular point based on the points around it.

49

2.1 Rates of Change and Limits – Determining Limits Graphically The concept of a limit is CENTRAL to calculus. Limits can be used to describe how a function behaves as the independent variable moves towards a certain value.

Consider the functions

and

Graph

, using an appropriate range. On the same

grid graph

on

.

. (Be careful…when does x = 1?)

When does What are the values of How do

and

and

at the point of intersection?

behave near x = 0?

What do you think the value of the quotient function point of intersection?

50

is at the

Look graphically and numerically: Using your calculator, graph both and

, then complete the tables of values below. x

x

0.3

0.03

0.2

0.02

0.1

0.01

0

0

0.1

0.01

0.2

0.02

0.3

0.03

Consider what happens to the y-value as the x-value gets closer and closer to 0.

How do the functions behave as x approaches 0? (use the ZOOM function on your calculator to explore)

Definition: If functions

and

are both 0 at

found by substituting

as the result would be

What value will the quotient approach as the values of the functions become more and more alike?

then

cannot be

. Substitution produces a

meaningless expression known as an indeterminate form.

Other indeterminate forms include:

.

To determine the value of an indeterminate form, other methods, such as graphing, analyzing numerically or using algebraic manipulations, must be used.

51

Example 1: Determining a limit graphically and numerically Find the value that

approaches as x approaches

.

Use a graph and a table of values to justify your answer.

How to Do It What to Think About How does the function behave as x gets closer to zero from the left? From the right?

What is the value of the function at ?

What value does the function approach as x gets closer to zero from the left? From the right?

Definition:

is read, “the limit of f of x as x approaches a equals L”. It

means that as x gets closer and closer to a (from the left-side and the right-side of a), the function’s y – value gets closer and closer to the number L.

52

Your Turn Compare that

and approaches as

graphically then graph

and find the value

. Confirm using a table of values.

x

y

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Example 2: The existence of a limit versus the existence of the function at a point.

Given How to Do It

determine

. Is

equal to

?

What to Think About How does the function behave as x gets closer to 2 from the left? From the right?

What is the value of the function at ?

Is the function continuous at x = 2?

53

Your Turn For each of the following functions, determine if a)

at

b)

at

Did You Know: The modern notation of placing the arrow below the limit symbol is attributed to the English mathematician G.H. Hardy, who used it in his 1908 book A Course of Pure Mathematics.

Summary: The value of a limit is the y – value that the function approaches as

.

The existence of a limit as never depends on if the function is defined or undefined at . The limit exists if the function approaches the same y – value on both sides of .

54

Practice For questions 1 to 8, determine the limit graphically.

1.

2.

3.

4.

5.

6.

55

7.

8.

For questions 9 to 12, use a graph to show that the limit does or does not exist

56

9.

10.

11.

12.

2.2 Exploring Limits Warm Up Consider the following functions:

a)

b)

c)

Without a graphing calculator sketch the graph of each function. Make sure to consider the points where the function does not exist. a) b) c)

Using the graphs, determine each of the following limits and confirm algebraically. a)

b)

c)

Why does factoring help to determine the limit algebraically?

57

Did You Know A removable discontinuity (or point discontinuity) occurs when a function has a hole at one point c on an open interval such that , but or does not exist. You will explore the concept of continuity in Section 2.5.

Example 1: Determining the Limit of a Function Find the limit of each of the following functions, if it exists. a)

How to Do It

b)

c)

d)

x 5 x 25 x  25 lim

What to Think About What is the value of the function if you substitute x = 2?

What is the value of the function if you substitute x = 1?

Can the function be simplified if the numerator or denominator are factored?

What happens on the graph of b) at the point (1,3)?

What is the value if the limit involves division by zero?

58

Your Turn Find the limit of each of the following functions, if it exists. a)

d)

b)

e)

c)

Look at b) and c) graphically. What do you think makes a limit exist?

59

Definition: The existence of a limit depends on the one-sided limits about x = a.

So when

, then

exists and

, where

.

When the right-hand limit equals the left-hand limit at a point, the limit exists.

Example 2: One-sided and Two-sided Limits Given the graph of f determine each limit

How to Do It

What to Think About What does the notation “ mean?

What does the notation “ mean?

”

”

Is the left side limit the same as the right-side limit?

What does the abbreviation “DNE” represent?

60

Did You Know A jump discontinuity occurs when, for some point c on an open interval, , but . The function may or may not exist at

.

Your Turn 1) Given the graph of g determine each limit. a)

d)

g)

b)

e)

h)

c)

f)

2) Given the graph of f determine each limit. a)

b)

c)

61

3) Given the following function, sketch the graph of the function and then use one- sided j

limits to show that the

62

does not exist.

Example 3: The Greatest Integer Function or

is called the greatest integer function and is defined to be the

largest integer to the “left” of x on the number line (or the greatest integral part of x).

What is the result when the greatest integer function is applied? a)

b)

How to Do It

c)

d)

e)

f)

g)

What to Think About What is the greatest integer less than the number 2.3?

What is the greatest integer less than 1.2? Try plotting this on a number line.

Did You Know: The greatest integer function is commonly used when calculating utility bills (gas, electricity, water, etc.) as well as postage rates.

63

Your Turn 1. Graph the greatest integer function (also called a step function):

2. Use the graph to help determine the following limits: a

64

a)

d)

b)

e)

c)

f)

Summary: For the greatest integer function, (

n is an integer)

Practice 1. What is

?

Determine each limit by substitution and support the result graphically.

2.

3.

65

4.

5.

6.

66

7. Explain why you cannot use substitution to determine each limit. Use a graph to support your explanation a.

b.

Did You Know: The signum function is a mathematical function that extracts the sign of a real number. It is often represented as

Determine each limit.

8.

10.

9.

67

Which of the statements are true about the function

graphed below and

which are false?

11.

16.

12.

17.

13.

18.

14.

19. There is a removable

discontinuity at 15.

20. There is a jump discontinuity at .

68

.

Given that

and

determine the following limits.

21.

23.

22.

24.

Complete the following parts for each of questions 25 and 26. a) Draw the graph of f. b) Determine c) Does

and

.

exist? If so, what is it? If it does not exist explain why.

25. c = 1,

69

26. c = 2,

Complete the following parts for question 27. a) Draw the graph of f. b) At what points c in the domain of f does c) At what point “c” does the limit exist and why?

27.

70

exist?

2.3 Asymptotic Behavior

Sketch the graph of

and determine each limit: a)

d)

b)

e)

c)

Now sketch the graph of

and determine each limit:

a)

d)

b)

e)

c)

What can you conclude are the conditions necessary for a vertical or a horizontal asymptote to exist?

71

or

Example 1: Finding a Horizontal Asymptote Find the horizontal asymptote(s) for each function. Use a graph and limits to help. a) How to Do It

b) What to Think About Numerically, what happens to the value of the rational expression as x becomes very large?

What value does the function approach as the denominator gets positively or negatively very large?

Does the +1 have an effect on the value of the denominator as ?

72

Your Turn Find the horizontal asymptote(s) for each function using a graph of the function and limits. a)

b)

, where

.

73

.

Example 2: Finding a Vertical Asymptote Find the vertical asymptote(s) for each function. Use a graph and limits to help a) How to Do It

b) What to Think About Are there any values of x that make the function undefined?

As x approaches this value, what happens to the value of the denominator?

What is the limit of the function at the value(s) where the denominator equals zero?

What is the shortcut for determining if there is a vertical asymptote?

74

Your Turn Find the vertical asymptote(s) for each function using a graph of the function and limits. a)

b)

75

Example 3: Finding the asymptotes of a function.

Determine the asymptote(s) of How to Do It

. Justify your answer.

What to Think About Are there values that make the denominator equal to zero?

Does the numerator also equal zero at any of these points?

What happens at x  2 ?

Is it necessary to show both sides of an infinite limit to justify a vertical asymptote?

Do either – 4 or + 2x affect the numerator or denominator as ?

76

Your Turn Determine the asymptote(s) of each function. Justify your answer. a)

b)

77

Practice For questions 1 to 4 determine each limit graphically and numerically.

1.

2.

78

3.

4.

79

For questions 5 to 8: Determine the asymptote(s) of f(x). Justify your answer using limits. Support your solution graphically and/or numerically.

5.

6.

80

7.

8.

81

2.4 End Behavior of Functions

Imagine standing 2 meters away from any wall in your classroom. Now, take a step that is exactly of the way towards the wall.

Next, repeat that process – take another step that is exactly of the way towards the wall. Then, keep repeating this step process until you touch the wall with your foot.

Graph what this looks like using the number of steps, n, as the independent variable and the distance from the wall, d, as the dependent variable.

Express this process as limit. What number of steps would you use in the limit?

In theory, if the pattern is followed forever, will you ever reach the wall?

Did You Know The end behavior of a function describes the shape of the curve as It can be used to predict behavior in the distant future or distant past.

82

.

Example 1: End Behavior Function Models Let and . Show that while f and g are quite different for small values of x, they are essentially identical for |x| large.

How to Do It

What to Think About Compare the graphs of the functions for small and large values of x.

What happens to the shape of the graphs as increases?

Does

represent a good

model for the behavior of either end or both?

at

83

Definition: End Behavior Model The function g is a good approximation for the behavior of f and is called a right end behavior model for f, if and only if

.

The function g is a good approximation for the behavior of f and is left end behavior model for f, if and only if

called a

.

If g approximates f well at either end, then they behave the same way as . By dividing them and taking the limit, the quotient must equal 1 because they behave the same.

Your Turn Let

. Show graphically and analytically that

model for f and that

84

is a left-end behavior model for f.

is a right-end behavior

Example 2: Finding End Behavior Models Find a function that is an end behavior model for each function. a) How to Do It

b) What to Think About Which are the dominant terms in the numerator and denominator?

When x is very large, what happens to the rest of the terms

Does the function have a limit as ?

What does this limit tell you about the function?

85

Your Turn For each function determine the end behavior model to find the limits as Identify any horizontal asymptotes. a)

b)

c)

86

.

Practice For questions 1 to 4 match the function with the graph of its end behavior model. a) 1.

2.

3.

b)

c)

d) 4.

87

For questions 5 to 10: a) Determine a power function,

, end behavior model for f; and,

b) Identify any horizontal asymptotes, if they exist.

5.

88

8.

6.

9.

7.

10.

For questions 11 and 12 sketch a graph of a function

that satisfies the

stated conditions. Include any asymptotes.

11.

,

12.

,

,

,

,

,

,

,

,

89

2.5 Function Continuity

For what intervals is the function f continuous? Use interval notation to express your answer.

For which points is f discontinuous?

For each point of discontinuity determine the value of the function.

Find the limit at each point of discontinuity or state that it does not exist.

Compare the value of the function at each discontinuity to the limit of the function.

What do you notice?

Determine

and compare this to the value of the function at x = 3.

What is the relationship between the limit and the value of the function at each point where the graph is continuous?

90

Definition: Continuity at a Point Interior Point: A function y = f(x) is continuous at an interior point c of its domain if EndPoint: A function y = f(x) is continuous at its left endpoint a of its domain if

Similarily, a function y = f(x) is continuous at its right endpoint b of its domain if

NB: If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Sometimes a point c may be classified as a point of discontinuity even though it is not in the domain of f. Can you provide an example?

Example 1: Finding Points of Continuity and Discontinuity Find the intervals of continuity and the points of discontinuity of the greatest integer function.

How to Do It

What to Think About Where is the function discontinuous?

Can you generalize the statement?

What does this mean in terms of the intervals of continuity and points of discontinuity?

91

Your Turn Find the points of discontinuity and intervals of continuity for each function. a)

b)

92

Example 2: Types of Discontinuities Write the equation of each function. Use a piece-wise function if necessary. Then, state the type of discontinuity or state that it is continuous.

How to Do It

What to Think About What function will generate each graph?

What algebraic expression results in a hole in a graph?

Which graphs can be represented using a piecewise function?

What happens to these functions as ?

93

Your Turn For each function identify the point(s) of discontinuity and the type of the corresponding discontinuity. Determine the interval(s) of continuity.

94

a)

d)

b)

e)

c)

f)

Example 3: Extending a function to create a continuous function Extend How to Do It

to create a new function, g(x) that is continuous at x = 3.

What to Think About Can you factor the denominator?

What is the domain of f?

What is happening around x = 3?

Can you graph it and use your calculator zoom to determine numerically if there is a removable discontinuity at x = 3?

Is a factor of the numerator of f?

What is the limit of the simplified function as ?

95

Your Turn a) For each function find the value of a that makes the function continuous.

b) Determine the extended function for

96

.

Definition: A function is continuous on an interval if and only if it is continuous at every point of the interval.

A function is called a continuous function if it is continuous at every point of its domain.

Note: A continuous function need not be continuous on every interval, since there may be a point of discontinuity at a point outside of the domain of the function.

For example,

is not continuous on

since it has a point of

discontinuity at x = 0; but the function is still considered to be a continuous function, because x = 0 is not part of the domain of the function!

Example 4: Continuous Functions Is each of the following functions considered a continuous function? Justify your answer. a)

How to Do It

b)

What to Think About Are there any values of x where the function does not exist?

Is the x-value part of the domain of the function?

What do you know about each part of the piecewise function?

Which value(s) of the domain need to be checked?

97

What are the limits of each part of the function at x = 2 and what is the value of the function at x=2

What do the results tell you?

Your Turn Determine if each function is a continuous function. Justify your reasoning. a)

b)

Summary: Properties of Continuous Functions If functions f and g are continuous at , then the following combinations are continuous at x = a.

1. Sum rule:

f+g

2. Difference rule:

fg

3. Product rule: 4. Constant multiples: 5. Quotient rule:

98

, for any number c

Practice For questions 1 to 6 identify the interval(s) of continuity, the point(s) of discontinuity, and each type of discontinuity.

1.

4.

2.

5.

3.

6.

99

For questions 7 to 8 use the following function f and the corresponding graph of f.

7. a) Does

exist?

exist?

b) Does

exist?

b) Does

exist?

c) Does

exist?

c) Does

exist?

d) Is f continuous at x = 1?

100

8. a) Does

d) Is f continuous at x = 1?

For questions 9 to 11: a) determine each point of discontinuity, b) identify which discontinuities are removable or not removable and state your reasons.

9.

10.

11.

101

For questions 12 to 13 identify the extended function, g(x), that makes the given function continuous at the indicated point.

12.

13.

102

For questions 14 and 15 sketch a possible graph for a function f that has the stated properties.

14. f(5) exists but

does not.

15. f(x) is continuous for all x except x = 2, where f has a non-removable discontinuity.

103

2.6 Intermediate Value Theorem a) On the grid provided below draw a graph of f(x) = x +1 using a scale and domain [a, b] of your choice. Label the left endpoint of your graph as (a, f(a)) and the right end point as (b, f(b)).

b) Place a ruler on your graph so that it is parallel to the x-axis and passes through (a, f(a)). c) Keeping your ruler parallel to the x-axis slide it upwards until you reach the point (b, f(b)).

d) Is your graph continuous on the interval [a, b]?

e) Was there any y-value (i.e., any value of f(x)) between f(a) and f(b) that your ruler did not pass through?

f) Do you think the same observations would be made for any function? Can you think of a function where parts d) and e) do not hold true? Explain

104

Example 1: Applying the Intermediate Value Theorem to Determine if a Function Has a Root on a Given Interval Does How to Do It

have at least one root in [0, 1]?

What to Think About What conditions do you know exist?

What can you conclude from the conditions?

What do you know about the yvalues of the end points and what does this tell you?

105

Your Turn Is any real number on the closed interval [1, 2] exactly 1 less than its cube? (Hint: Is there a solution to ?)

Example 2: Applying the EVT to Determine if a Function Has a Maximum or Minimum Let

. Although f is continuous on

, it has no minimum value on the

interval. Does this contradict the EVT? Justify your answer.

How to Do It

What to Think About What conditions are stated?

What can you conclude from the conditions?

How does this affect the given statement?

106

Your Turn For each graph identify the x-value at which a maximum or minimum occurs. Explain how your answer is consistent with the Extreme Value Theorem. a )

b)

c)

107

Practice 1. The function k(x) is continuous on the interval [8, 11]. If k(8) = 3 and k(11) = 2, can you conclude that k(x) is ever equal to 0? Justify your answer.

2. Is there a solution to

3. Let

4. If

. Show that there is a number c such that f(c) = 1.

, is there a number in the interval

your reasoning.

108

? Justify your answer.

where f(x) = 0.4? Justify

5. Suppose that function f has every value between y = 0 and y = 1 on the interval [0, 1]. Must f be continuous on that interval? Explain your reasoning.

6. Between which of the following two values does the equation have a solution? a) Between 2 and 1 b) Between 3 and 2 c) Between 0 and 1 d) Between 1 and 0

7. Prove that the function

intersects the x-axis on the interval [0,

2]. Can the same be said for the function

? Explain your reasoning.

109

2.7 Chapter Review For questions 1 to 3, use your calculator to graph each function and estimate the limit numerically to two decimal places or state that the limit does not exist.

1.

2.

3.

For questions 4 to 11, evaluate the limit if it exists. If not, determine whether the onesided limit exists.

110

4.

5.

6.

7.

8.

9.

10.

11.

12. Determine the left and right-hand limits of the function f(x) shown below at x = 0, 2, and 5.

13. Determine: a) b) c) whether g(x) is continuous at x = 3. d) the points of discontinuity of g(x). e) whether any points of discontinuity are removable.

111

14. Find all points of discontinuity for the given functions. a)

b)

For questions 15 & 16 find i) a right-end behavior model and ii) a left-end behavior model for the function.

15.

112

16.

17. What value of k makes f a continuous function?

18. Show that

has a root somewhere in the interval [1, 2].

113

Unit 1 Project: Limits and Discontinuity 1. Create three piecewise functions over the domain of all real numbers. Each function must contain at least 3 different pieces and may not be continuous. One function must include a jump discontinuity; the second function must contain an infinite discontinuity; and, the third function must include a point discontinuity. For each function: 

Provide a graph.



Explain why the function is not continuous at the specific point.



Explain whether the limit exists or does not exist at the point above. Explain how you could change your function such that there is a limit.



Calculate the left- and right-side limits at each point of discontinuity.

2. Create three functions over the domain of all real numbers: 

one which approaches infinity as x approaches infinity;



one which approaches 0 as x approaches infinity



one which approaches a line which is not 0 as x approaches infinity



Explain how you can determine the limit as x approaches infinity, for each of your functions.

3. Describe an experiment that you could create that would showcase the concept of a limit in a real-world application. Explain why your situation would be either continuous or discontinuous.

4. Describe a real-world example of one type of discontinuity. Identify which type and explain why the situation becomes discontinuous. What are the real-life implications of the discontinuity for your situation?

114

Unit 2: Differential Calculus This unit focuses on the following Big Idea from the British Columbia provincial Calculus 12 curriculum: 

The concept of a limit is foundational to calculus

By the end of this unit, you should be able to: 



Apply the definition of the derivative to determine the instantaneous rate of change, slope of the tangent and to analyze the graph of a function (including identifying intervals in which the function increases, decreases, its concavity and extreme values).

Use different definitions of the derivative to determine its value graphically, numerically and analytically.

By the end of this unit, you should know the meaning of these key terms:           

Chain Rule Composite Definition of a Derivative Using Limits Corner Cusp Definition of a Derivative Using Limits Differentiability Higher Order Derivative Horizontal Inflection Point Jerk Leibniz Notation Local Linearity

           

Local Maximum Local Minimum Logarithmic Differentiation Newton Notation nth Derivative One-Sided Derivatives Product Rule Quotient Rule Rectilinear Motion Simple Harmonic Motion Vertical Inflection Point Vertical Tangent

Top fuel dragsters are the fastest sanctioned category of drag racers in the world reaching speeds of over 500 km/h to complete runs of 1000 feet in less than 4 seconds! They are the quickest accelerating cars on the planet. The application of derivatives can calculate their instantaneous velocity and acceleration.

115

Chapter 3: The Derivative The average velocity of a dragster can be calculated given its position-distance time graph or a table of values.

Time 0 1 2 3 4 5 6 7

Distance 0 30 65 160 290 480 720 1000

What is the car’s average speed between 0 and 5 seconds as it begins to reach top speed? How could you calculate it?

What is the car’s speed exactly at 4.3 seconds? How would you calculate it?

By the end of this chapter you will be able to calculate the instantaneous speed of the dragster.

116

3.1 Slope of a Function Warm Up On a road trip through Tennessee, a group of college students were 80 miles from South Padre Island at 3pm. By 4:15pm they were only 10 miles away.

What was their average speed?

Do you believe this speed accurately reflects their trip? Why?

How might you estimate the instantaneous velocity of a dragster at exactly 3.5 seconds given the finite data in the graph?

Note to student: Refer to applet that follows the curve of a function and solves for the instantaneous rate of change. https://www.geogebra.org/m/STHtFSzX#material/aUjja5NV

117

In general, the average rate of change of a function over an interval, is the amount of change divided by the length of the interval, written as a difference quotient.

Average rate of change =

or

or

Example 1: Finding Average Rate of Change Find the average rate of change of How to Do It

over the interval [1, 3] What to Think About What are values of the function at the endpoints?

How are these used to calculate the difference quotient?

Graphically, what does the value represent?

118

Your Turn Find the average rate of change of

over the interval [2, 5]

Graph any function f(x) and illustrate the slope between any two points evaluated at x = a and x = b.

Definition: A line through two points on a curve is called a secant line. The slope of the secant line of a continuous function on the closed interval [a, b]

The slope of the secant line represents the average rate of change of f on [a, b].

.

119

Example 2: Growth in Technology According to Moore’s Law, the computing power, related to the number of transistors in a dense integrated circuit, doubles every 2 years indicating that the speed of computer processing has been doubling during that time frame.

This trend has been used to set goals within the industry.

Given the data listed above, determine the average rate of change in transistor growth between 1993 and 2000 once the Pentium processor was introduced.

How to Do It

What to Think About What are the applicable units for this rate of change?

When is the processor computing speed growing at its slowest speed? Fastest speed? Why?

120

Your Turn The rise of bitcoin as an alternate currency has been a highly volatile and intriguing investment. Over the long term, we can assess its average rate of change. Here is a graphical representation of its value. Estimate the average rate of change from June 2017 to January 2018. Estimate the average rate of change from January 2018 to March 2018. How does this compare to the Standard & Poor’s 500 which is an American stock market index based on the market capitalizations of 500 large companies having common stock listed on the NYSE or NASDAQ.

121

Slope of a Function at a point To calculate the exact value of the instantaneous change of a function we need to use the concept of infinitesimals and limits. In the development of calculus, Newton and Leibniz used the concept of infinitesimals but did not define them explicitly.

An infinitesimal is now defined as an indefinitely small quantity approaching zero. The idea is to break down a changing function into tiny pieces, considering each piece to occur over an instantaneously or infinitesimal change in the independent variable. Over an infinitely small interval we can consider the change to have a constant slope.

Did You Know - Leibniz Notation

Gottfried Leibniz (1646-1716) discovered calculus through his use of inifinitesimals, which is expressed in differentials. Thus, the derivative in Leibniz notation is expressed as and read as “dy by dx.”

Leibniz notation

Note to student: Refer to this applet for a visualization of two points getting closer and closer together at the point of tangency: https://www.geogebra.org/m/WGCsKBeM

Definition: Slope of a Curve at a Point The slope of a curve limit

, at a point is the value

provided the

exists.

The tangent to the curve at P is the line through P with this slope.

The slope of the tangent indicates the instantaneous rate of change at that point.

122

Example 3: Exploring Slope and Tangent a) Find the slope of b) Given the function

between x = a and x = a+h. , find the slope of the tangent at any point x = a. Use a

limit as to calculate the slope at x = a. c) Use your limit expression to from part b) to calculate the slope of the function at x=2 How to Do It What to Think About What is the significance 2x as it relates to the original function?

What is the significance of the slope of the tangent to the function at x = 2?

Draw a rough sketch to illustrate your calculations.

123

Your Turn Given the function y  x  3 , 2

a) Determine the slope of the tangent any point x=a.

b) Evaluate the slope at x  3

Practice Find the average rate of change of the function over each interval.

124

1.

2.

3.

4.

5.

6.

7. You’ve developed a new algorithm for machine learning that is quickly receiving interest in the AI marketplace. Your customer base for your new company has risen at a record pace over the last few months as illustrated in the table below.

What is the average growth rate per month for your consumer interest over the months of September and December?

Month

Aug 2017

Sept 2017

Oct 2017

Nov 2017

Dec 2017

Jan 2018

Feb 2018

# of Customers

256

512

1,024

2,048

4,096

8,192

16,384 32,768

8. Describe what happens to the line tangent to the function of

Mar 2018

over the

interval [1, 5] between consecutive points with a whole value of x.

125

9. The height, in feet, of a roller coaster entering a full corkscrew is given by the graph below. Estimate its instantaneous rate of change at 6 seconds by finding the rate of change using the intervals , and

10. The Dow Jones Industrial Average rises and falls during the day. Use the data below to estimate its instantaneous rate of change at 2:45pm.

126

11. Determine the instantaneous rate of change to the graph

12. Determine the instantaneous rate of change to the graph

13. Determine the instantaneous rate of change to the graph

at

.

at

at

.

.

127

3.2 Definition of the Derivative Warm Up Sketch a graph of the function

and calculate the slope of the tangent

x=2

Calculate the slope of the curve at x=a using a limit. Use this information and the value of the function at x= 2 to write the equation of a line that is tangent to the curve.

Definition: Tangent line equation at a point

with a slope of

Note: Newton notation for the slope function uses an apostrophe.

128

Definition: The derivative function, point of the domain.

represents the slope of the curve of

at any

Note: Newton notation for the derivative function uses an apostrophe.

Example 1: Exploring Slope and Tangent a) Given the function , find the equation of derivative function. b) What is the equation of the tangent line at x = 1?

How to Do It

What to Think About Did you apply any shortcuts to expand the brackets?

What is the significance of the expression ?

Draw a rough sketch to illustrate your calculations.

129

Your Turn a) Given the function , find the equation of the slope of the tangent any point x = a.

Did You Know – Newton Notation

Sir Isaac Newton (1642-1727) invented calculus through his method of fluxions. His notation is useful when the independent variable is time.

Newton’s notation for derivatives is expressed as y’ and read as “y prime.” y’ or represents the slope of the equation of the tangent for the graph of y or respectively.

b) What is the equation of the tangent line at x = 2?

130

Example 2: Finding the derivative of a Functions a) Use the definition of the derivative to differentiate

.

b) What is the equation of the tangent when the slope of the tangent equals How to Do It

?

What to Think About What does differentiate mean?

How can we deal with the complex fraction?

Which method do you prefer?

131

How to Do It

What to Think About What does it mean if there are two x-values where the slope is ?

What does this look like graphically?

132

Your Turn Differentiate

at

and determine the equation of the tangent line.

Example 3: Differentiating Trigonometric Functions at a point What is the slope of the function derivative to justify your result.

at

? Use the definition of the

How to Do It

What to Think About Differentiate the function using the concept of limits.

Do you recognize this limit?

What does this look like graphically?

Your Turn What is the slope of the function

at x  0 ?

133

Example 4: Differentiating Piece-wise Functions Consider the function How to Do It

. Evaluate

using two different methods.

What to Think About How do you write the function using piece-wise notation?

What are two ways to identify the slope of the tangent at x = 2?

Why must the derivative of an absolute value function be written in a piece-wise fashion?

Consider this graphically and algebraically.

134

Your Turn Consider the function

. Evaluate

using two different methods.

Practice In questions #1-4, at the indicated point, identify the slope of the curve, the equation of the tangent, and draw a graph representing the relationship between the tangent and function 1.

at

2.

at

135

3.

at

4.

at

In questions #5-6, describe what happens to the slope of the tangent over the entire domain of the following functions. Consider whether there are horizontal tangents, vertical tangents, increasing or decreasing slopes, and the intervals where the slope of the tangent is increasing or decreasing.

5.

7.

6.

At Wrigley Field in Chicago, IL it is customary for a fan to throw an opposing team’s home run ball back onto the field.

Suppose an eager fan in the left field stands throws a ball that follows a flight pattern where its height is measured as a function of time where , how fast is the ball falling 2.5 seconds after it leaves their hand? The rate of change of height represents the velocity of the ball. Use the definition of a derivative to find how fast the ball is falling at 2.5 s.

136

8.

Cliff jumping is a very popular pastime in Mexico.

Suppose that a diver jumps from a 100ft cliff, what will his speed be when he enters the water if his height above the water is measured as a function of time and ?The rate of change of height

represents the velocity of the jumper.

Use the definition of a derivative to find how fast the ball is falling at 2.5 s.

9.

At what point is the slope of the tangent to the graph of

10. Find an equation for each line tangent to the curve

horizontal?

that has a slope of 1.

137

Did You Know “The Controversy” 1709-1716 Two 17th century mathematicians both claimed that they invented calculus.

Isaac Newton began his work on the development of calculus in 1666 with manuscripts that highlighted his method of fluxions and fluents, which he called the derivative of a continuous function. Newton published his work in 1687 in his book the Philosophiae Naturalis Principia Mathematica.

Leibniz first produced his findings in 1684 and 1686 with his writings on differential calculus and integral calculus in his book Nova Methodus pro Maximis et Minimus.

Both men claimed that the other had stolen his idea.

In the end, both men have been noted as independent inventors of calculus after a decades-old controversy about the intellectual property related to the invention of modern calculus.

138

3.3 Differentiability of a Function Warm Up Zooming in to “See” Differentiability Graph the following functions and assess the value of their derivative at x = 0.

What do you notice happens to your graphs as you zoom in at

Is it possible to calculate the slope at

?

in

What would the slope be at a point that is a corner?

Local Linearity – all functions that are differentiable at a point x = c are well modeled by a unique tangent line in a neighborhood of c and are thus considered locally linear.

139

Definitions: The Right-hand derivative of a function f(x) at x

The Left-hand derivative of a function f(x) at x

Example 1 - One-Sided Derivatives Can Differ at a Point Show that the following function has left-hand and right-hand derivatives at x  0 .

How to Do It

What to Think About What does the graph of the function look like?

What can you conclude about the derivative x = 0?

140

Your Turn Show that the following function has left-hand and right-hand derivatives at definition of the derivative equation.

using the

Graphically what does this look like?

What happens when the left-hand and righthand derivatives are not the same?

What is the left-hand derivative at x = 0?

What is the right-hand derivative at x = 0?

What is the derivative at x = 0?

141

Four ways in which a function fails to be differentiable: Example 2: Corners Graph and analyze its derivative (slope of the tangent) on either side of . What do you notice? What can you conclude about the slope at ?

How to Do It

What to Think About What is the relationship between the left and righthand derivatives at the corner?

Your Turn Graph

142

and analyze its derivative

Example 3: Cusp Graph How to Do It

and analyze its derivative graphically What to Think About What is the relationship between the left and righthand derivatives at the cusp?

How does the graph of the derivative relate to the graph of the function?

Did You Know Your graphing calculator can graph the derivative function.

In the graphing screen, choose 8: nderiv( from the MATH menu.

Use x as the independent variable, then enter Y1 or the function of your choice.

Finally, choose “for all x”.

On a TI-83 calculator enter

143

Your Turn Graph

and analyze the derivative graphically (specifically at x = 2)

Example 4: Vertical Tangent Graph How to Do It

and analyze the derivative graphically What to Think About What is the relationship between the left and righthand derivatives at the vertical tangent?

Does the function have a tangent line at ?

144

Your Turn Graph

and analyze the derivative graphically

Example 5: Discontinuity a) Graph b) Graph How to Do It

and analyze the derivative graphically and analyze the derivative graphically What to Think About What is the relationship between the left and righthand derivatives at the discontinuity?

145

Your Turn Graph

and analyze the derivative graphically

Note: Continuity at a point does not guarantee differentiability as there may be a

corner, cusp or vertical tangent. Conversely, however, if a function is differentiable at a point, then it must be continuous at the point.

146

Practice For the following graphs, identify the points at which the graph is (a) differentiable (b) continuous but not differentiable (c) neither continuous nor differentiable. Write your solution using set notation or interval notation 1.

2.

3.

4.

5.

147

For what values of x do the following functions not have a derivative?

6.

7.

8.

9.

For questions 10 to 13 compare the left-hand derivative and right-hand derivatives to justify that the function is differentiable at point P.

148

10.

,P=0

11.

12.

,P=0

13.

,P=1

,P=0

3.4 Graphing the Derivative of a Function Warm Up

Graph the piece-wise function

Graph the derivative of f on the second grid.

What would the vertical axis be labelled?

How could you draw a function if you were given the graph of its derivative?

Note to student: Refer to this applet that identifies the value of the derivative: https://mathinsight.org/applet/derivative_function 149

Example 1: Sketch f ’ Given the Function f Given the graph of a function sketch a graph of the derivative.

How to Do It

What to Think About

Given graph:

When f is increasing what do we know about f ’? f

When f is decreasing what do we know about f ’?

What happens to the slope of f over ?

What happens to the slope at the vertex?

What happens to the slope of f over ?

150

Your Turn Sketch a graph of f ‘ for each function shown below

151

Example 2 - Sketch f Given the Derivative Function.

Given the graph of

, graph f. Assume

How to Do It

What to Think About

Derivative graph

If , then what can you assume about the graph of f? f’

If then what can we assume is true of f ?

If f’ = 0 then what can we assume of the original graph?

What is the connection between the degree of the polynomial function and the degree of its derivative?

152

Your Turn Sketch f given the graph of f ‘

153

Practice Match the graph of the function with its derivative.

Derivative Graph

154

Original Function

1.

a.

2.

b.

3.

c.

4.

d.

5. Given the function shown below sketch a possible graph of f(x).

6. Sketch a possible graph of f(x).

155

7. The declination of a celestial body is its angular distance north or south of the equator. The declination of the Sun changes from 23.5 N to 23.5 S and back again during the course of the year. Answer the following questions by estimating the slopes of the graph given below. (Graph is for Northern Hemisphere)

a) On what day is the angle of the sun increasing at the greatest rate?

b) On what day does the angle of the sun have no rate of change?

c) On what day is the angle of the sun decreasing at the greatest rate?

156

8. Over the course of a 12-hour day, the value of a stock on the TSX was charted on the graph below. a) What is the value of the stock when the value of the derivative is at its largest? Most negative?

Stock Value

Hours

b) What is the value of the stock when its derivative is zero?

157

9. A father and son have a foot race at the family picnic. The father orchestrates a photo finish where the race is a tie and he meets his son at the finish line. The position of each runner in 10s of metres is given in relation to their time in seconds in the graph below. The father’s progress is graphed in red. a) Describe the speed of each runner throughout the race.

b) At what point are they the farthest apart?

c) When are they running at the same speed?

d) When does the father start running faster than his son and closing the gap?

158

10. Given the graph of each function f below, sketch the graph of f’ below it a. b. c.

d.

e.

f.

159

11. Given the graph of the derivative, f’, below, sketch the graph of the function f.

160

a.

b.

c.

d.

3.5 Chapter Review Find the average rate of change of each function over the given interval.

1.

2.

3. The following graph shows the number of active Uber drivers in the USA. Find the average rate of change related to this increase between July 2013 and July 2014.

161

4. An object in free fall begins to pick up speed according to the data below. Use the concept of infinitesimals to estimate the speed of the object at 3 seconds.

5. The intensity of a sound as it moves away from its source will resemble an inverse square law. This is useful in measuring the strength of a radio signal.

Estimate how fast the signal is losing its strength at 3 metres away from its source

162

6. Use the definition of the derivative to determine the slope of the tangent to the graph .

7. Use the definition of the derivative to determine the slope of the tangent to the graph

.

8. Use the definition of the derivative to determine the slope of the tangent to the graph .

163

9. At what point is the slope of the tangent to the graph of

10. At what point does the tangent to the curve

have a slope of

horizontal?

?

11. Determine whether the function has a tangent at the indicated points. If it does, indicate the slope. If not, indicate why.

164

12. Identify points on the graph where the function is not differentiable

13. Sketch the graph of the function that satisfies the following conditions

165

14. Given the graph of the function f, below, sketch the graph of the derivative f’.

166

Chapter 4: Differentiation Techniques Although the concept of a derivative as the slope of the tangent to a function can be attributed back to mathematicians such as Euclid, Archimedes and Apollonius in the 3rd century BC, their differentiation techniques were limited to the use of infinitesimals. Modern techniques of differentiation are generally credited to Leibniz and Newton, though these are largely built on earlier works submitted by other great minds in the field of mathematics. Newton was the first to apply his techniques to theoretical physics and Leibniz’s notation is still widely used today.

Today derivatives are used by: 

governments in population censuses



economists to ascertain marginal cost



pharmaceutical scientists as they test the body’s reaction to certain medications.

Any time you are discussing a rate of change, you are discussing derivatives.

167

4.1 Derivatives of Polynomial Functions

Warm up: Use your knowledge of the definition of derivatives to find the derivative of each function:

Can you find a pattern to avoid using the definition?

Definition of the Derivative (the derivative function):

The Derivative at a Point (2 formulas)

Right-Hand Derivative at a

168

Left-Hand Derivative at a

The Power Rule, as with all other derivative formulas, has its origins using limits.

Power rule Proof Using Limits

Proof #2 of the Power Rule using Limits

169

Example 1: Application of the Power Rule for Differentiation Find

for the function

How to Do It

What to Think About How does this connect the graph of the function and its derivative?

Your Turn a.

e.

b.

f.

c.

g.

d.

h.

Sum rule Difference rule Constant multiple Constant rule

170

Example 2: Application of the Power Rule Within the Sum, Difference & Constant Rules of Differentiation Find

if

How To Do It

What to Think About Application of the power rule How does this connect the graph of the function and its derivative?

Your Turn Find

a)

b)

if: c)

d)

171

Example 3: Determining Tangent Lines Find the equation of the tangent to the curve

at the point

.

Support your answer graphically.

How to Do It

What to Think About What does

represent?

Are there any points where can’t exist? Why?

To illustrate, use 2nd Draw 5: Tangent To get the Y1 function on the calculator use VARS, YVARS, 1:Function, 1: Y1

172

To illustrate, use 2nd Calc 6: and enter value for x

Your Turn Find the equation of the tangent to the curve

at the point

. Support

your answer graphically.

173

Example 4: Finding Horizontal Tangents Does the curve have any horizontal tangents? If so, at which xvalues?

What are the equations of the horizontal tangents?

How to Do It What to Think About What condition must exist for a graph to have a horizontal tangent?

How can you use the graph of the Remember Your graphing calculator can graph the derivative function.

In the graphing screen, choose 8: nderiv( from the MATH menu.

Use x as the independent variable, then enter Y1 or the function of your choice.

Finally, choose “for all x”.

On a TI-83 calculator enter

derivative to find where a horizontal tangent line exists on f?

174

Your Turn Does the curve have any horizontal tangents? If so, at which xvalues? What are the equations of the horizontal tangents?

Example 5: Finding Horizontal Tangents using Technology Determine the values of x where the curve tangents.

How to Do It

has horizontal What to Think About What is the slope of a horizontal tangent?

Your Turn Determine the function values where the curve has horizontal tangents.

175

Practice Find

for each equation:

1.

4.

2.

5.

3.

6.

Determine the values of x for which each curve has a horizontal tangent.

7.

176

8.

9.

11.

10.

12.

13. What is the value of the slope of the tangent to the equation ?

14. Find the equation of the line tangent to the equation

15. What is the value of the slope of the tangent to the equation

at

at

?

at

?

177

16. Find the equation of the line perpendicular to the tangent to the curve at the point

17. Find the points on the curve

when the slope of the tangent is 5.

What feature does this point correspond to on the original function?

18. Find the intercepts of the tangent to the curve tangency is (2,16).

19. Where are the tangent lines of

20. Find the points where the curve x-axis.

178

and

when the point of

parallel?

has tangents parallel to the

4.2 Higher Order Derivatives Warm Up The heights in feet of a water skier’s jumps are modelled by the equation . How would you determine an equation that represents how the height is changing over time in seconds? What are the units of this equation?

How would you determine an equation that represents how fast the change in height is changing? What are the units of this equation?

What real-world measurements do each of these new equations represent?

179

Example 1: Second and Higher Order Derivatives Find the first four derivatives of How to Do It

What to Think About How do you denote the different derivatives in both Leibniz and Newton notation?

Did You Know The 4th derivative is known as the “jounce” and is useful in determining the cosmological equation of state, which is characterized by a dimensionless number w. It is defined as the ratio of its pressure to its energy density.

The 5th and 6th derivatives, while useful in applications of theoretical physics, have no universally accepted name.

Recently, the use of the fifth derivative and curve fitting were used to perform DNA analysis and population matching.

180

Your Turn Find the first four derivatives of the following functions. a) c)

b)

d)

Example 2: Differentiating a Product Does the derivative of How to Do It

equal

?

What to Think About Is it possible to rewrite the function in order to have only power terms?

Your Turn Find the derivative of

181

Product Rule – the product of two differentiable functions u and v is differentiable

Example 2: Differentiating a Product of Functions Find

if

How to Do It

What to Think About Does it matter which function is u and which is v?

For the product of polynomials, how can we verify the derivative?

Your Turn Differentiate

182

Example 3: Working with Numerical Values Let

be the product of the functions f and g. Find ,

,

, and

How to Do It

What to Think About What differentiation rule must you use for this function?

Your Turn Let be the product of the functions f and g. Find ,

if

,

if

and

Quotient Rule – at a point where

, the quotient

of two differentiable

functions is differentiable

183

Example 4: Differentiating a Quotient Find the derivative of a)

and b)

How to Do It

.

What to Think About What differentiation rule must you use for this function?

Does it matter which term goes first in the numerator?

Is there an advantage to keeping your denominator in a factored form?

Your Turn Find the derivative of the following function

184

Example 5: The Quotient Rule with an Application of Numerical Values Let

be the quotient of the functions f and g.

Find

if

.

How to Do It

What to Think About How would you determine the derivative in terms of x for this function?

Your Turn Let

be the quotient of the functions f and g. Determine , and

if

,

,

.

185

Practice For questions 1 to 4 determine the first four derivatives of each function.

1.

3.

2.

4.

For questions 5 to 12 determine 5.

186

6.

7.

10.

8.

11.

9.

12.

187

Given that u and v are functions that are continuous and differentiable, and that , determine the values of the derivative at x = 3.

13.

14.

15. Determine the derivative of

using the quotient rule.

For questions 16 & 17, determine the equation of the tangent line to the curve at the given point.

16.

188

at (1,16)

17.

at (1,2)

4.3 Velocity & Other Rates of Change Warm Up Analyzing a car’s Performance Suppose you drive your car for 3 hours and travel exactly 180 mi. What is your average velocity?

If the car’s velocity function is during the first 3 hours, is there a time that the car’s velocity is exactly 60 mph? Justify your answer.

Instantaneous Velocity = first derivative of the position function = s'(t) How does Liebniz notation help to determine the units of a derivative?

=

Definitions Let s(t) be a position function for an object. Then the velocity function is which represents the change in position over time.

When

, the object is not moving. The object is stopped or at rest.

When right/up).

When left/down).

, the object is moving in the positive direction (forward/to the , the object is moving in the negative direction (backward/to the

The net change in position over [a, b] is called the displacement and equals .

The average velocity over [a, b] equals

.

Some objects move one-dimensionally. This motion is called rectilinear motion.

189

Example 1: Relating Velocity & Position Given the position function

(t is measured in sec, s is measured in ft)

a) find the velocity function.

d) find the velocity at 2 secs and at 4 secs.

b) find the displacement over the first 2 secs.

e) determine when the object is at rest.

c) find the average velocity after 2 secs.

How to Do It

f) determine when the object is moving in a positive direction and in a negative direction.

What to Think About What is the relationship between the position function and its velocity?

What is the relationship between the position function and displacement?

What is the relationship between average velocity and displacement?

What is the relationship between the velocity at a point and the position function?

What do zero, positive and negative values of the derivative imply about the function?

190

Your Turn Baby Kaya’s first steps were filmed last week in a 5 second video. The position of her right foot, relative to her left, was recorded as the following function (t is measured in sec, s is measured in cm) a) What is the velocity at time t?

b) When is the right foot at rest?

c) What is the velocity after 1 seconds? 1.5 seconds? 3 seconds?

d) When is the right foot moving forward (in a positive direction)?

e) When is the right foot moving backward (in a negative direction)?

191

Example 2: Relating Velocity to a Position-time graph Given the position function to determine the following.

from example 1, use the previous solution

a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero.

c) graph the position-time graph.

b) Draw a diagram to represent the rectilinear motion of the object for How to Do It

.

d) find the displacement after the first 5 secs. e) find the total distance traveled over the first 5 secs.

What to Think About What direction is the object moving during the first second?

When does the object change directions?

In between stopping points, is it possible for an object to change direction?

Does the position-time graph indicate which direction the object is moving?

Does the position-time graph indicate when an object is stopped?

192

How do displacement and total distance differ?

Your Turn Consider baby Kaya’s first steps

from the last example.

a) Complete the table of values. For position, find the value at each time. For velocity, determine whether v(t) is positive, negative or zero. Does this table of values describe all important intervals? Please add any additional points that would be useful. t 0 1 2 3 4 5

s

v

t

s

v

b) Draw a diagram to represent the rectilinear motion of the object for

.

193

c) graph the position-time graph.

d) find the displacement after the first 5 secs.

e) find the total distance traveled over the first 5 secs.

194

Example 3: Free-Falling Object The vertical height y (in feet) of a coin thrown upwards from the observation deck of the Empire State Building, in time t seconds, is described by the position function: . a) What is the initial velocity at t = 0? b) What is the height of the coin when the velocity is zero? c) What is the speed of the coin when it returns to earth?

How to Do It

What to Think About What does the word initial imply?

What is important about the height when the velocity is zero in this situation?

When does the coin return to earth (hit the ground)?

How does speed relate to velocity?

195

Your Turn On a construction site, a dynamic blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of ft after t seconds. a) How high does the rock go?

Did You Know Free Falling to Maximum Velocity Skydivers are familiar with the term “terminal velocity” which states that terminal velocity is the highest velocity attainable by an object as it falls through a fluid (air being the most common example)

b) What is the velocity of the rock when it is 256 ft above the ground on the way up? on the way down?

c) What is the speed of the rock when it is 256 ft above the ground on the way up? on the way down?

d) When does the rock hit the ground?

196

As a skydiver falls towards the earth, his velocity continually gets faster. This change in velocity is known as acceleration. Acceleration is the rate of change of the velocity

Example 4: Relating Position to Velocity and Acceleration Consider the position function for a particle at time, t. a) Determine v(t) and a(t) b) Graph s(t), v(t) and a(t) over the interval 0 < t < 8.

Use a vertical scale between –40 and 60. c) Describe the motion of the particle.

How to Do It What to Think About What is the relationship between velocity, acceleration and position?

What points on the velocity function correspond to minimum and maximum points on the position function?

What feature of the velocity function do the zeros of the acceleration function correspond to?

When the acceleration function is zero and the velocity function is at an extreme value, what is happening on the position function?

197

How do you determine which direction the particle is moving?

How do you determine if the particle is speeding up or slowing down?

Summary A particle is speeding up when its velocity and acceleration move in the same direction (both v and a have the same sign).

A particle is slowing down when its velocity and acceleration move in the opposite directions (v and a have different signs).

198

Your Turn Consider the horizontal position function of a hummingbird as it approaches a feeder

a) Determine v(t) and a(t)

b) Graph s(t), v(t) and a(t) and describe the horizontal motion of the hummingbird over the interval 0 < t < 6.

199

Practice 1. You are training your new puppy Hugo to walk with a leash. His velocity in the first few seconds is shown in the graph below. a) When is he moving forwards?

b) When is he moving backwards?

c) When did he stop?

d) When is his acceleration negative?

e) When is his acceleration positive?

f) When is he moving at his greatest speed?

g) When is he speeding up?

h) When is he slowing down?

200

2. The height of a football of an NFL punt is given by the function .

a) What is the ball’s velocity as a function of time?

b) What is the ball’s acceleration as a function of time?

c) What was the maximum height of the ball?

d) When did the ball hit the turf?

e) When was the ball at a speed of 4.4mph?

201

3. A golfer in a long-distance drive contest hits a ball whose height is given by the equation . What is the velocity of the ball at 2 seconds?

4. A bungee jumper leaps off a bridge with an initial velocity of 3.6mph and reaches heights of . a) What is his velocity at time t?

b) When does he reach the peak of his jump?

202

c) When does he hit the water below?

d) What is his velocity when he hits the water?

5. On the show American Ninja Warrior, during the first hooks of the Sky Hang obstacle, a contestant’s body is found at the speeds given by the function .

Find the contest’s velocity every time their acceleration is zero.

203

6. An NFL football offensive lineman, while doing a wave drill moving forwards and backwards to work on his foot speed, had his position from the goal line marked with the function a) What is the displacement during the first 5 seconds?

b) What is the average velocity in the first 5 seconds?

c) What is the instantaneous velocity at 3 seconds?

d) What is his acceleration at 3 seconds?

e) When did he change direction?

204

7. Given the graph of functions a, b, and c, indicate which represents the function and its first two derivatives.

a

b c

8. Given the functions a, b, c and d, find indicate which represents the original function and its first three derivatives.

b a

d

c

205

4.4 Derivatives of Trigonometric Functions Warm Up Sketch the graph of

Create a sketch of the derivative graph for

by considering the slope

for several points along the graph above.

Repeat the process again, this time by considering the slope for several points along the graph that you just created.

What can you conclude about the derivative of the sine function?

206

See the applet to help identify the value of the derivative of the sine function: https://www.intmath.com/differentiation-transcendental/differentiation-trigonometricfunctions-interactive-applet.php Derivatives of basic trigonometric functions:

Example 1: Derivatives Involving Trigonometric Functions Differentiate How to Do It

What to Think About What differentiation rule must be used?

Your Turn Find the derivative of: a)

c)

b)

207

Example 2: The Motion of a Weight on a Spring (Simple Harmonic Motion) A weight hanging from a spring is stretched 5 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is .

What is its velocity and acceleration at time t?

How to Do It

What to Think About What are other examples of periodic motion?

Did You Know Simple Harmonic Motion is periodic motion that can be modeled with a sinusoidal position function.

Your Turn A dog in a swing with an appetite for extreme sports sees his height above the ground follow the parameters of the simple harmonic function .

What is its velocity and acceleration at time t?

208

Given his thrill-seeking nature and desire for the rush of adrenaline, your dog emphatically requests that you push him through each swing from the back.

Mathematically, how do we account for this external factor?

Definition: Jerk A sudden change in acceleration is called a “jerk”.

Therefore, a Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is

Example 3: Deriving the Jerk a) Determine the jerk caused by the constant acceleration of gravity. b) Determine the jerk of the simple harmonic motion in Example 2.

How to Do It

What to Think About Does it make sense that gravity has a jerk of zero?

Your Turn Determine the jerk felt by the dog in the swing

209

Did You Know Jerk is felt as a change in force such as the boost that a space shuttle gets during its launch as it exits the atmosphere.

For those of us that will never visit the moon, we can still experience the jerk.

Roller coasters are designed to stimulate the nervous system by changing the acceleration by controlling its rate of change, magnitude and duration. These changes will prey upon your body’s ability to handle g-forces as it demands the designers to have an impeccable understanding of the jerk!

Example 4: Determine How to Do It

What to Think About How can you rewrite the tangent function in order to derive it?

Which differentiation rule must we apply to find the derivative?

Your Turn Find

210

.

Derivatives of Other Basic Trigonometric Functions

Example 5: Finding Tangent and Normal Lines Find the equations for the lines that are tangent and normal to the graph of

at x = 2. Use a calculator to round all significant values to the nearest hundredth and support your conclusion graphically. Use the window [-5,5] by [-4,4] How to Do It

What to Think About How do you enter your calculator?

into

211

Your Turn Find the equations for the lines that are tangent and normal(perpendicular) to the graph of

at

x=

.

Practice For questions 1 to 10 determine

212

1.

4.

2.

5.

3.

6.

7. A bungee jumper is bouncing in simple harmonic motion with her position . What is the jerk at time t?

8. What is the equation of the line that is tangent to the graph of ?

9. Over the domain

, find the point(s) on the curve

at

that have

tangents with a slope of 2.

213

10. What is the equation of the horizontal tangent to the curve

11. What is the value of a in which the function

?

will be differentiable

at x = 0?

12. Evaluate the derivative of

214

at

. (Hint: use trigonometric identities)

13. A child is playing with a yoyo has position at a) Is the yoyo traveling upwards or downwards at

.

?

b) Is the yoyo speeding up or slowing down at

?

215

4.5 Introducing the Chain Rule Warm Up A composite function is a nested function of the form .

Identify the inside and the outside function in each composition

a)

Outside Function y=

Inside Function u=

b)

y=

u=

c)

y=

u=

d)

y=

u=

e)

y=

u=

Find

Why can’t you simply use the power rule on the original function?

Find

216

where

Definition: The Chain Rule:

Example 1: Taking the Derivative of a Composite function Given the function

what is

?

How to Do It

What to Think About What differentiation rule can you use to find the derivative of this function?

Your Turn Determine the derivative of the following functions a)

b)

217

Example 2: Application of the Chain Rule in the Context of a Trigonometric Derivative a) Graph the function and its derivative using your graphing calculator b) Graph to see the graph of the derivative of this function How to Do It What to Think About What should you input into your calculator?

From the graphs what do you think the derivatives are?

What should you input into your calculator?

From the graphs what do you think the derivatives are?

Why does the amplitude of the derivative graph change and what does it mean?

218

Your Turn a) Plot the function

and its derivative using your graphing calculator and show

the graph on the axes below:

b) Now graph

and its derivative

c) Why does the amplitude increase by a factor of 3? Explain in your own words.

219

New Rules for Composite Trigonometric Functions:

Practice For questions 1 to 15 determine 1.

2.

3.

220

.

4.

5.

6.

7.

8.

221

9. An Olympic swimmer doing the butterfly has his center of gravity found by the function

.

What is the velocity of the swimmer as a function of time?

For questions 10 and 11 determine 10.

11.

222

4.6 Chapter Review Differentiate each equation: 1.

5.

2.

6.

3.

7.

4.

223

For questions 8 and 9 determine the values of x for which the function has horizontal tangents 8.

9.

For questions 10 and 11 determine the slope of the tangent at the indicated point 10.

11.

For questions 12 & 13 determine the equation of the tangent line and the normal line at the given point.

12.

13.

224

For questions 14 and 15 determine the first four derivatives of the given equation.

14.

15.

For questions 16 and 17 differentiate the given equation.

16.

18.

17.

19.

225

20. Use the following information to answer questions a to h: A particle is moving along the x-axis with its velocity function, v(t), shown in the figure below. a) A particle is moving along the x-axis When is the particle moving to the right?

b) When is the particle moving to the left?

c) When did it stop?

d) When is the particle’s acceleration negative?

e) When is the particle’s acceleration positive?

f) When is the particle moving at its greatest speed?

g) When is the particle speeding up?

h) When is the particle slowing down?

226

21. Use the following information to answer questions a to f: A particle in rectilinear motion follows the position function in the first 7 seconds of its journey. a) What is the velocity at time t?

b) What is the velocity after 2 seconds? 5 seconds?

c) When is the particle at rest?

d) When is the particle moving to the right (in a positive direction)?

e) What is the displacement during the 7 seconds?

f) What is the total distance traveled from time t = 0 to t = 7 sec?

227

22. Use the following information to answer questions 35 to 37: The outside sphere on a pendulum on a desk continues in a simple harmonic motion in which its height is measured as . a) Determine its velocity and acceleration at time t.

b) Find the velocity, acceleration at time

c) Identify the direction of the pendulum and whether it is speeding up or slowing down.

228

Chapter 5: Advanced Differentiation Techniques Noise cancelling headphones take the noise accumulated, measured in decibels, outside of the ear phones and constructs a function to represent it. It order to cancel it, that function is embedded within a function that builds its opposite. It would require an application of the chain rule to determine the instantaneous change in decibels required to cancel out random sounds.

Entering a corkscrew turn in a hanging roller coaster, the center of gravity of one of the riders follows the pathway defined by … it requires an application of implicit differentiation to determine how fast the height of the person is changing (and the G forces applied on the body!)

229

5.1 The Chain Rule Warm Up Determine the derivative of the following functions a) b)

Which rule must be applied to differentiate each of these?

Write each function as a composition of functions. Write each with an outer function of and an inner function of . a)

b)

Example 1: Using the New Rules and Beyond Determine How to Do It

230

for

and What to Think About Which differentiation rule is needed for this type of function?

Your Turn Differentiate the following functions a)

c)

b)

Example 2: Using Multiple Laws of Differentiation Differentiate How to Do It

What to Think About Which laws of differentiation are applicable?

231

Your Turn Differentiate each composite function. a)

c)

b)

d)

Example 3: Rectilinear Motion An object moves along the x-axis so that its position at any time is given by . What are the velocity and acceleration of the object as a function of t?

How to Do It

What to Think About How are position, velocity and acceleration functions related?

232

Your Turn An object moves along the x-axis so that its position at any time is given by . Find the velocity and acceleration of the object as a function of t.

Example 4: Equation of the Tangent Line What is the equation of the line tangent to the curve

at the point where

?

How to Do It

What to Think About What is the tangent line equation in point slope form?

Which special triangle will apply in the evaluation of the point of tangency?

233

Your Turn a) What is the equation of the line tangent to the curve

at the point where

?

b) Show that the slope of every line tangent to the curve

is positive.

Did You Know Biologists employ the Chain Rule when they determine the speed of growth of a bacteria population

234

Example 5: Composition of Multiple Functions What is the derivative of

?

How to Do It

What to Think About How many times will the chain rule be applied?

Could we use a substitution method to make the problem simpler?

Your Turn Differentiate the following functions a)

b)

235

Practice For questions 1 to 8 determine 1.

.

4.

2.

5.

3.

6. What is the equation of the tangent line to the curve

236

at x = 1?

5.2 Composite Functions and Function Notation Warm Up Given

and

, determine

a)

b)

Given

and

, determine

a)

b)

Does the order of function matter when using this notation?

237

Example 1: Derivative of a Linear Composite Function Using two different methods, determine

if

How to Do It

Your Turn Determine

if

,

Given the composite function

238

and

What to Think About Is it advantageous to determine the expression for the composite function before differentiating?

More Notation for the Chain Rule:

Then:

,

and

Example 2: Derivative of a Non-Linear Composite Function Given the following information determine the value of at and How to Do It

given

.

What to Think About What differentiation rule is required for this function?

Which notation would you prefer to use Leibniz or Newton?

What are the advantages of both?

Your Turn Determine

at the given value of x. at

239

Example 3: Calculate the Derivative Given a Table of Values Suppose: Function

and the following table of values is true: x=

1 5

2 4

3 7

3

3

2

7

7

8

5

9

12

Determine: a) b) c) How to Do It

What to Think About As is a composite function, what must be applied to derive it?

How can you write solely in terms of a composite function in terms of and

240

Your Turn Suppose the functions f and g and their derivatives have the following values at and x

f(x)

g(x)

2

8

2

3

6

4

f’(x)

g’(x)

3π

5

Evaluate the derivatives with respect to x of the following combinations at the given value of x. a)

at x = 3

b)

c)

at

d)

at

e)

at

at

241

Example 4: Calculate the Derivative Given Specific Values Suppose

,

,

, and

How to Do It

Your Turn If

,

If

,

, and

Did You Know In a memoir by Gottfried W. Leibniz, it is apparent that he attempted to use the Chain Rule to differentiate a polynomial inside of a square root.

Although the memoir contained various errors, it is believed to be the place in which the Chain Rule originated. The Chain Rule is a relationship between three rates of change, stating that the first is a product of the other two.

, and

In Leibniz notation it is

?

, , what is

, ?

For the same function given in Example 5

242

?

What to Think About Do you need to know what the functions are to solve this type of question?

, what is

determine

. What is

.

where y is a function of u and u is a function of x.

Practice Determine the value of

at the given value of x.

1.

3.

2.

4.

243

Given that the functions f and g and their derivatives have the following values at x = 1 and x = 2. x 1

8

2

3

2

3

4

5

Calculate:

244

5.

7.

6.

8.

5.3 Implicit Differentiation Warm Up Graph the following two functions on the grids provided. Determine the equations of their tangents at and sketch them on the graph.

What do you notice about the slopes of these two functions?

245

Example 1: Implicit Differentiation of a Square Root Relation Given the relation , how would you determine the slope of the tangent at the points (4,2) and (4,2)? Graphically support your findings.

How to Do It

What to Think About Is this equation a function?

What are the obstacles in differentiating a relation?

How can you differentiate a relation that has more than one point at an x value?

Is it possible to simply move left-to-right along the equation and differentiate each term with respect to x?

How is differentiating y2 with respect to x similar to the chain rule?

What are the advantages of deriving this relation using implicit differentiation?

How would you graph this relation using a graphing calculator?

246

Your Turn Given the relation

, how would you determine the slope of the tangent at the

points (16,2) and (16, 2)? Graphically support your findings.

247

Example 2: Implicit Differentiation Involving a Circle Find the slope of the circle

at the point (4, –5) using implicit differentiation.

How to Do It

What to Think About How do you write y explicitly in terms of x in order to derive it?

What does this look like graphically?

Your Turn Find the slope of the circle

248

at the point

using implicit differentiation.

Note: Implicit differentiation requires differentiating each term with respect to x.

Whenever a variable is involved in the equation that is a function of x, you can consider it as a function inside the relation and apply the CHAIN RULE. This is another way to think of implicit differentiation.

Example 3: Implicit Differentiation Involving an Ellipse Find the tangent and normal to the ellipse

How to Do It

at the point (2,3)

What to Think About What differentiation rules must you apply to find the derivative implicitly?

What is the relationship between the slope of the tangent and the slope of the normal?

249

Your Turn What are the tangent and normal to the ellipse (1,3)?

at the point

Example 4: Using Implicit Differentiation in a Proof Show that the slope How to Do It

is defined at every point on the graph of What to Think About Which differentiation technique must be applied?

What is the link between continuity and differentiability?

250

Your Turn Find a)

of the following relations b)

Example 5: Implicit Differentiation Involving a Hyperbola Find if How to Do It What to Think About Moving left-to-right and differentiating as you go, which rules do you need to apply to differentiate with respect to x?

What differentiation rule must you apply when taking the second derivative?

How could you simplify the second derivative?

251

Your Turn Find

if

Practice For questions 1 to 5 determine 1.

4.

2.

5.

3.

252

For questions 6 and 7 determine the slope of the curve at the indicated point.

6.

7.

For questions 8 to 10 determine the equation of the tangent line to the curve at the indicated point.

8.

10.

9.

253

11. Determine

12. On the circle

254

for the function

, what is the value of

at the point

?

5.4 Derivatives of Exponential and Logarithmic Functions Warm Up Graph . Then graph in bold the derivative of this function using the numerical derivative feature of your calculator:

What do you notice about the function and its derivative?

255

Did You Know

e is an irrational number. This constant, called Euler’s Number, named after Leonhard Euler, to 9 decimal places equals 2.71828128. More formally,

256

Derivative of base e exponential functions

Example 1: Derivative of the Exponential Function as Part of a Composite Function Find the derivative of How to Do It

What to Think About What is the embedded function?

Your Turn Derive the following functions a)

d)

b)

e)

c)

f)

257

Did You Know The Relationship Between e & Compound Interest

Compound Interest is found given the formula

where F = Future

Value, P = Present Value, r = rate, n = number of payments. Substitute

and you will get

This is why e appears in many banking formulas. It also hints at the idea that a banker, prior to a mathematician, discovered the value of e! Necessity is the mother of all inventions.

Example 2: How Fast Does a Flu Spread?

The spread of a flu virus in a major city is modeled by the equation

where is the total number of citizens infected t days after the flu was first discovered. a) Graph the function with the dimensions b) Estimate the initial number of citizens infected with the flu. c) How fast was the flu spreading after 3 days? d) Graph the derivative with the dimensions spread at its maximum rate? What is this rate?

How to Do It

258

.

; when will the flu

What to Think About What point on the original graph identifies the maximum rate of the spread of the flu?

What are two methods to find the maximum rate of the speed of the spread of the flu?

How fast is the flu spreading by day 30?

259

Your Turn

The growth of the Spanish Influenza (1918), in its first 4 months spread at a rate that can be modelled by the equation

where P is the number of citizens and t is

the time in months. a) Graph the function

260

with the dimensions

.

b) Estimate the initial number of citizens infected with the flu.

Did You Know The Natural Logarithm In Latin, “logarithmus naturali,” denoted by the button ln on your calculator, is the inverse of e.

c) How fast was the flu spreading after 3 days?

d) Graph the derivative with the dimensions ; when will the flu spread at its maximum rate? What is this rate?

The Scottish mathematician, John Napier (1550-1617), nicknamed the Marvellous Merchiston, was studying the motion of someone covering a distance by dividing the time into short intervals of length.

He coined the name for this relationship using the Greek words logos (ratio) and arithmos (number). This term became Latinized into the current version logarithm. Napier is best known as the discoverer of the concept of logarithms which provided a platform for later scientific advances in astronomy, dynamics and other branches of physics.

Napier’s counting tables

261

Example 3: Prove the Derivative of the Natural Log Derive How to Do It

What to Think About How can we rewrite the natural logarithm into a form that we can differentiate?

As the function y is not written explicitly in terms of x while in exponential form, what type of differentiation must be applied?

Definition

Example 4: Application of the Derivative of the Natural Logarithm Find the derivative of How to Do It

What to Think About What is the embedded function?

Identify any restrictions

262

Your Turn Determine the derivative of each function: a)

e)

b)

f)

c)

g)

d)

263

Practice Determine 1.

7.

2.

8.

3.

9.

4.

10.

5.

11.

6.

264

12. At what point on the graph of line

is the tangent line perpendicular to the

?

13. A line with slope m passes through the origin and is a tangent to the curve . What is the value of m?

14. Find the equation of the tangent line to the curve origin.

and passes through the

265

15. Find the equation of the normal line to the curve the origin.

Laws of Logarithms

266

that passes through

5.5 Logarithmic Differentiation Warm Up Use the applet below to visually identify the graph of the following exponential functions http://webspace.ship.edu/msrenault/geogebracalculus/derivative_exponential_functi ons.html

All of the graphs have one common coordinate. What is the coordinate?

What happens to the graph of the derivative for the functions mentioned above as you change its base?

Why would the derivatives of their graphs be slightly different?

Logarithmic Differentiation – the process of taking the natural logarithm of both

sides of an equation, differentiating, and then solving for the desired derivative.

This is applicable when

267

Example 1: Both the Base and the Exponent are Variables What is the derivative of

?

How to Do It

What to Think About When is logarithmic differentiation applied?

How is this function different from the function ?

Your Turn What is the derivative of

268

?

Example 2: The Derivative of an Exponential Function with a Constant Base and a Variable Exponent What is the derivative of

?

How to Do It

Your Turn What is the derivative of

What to Think About How does this function differ from the previous example?

?

We now have a new “short-cut” formula for taking the derivative of a function in the form where a is a constant.

Definition

269

Example 3: Using the Laws of Logarithms At what point on the graph of the function

does the tangent line have a slope

of 10? Round all values to the nearest hundredth.

How to Do It

270

What to Think About What will we have to equate to each other in order to solve this problem?

Your Turn At what x-value on the graph of the function

does the tangent line have a

slope of 8? Round all values to the nearest hundredth.

271

Definition

Example 4: Develop the Derivative of a Common Logarithm Develop the derivative of How to Do It

What to Think About How can we determine the derivative of a logarithm without being given the formula?

What type of differentiation is required?

Your Turn What is the derivative of

?

Did You Know At a rock concert, the sound pressure can be measured with the logarithmic function where W represents the sound (in watts) and represents the lowest threshold of sound that humans can detect.

The application of the derivative of a common logarithm would determine the change in sound pressure (in decibels) over time.

272

Example 5: The Derivative of a Logarithmic Composite Function What is the derivative of

?

How to Do It

What to Think About Which is the embedded function?

Your Turn Determine the derivative of each function. a)

d)

b)

e)

c)

273

Example 6: A Tangent Through the Origin A line with slope m passes through the origin and is tangent to the graph of . What is the value of m?

How to Do It

What to Think About What are two methods to determine the slope of the tangent line?

Your Turn A line with slope m passes through the origin and is tangent to the graph of the natural logarithm function . What is the value of m?

274

Practice Determine

for each equation.

1.

7.

2.

8.

3.

9.

4.

10.

5.

11.

6.

12.

275

13.

14.

15. Prove that the graph of

approaches a horizontal tangent as

16. A glass of cold milk from the refrigerator is left on the counter on a warm summer day. Its temperature y (in degrees Fahrenheit) after sitting on the counter t minutes is

a. What is the temperature of the refrigerator? How can you tell?

b. What is the temperature of the room? How can you tell?

276

.

5.6 Chapter Review For questions 1 to 8 determine 1.

6.

2.

3.

7.

4.

5.

8.

277

Use the following information to answer questions 9 to 14. Given that the functions f and g and their derivatives have the following values at x = 3 and x = 4 x 3

-6

4

3

4

3

4

-4

16

1

7

5

67

6

11

-7

-4

-8

3

-3

9.

12.

278

10.

13.

11.

14.

8

1

For questions 15 and 16 determine 16.

15.

For questions 17 and 18 determine 17.

: 18.

279

For questions 19 and 20 determine the equation of the tangent line to the curve at the indicated point.

19. at 20. at

21. On what interval(s) is

for

22. At what point is the tangent of

23. For what values of x is

280

?

parallel to the line

parallel to

?

?

24. You are attempting an inside loop with your new RC model airplane. It is following the route represented by . At what rate is it climbing when the coordinates are ?

281

Unit 2 Project - Design a Roller Coaster

You have been hired to design a new roller coaster for Belmont Park in San Diego.

Your coaster must satisfy the following safety conditions: no descent can be over , the design must begin with a

incline, never get as high as its previous

high point, and end at the same height it starts. Write a summary and present your results to the class as if you were trying to convince them to invest in and build your coaster, since your design is the best!

In small groups discuss some possible themes for your design. From your experiences discuss what elements you think make a good roller coaster. Come up with a plan to design the world’s newest roller coaster! You must include the following:

282



Functions you use must be differentiable and continuous at all points along the curve of the roller coaster. Use at least 5 distinct functions.



You need to show the graph and a proof that the function you generate is continuous and differentiable at the points where each pair of curves meet.

You may use any type of function including linear, trigonometric, exponential, logarithmic, and polynomial.

Research the best roller coasters in the world, answer some of the following questions (and get some inspiration for designing your own roller coaster):  What makes them great?  Where are they?  What kind of names do they have?  How much does it cost to ride them? How much did it cost to build them?  How fast do they go?  If you could ride one roller coaster in the world, which one would you most like to ride and why?

Choose a theme for your coaster and transfer your design to a piece of poster paper. It can be drawn freehand, but it should look like what you have drawn on Desmos. Tape or glue several pieces of poster paper together horizontally. You are encouraged to use Desmos or any other graphing application to print your functions.

Make sure that you use proper scaling to ensure the shape is accurate when enlarged  Cut out the roller coaster so that the top of it is your track.  Then tape or glue one end of the poster paper to the other end. You will end up with a 3-dimensional representation of your roller coaster.  Below each of your curves, write its function. Use a heavy dot to show the transition points. On the poster paper itself, show that the curves are continuous and differentiable at those points.

Answer the following questions:  Where is the path increasing and decreasing?  Where is it concave up and down?  For each fall, where is the steepest descent and how steep is the angle?  Draw a graph of the slope of the path verses the distance along the ground from the start (graph the first derivative).  Draw a graph of the rate of change of the slope (graph the second derivative).  The thrill of the roller coaster is defined as the sum of the angle of steepest descent in each fall (in radians) plus the number of tops. Calculate the thrill of each coaster.  Approximate how fast your best roller coaster will travel, showing your assumptions and calculations.

283

284

Unit 3: Applications of Derivatives This unit focuses on the following Big Idea from the British Columbia provincial Calculus 12 curriculum: 

Differential calculus develops the concept of instantaneous rate of change

By the end of this unit, you should be able to:  Use derivatives to analyze properties of a function  Solve problems involving optimization  Understand and apply the Mean Value Theorem  Connect the behavior of a differentiable function over an interval to the behavior of the derivative of that function at a particular point in the interval through the Mean Value Theorem.

By the end of this unit, you should know the meaning of these key terms:        

      

Increasing Decreasing Maximum Minimum Horizontal inflection point Vertical inflection point Critical point Concavity

Extreme Value Inflection point Mean Value Theorem Rolle’s Theorem Optimization First derivative test Second derivative test

At what rate does the surface area of a spherical object to the change in its volume

change in relation

?

285

Chapter 6: Analyzing Functions Using Derivatives We understand our world by observing change. In almost every field of study, an observer can collect data and analyze the change behavior of the situation. Knowing how something is changing over time allows the researcher to make predictions about the future and the past. Mathematicians can create models (or functions) of the rates of change data; and these can be used to discover the conditions that would optimize the situation. Optimization is a key application of the derivative.

286

6.1 The First Derivative Test Warm Up Given the graph of

Determine the following characteristics of f Characteristics Interval or Characteristics of f values of f’ f is increasing on f is decreasing on f is neither increasing nor decreasing at

since since since

Sketch a graph of f that passes through (0, 0) and (2, 4) with zeroes at

.

Is there a way to determine where maximum and minimum values of a function occur?

Definition:  An extreme value of a function f is called a relative (local) maximum if there is an open interval containing c on which is a maximum value.  An extreme value of a function f is called a relative (local) minimum if there is an open interval containing c on which is a minimum value.  If or if is undefined at c, then c is called a critical point of f.

287

Example 1: Determining the Extreme Values of a Function Given How to Do It

, find the maximum and minimum values of f using the derivative.

What to Think About What information do we need to determine the behavior of f?

What are the critical points?

What intervals are defined by the critical points?

How do you determine whether f is increasing or decreasing over the interval?

Could you use the graph of to determine the intervals where f is increasing or decreasing?

What are the extreme values of at each critical point?

288

Theorem: The First Derivative Test

Let c be a critical point of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then is a: o relative (local) maximum if changes from positive to negative at c. o relative (local) minimum if changes from negative to positive at c.

Let a be the left endpoint of a function f that is continuous on a closed interval . o

is a relative (local) maximum if

.

o

is a relative (local) minimum if

.

Let b be the right endpoint of a function f that is continuous on a closed interval . o

is a relative (local) maximum if

.

o

is a relative (local) minimum if

.

Your Turn Find the extreme values for each of the following functions. a) b)

289

Example 2: Determining the extreme values of a function on a closed interval Given defined on the closed interval find the extreme values of .

How to Do It

What to Think About What are the zeroes (xintercepts) of

?

What is the y-intercept of

?

What are the endpoints of

?

What does the rate of change information provided by the derivative function indicate about the behavior of at each value of x ?

290

Why is there not an extreme value at every critical point?

What happens at

?

Are there any values on where f has an overall maximum or minimum?

291

Definition: An extreme value of a function f is called an absolute (global) maximum if there is a point c in the domain D of f such that . Similarly, an absolute minimum occurs at c if .

Your Turn Find and identify each extreme value for the following functions. Justify your reasoning. a)

on

b)

Theorem: Extreme Value Theorem If a function f is continuous on a closed interval [a, b] then there exists both an absolute maximum and minimum on the interval.

292

Did You Know In 1637 mathematician Pierre de Fermat, in his work Methodus ad disquirendam maximam et minimam developed a method to find local maxima, minima and tangents to various curves that was equivalent to differential calculus. This was five years before Sir Isaac Newton was born!

Practice In questions 1 to 3, identify each x-value at which any absolute extrema occur.

1.

2.

3.

293

In questions 4 to 7, find the extreme values of the function and state the interval where the function is increasing or decreasing.

4.

6.

5.

7.

In questions 8 to 12, find the extreme values of the function 8.

11.

9.

12.

10.

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13. What is the absolute maximum on the function

?

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6.2 Modelling and Optimization Warm Up A rancher has 200m of fencing with which to enclose three adjacent rectangular pens. They border a straight river and do not require fencing along the river. What dimensions should be used so that the enclosed area should be a maximum?

DRAW a diagram and label the quantities relevant to the problem.

What are the possible quantities for the variables used for the dimensions?

(What are the restrictions on the variables?)

Write a system of equations to describe the situation and the quantity to be maximized or minimized.

Reduce the system to one equation involving the quantity to be maximized or minimized and one independent variable.

What kind of equation is the reduced system?

Change from standard form to vertex form

What is the maximum and when does it occur?

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What are the dimensions of the enclosed area?

How else could you find the extreme value of the situation?

Example 1: Area Optimization Problem A rancher has 200m of fencing with which to enclose three adjacent rectangular pens. They border a straight river and do not require fencing along the river. What dimensions should be used so that the enclosed area should be a maximum?

How to Do It

What to Think About What are the variables?

What equation can be used to determine the Area?

What does the first derivative test allow us to determine?

Are there any restrictions on the variables?

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Your Turn A farmer wants to fence an area of 750 000 m2 in a rectangular field and divide it in half with a fence parallel to one of the sides of the rectangle. How can this be done so the cost of the fence is kept as low as possible?

Example 2: Volume Optimization Problem An open topped box is created by taking a rectangular piece of cardboard measuring 16cm by 30cm and cutting out congruent squares from each corner so that the sides can be folded upwards to create the box. What size should the squares be to maximize the volume of the box?

How to Do It

What to Think About What does this situation look like?

What are the variables?

What equation can be used to determine the Volume?

The first derivative test allows us to determine extreme values.

Are there any restrictions on the variables?

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What happens at the remaining critical point?

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Your Turn The yearbook pages will have an area of 672 cm2 with top and side margins of 3 cm and bottom margin of 4 cm. What page dimensions will create the greatest possible area available for pictures.

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Example 3: Optimizing the Value of Numbers Find two numbers whose sum is 20 and whose product is as large as possible How to Do It

What to Think About How can we define the values?

What equations can be used to model the situation?

What are the restrictions?

Your Turn Find two positive numbers whose product is 10,000 and whose sum is a minimum.

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Practice 1. Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

2. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum.

3. A rectangular plot of land is bounded on one side by a river and on the other three sides by barbed-wire fence made with two strands of wire. If you have 2400 m of barbed wire what is the largest area that can be enclosed? What are the dimensions of the largest rectangular plot of land in this situation?

4. Your company has obtained a contract to build a square-based, open-topped rectangular tank for storing water. The contract states that the tank must have a volume of 500 ft3; be made from stainless steel; and, be as light as possible. What are the dimensions of the tank that meet these requirements?

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5. The combined perimeter of a circle and a square is 16. Find the dimensions of the circle and square that produce a minimum total area.

6. The combined perimeter of an equilateral triangle and a square is 10. Find the dimensions of the triangle and square that produce a minimum area.

7. A manufacturer wants to make an open top box having a square base and a surface area of 108 square centimeters. What dimensions will produce a box with maximum volume?

8. An aquarium with a rectangular base and rectangular sides is open at the top. It is to be constructed so that the width is 4 meters and its volume is 36 cubic meters. If the tank is to cost $10/sq m for the base and $5/sq m for the sides, what is the cost of the least expensive tank, and what are its dimensions?

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9. A cylindrical metal container, open at the top, is to have a capacity of cubic inches. The cost of the material used for the bottom of the container is $0.10/sq. in, and the cost of the material used for the curved part is $0.05/sq. in. Find the dimensions that will minimize the cost of the material, find the minimum cost.

10. A rectangle has a perimeter of 120 ft. What length and width should it have so that it’s area is a maximum?

11. A box with an open top is to be constructed from a square piece of aluminum, 4 m wide, by cutting out a square from each corner and bending up the sides. Find the largest volume that such a box can have.

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6.3 The Second Derivative Test Warm Up a) Given

b) Let

, graph

. If

and use the first derivative test to sketch

.

, use the first derivative test to sketch f.

At what point does f change from curving up to curving down?

Definition: Concavity is the word used to describe the curvature of a function. Intervals can be described as concave up (“smile”) or concave down (“frown”). The concavity of a function changes at a point of inflection.

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Example 1: Determining the Inflection Point(s) of a Function.

Given How to Do It

, find the point(s) of inflection of f.

What to Think About How does the graph of f behave?

What function tells us how the slope of f changes over time?

The second derivative of f is the first derivative of . We can find the critical points of to find the extreme rates of change of f.

What characteristics on f do the critical points of indicate?

What connection can you find between the second derivative and the behavior of f?

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Concavity, Points of Inflection and The Second Derivative Let be a critical point of a function that corresponds to a differentiable function f that is continuous on an open interval containing c. o If , then is increasing and f is concave up. o If , then is decreasing and f is concave down. o If , then has a critical point and f may have a point of inflection at .  If and , or vice versa, then changes sign at . 

Thus, is a point of inflection on f.

Note: when changes sign, changes from increasing to decreasing or vice versa and f changes from concave up to concave down or vice versa.

Your Turn Find the points of inflection and intervals of concavity for each of the following functions. b) a)

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Example 2: Determining the Extreme Values of a Function using the Second Derivative Test. a) Given

, find the extreme values of f using the second derivative test.

b) Find the inflection points of f.

How to Do It

What to Think About How can we use concavity to determine the behavior of f at each critical point?

If , then how are behaving?

and f

If , then how are behaving?

and f

What about at the critical point?

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How do you know if f will have an inflection point?

Theorem: Second Derivative Test for Finding Local Extrema Let f be a function such that and the second derivative of f exists on an open interval containing c.

1.

2.

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Your Turn Find and identify each extreme value of test. Justify your reasoning.

using the second derivative

Practice In questions #1 and 2, use the First Derivative Test to determine the extreme values of the function.

1.

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2.

In questions #3 and 4, use the Concavity Test to determine the intervals on which the function is concave up and concave down 3.

4.

Find the points of inflection for the following functions 5.

6.

For questions 7 to 9 use the graph to estimate where 7.

,

and

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8.

9.

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10. The graphs of the first and second derivative of a function are shown below. On the same diagram, sketch a possible graph of the function f.

313

11. Sketch the following function on x

2 0 4 0

3 3 0 -2

4 0 Æ Æ

5 -4 -5 0

12. How many inflection points does the function

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have?

6.4 Curve Sketching Warm Up Use the following information to sketch a possible graph of g. a) g is an even function and is continuous on x g g' g”

+  +

x g g' g”

0 2 DNE DNE

   1 0 0 0

  + 2 1 DNE DNE

3 3 1 0

What is happening at each point of inflection?

b) Determine the absolute extrema on

. Justify your reasoning

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Definition:  A horizontal inflection point occurs when changes sign at a point c and both .  A vertical inflection point is a special case where exists but both do not exist; and, where has a vertical tangent line at point c. Note that must also change sign at c.

Example 1: Determining the Inflection Points of a Function.

Given the graph of below a) Over what intervals is the graph of f is increasing? b) At what value(s) of should f have a relative maximum? c) Over what interval(s) is f concave up? d) Sketch a possible graph for f .

How to Do It

What to Think About

What are the critical points of f?

What does the behaviour of tell us about f?

What do the extreme values of tell us about ?

What does the behaviour of tell us about ?

What function tells us how the slope of f changes over time?

What assumptions was made when sketching the graph f?

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Your Turn Given the graph of to the right: a) Over what intervals is the graph of f increasing? f ‘(x) b) At what value(s) of x should f have a relative minimum?

c) What behavior does f have at

?

d) Over what interval(s) is f concave up?

e) What are the inflection points of f?

f) Sketch a rough graph of a possible f.

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Example 2: Sketch the Curve Without a Calculator Given , from example 2 in section 6.3, sketch the curve by including all characteristics of the function.

How to Do It

What to Think About What are the critical points?

What does the first derivative tell you about f?

What does the second derivative tell you about f?

Possible inflection points occur when .

How do you know if f will have an inflection point?

Which term of f dominates the behaviour as ?

Which term of f dominates the behaviour as ?

318

Your Turn Sketch the curve of

by including all characteristics of the function.

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Practice 1. Sketch a continuous curve label appropriate coordinates.    

320

with the following properties. Remember to    

2. Sketch a continuous curve label appropriate coordinates. x

y

with the following properties. Remember to

Curve Falling, concave up

3

2

Horizontal tangent Rising, concave up

6

5

Inflection point Rising, concave down

8

8

Horizontal tangent Falling, concave down

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3. Use the graph of f ’, where the domain is on which the continuous function f is: a. increasing

, to estimate the intervals

b. decreasing

Estimate the x-coordinates of all local extreme values.

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4. f is continuous on

and satisfies the following: x f

x f

1 2 3 4 0 2 0 3 2 0 Does not exist 1 0 1 Does not exist 0 + + 

+  

  

a. What are the absolute extrema of f and where do they occur?

b. What are the points of inflection?

c. Sketch a possible graph of f.

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6.5 The Mean Value Theorem Warm Up Sketch a differentiable function that passes through the points (2, 4) and (5, 4).

Is there at least one point on the graph where the derivative is zero?

What is the average rate of change on the interval?

Is it possible to graph a differentiable function that passes through both points without at least one point where the derivative is zero?

Sketch a differentiable function that passes though the points (9, 8) and (7, 4).

Is there at least one point on the graph where the derivative is zero?

What is the average rate of change on the interval?

Draw the secant line between both endpoints.

Is there at least one point on the graph where the derivative is equal to the slope of the secant line?

Is it possible to graph a differentiable function that passes through both points without at least one point where the slope of the curve is equal to the slope of the secant line?

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How could you write these two situations using a mathematical equation?

Theorem: Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least on x-value such that .

Theorem: Rolle’s Theorem (a special case of the MVT) If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and then there exists at least one x-value such that .

Did You Know Joseph Louis Lagrange lived from 1736-1813. By the age of 19, he was appointed professor of mathematics at the Royal Artillery School in Turin. Lagrange’s mean value theorem has a clear physical interpretation. If we assume that f(t) represents the position of a body moving along a line, depending on the time t, then the ratio of

is the average velocity of the

body in the period of

time b−a. Since f′(t) is the instantaneous velocity, this theorem means that there exists a moment of time in which the instantaneous speed is equal to the average speed.

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Example 1: Using the Mean Value Theorem Show that for on the interval [1, 2], there exists at least one point such that . Justify your reasoning.

How to Do It

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What to Think About What conditions must be met to use the Mean Value Theorem?

Your Turn Two stationary police vehicles equipped with radar are 8 kilometers apart on a highway.

As a logging truck passes the first police car, its speed is clocked at 90 km/hr. Four minutes later, when the truck passes the second police car, its speed is clocked at 80 km/hr. Prove that the truck driver must have exceeded the speed limit of 110km/hr at some point during the 4 minutes.

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Example 2: Necessary Conditions for the Mean Value Theorem and Rolle’s Theorem “If the graph of a function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal”. Is this statement always true, sometimes true or never true?

How to Do It

What to Think About What conditions results in the Mean Value Theorem not applying?

Your Turn True or False? (justify your answer) a) The Mean Value Theorem can be applied to

on the interval

.

b) If the graph of a polynomial function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal.

c) If

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in the domain of f, then f is a constant function.

Practice For questions #1-4, state whether the mean value theorem is applicable on the given interval and the value of c for which it applies 1.

2.

3.

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6.6 Chapter Review 1.

Find the local extrema of f using the first derivative test. Sketch the graph and indicate where the function is increasing and decreasing. a.

b.

330

2.

Find the local extrema of f using the second derivative test. Sketch the graph and indicate the points of inflection as well as intervals of concavity. a.

b.

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3.

A farmer wants to put a fence around a rectangular field and then sub-divide it into three smaller rectangular fields by placing two parallel fences to one of the sides. If he has a total of 1000 yards of fencing, what dimensions will give him the maximum area?

4.

If

of material is available to make a box with a square base and open

top, find the largest possible volume of the box.

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5.

The interior of a 400-meter track consists of a rectangle with semi-circles at two opposite ends. Finds the dimensions that will maximize the area of the rectangle.

6.

If

, find the number c that satisfies the mean value theorem on

the interval of

333

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Chapter 7: Solving Problems Using Derivatives In the previous chapter we looked at using derivatives to determine information about graphs of functions and optimization problems. These are not the only applications for derivatives. We will also see how derivatives can be used to solve the following types of problems:

Rates of Change – In this section you will take a look at problems that involve two quantities or variables. If the rate of change of one quantity such as the height of water in a conical tank, changes with respect to time, how do we you find the rate of change of the second quantity, the radius of the water in the tank, with respect to time?

Linear Approximations – In this section you will explore how to use the derivative to compute a linear approximation to a function. You can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when you have the function, it is much solve certain problems using a simpler “approximation”.

easier to

Differentials – In this section you will compute the differential for a function. You will use differentials to explore how much a change in one variable affects the amount of change in a second variable. For example Newton’s second law states that, for a rocket, the thrust is proportional to the exhaust velocity and the fuel burn rate, or

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7.1 Related Rates Involving Shape and Space

Warm Up You throw a rock into a calm pond. A circular ripple pattern emerges with the outermost circle’s radius growing at a rate of 8 cm per second.

Using Leibniz notation, write the rate of change of the radius as a derivative.

What is the formula for the area of the outermost circle?

Write an expression for the rate of change of the area?

What is the rate of change?

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Steps to solve problems involving related rates: 1. Draw a picture and label and name the variables 2. Consider which variables change with time. Consider which aspects are constant.

3. Consider any restrictions or constraints on your variables.

4. Write an equation (a mathematical model) that relates the variables.

5. Simplify the expression by substituting any constant values.

6. Differentiate with respect to time. You will likely need to use implicit differentiation.

7. Substitute the given rate information and solve for the unknown.

8. Ensure that your answer is reasonable and eliminate any extraneous solutions.

Example 1: Determine the Rate of Change in Volume Problems Involving Spheres and Cylinders a) A spherical balloon is being filled with air. The radius of the balloon is increasing at 2 cm/s. What is the rate of increase in the volume of air when the radius is 3 cm? b) Water is being poured into a cylindrical water tower in Shaunavon, Saskatchewan that is 8 m tall and 1 m in radius at a rate of 6 m 3/s. At what rate does the surface of the water rise?

How to Do It

What to Think About What equation can be used as a mathematical model for this situation?

Which quantities are changing, and which are not?

With respect to which variable should we differentiate?

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Which equation can be used as a mathematical model for this situation?

Which quantities are changing, and which are not?

Your Turn You are working at a local outdoor recreation center. To get ready for the summer, your boss had you clean the bottom of the swimming pool. The pool opens tomorrow, and you need to fill the pool back up. The hose you are using lets water in at a rate of . If the pool is in the shape of a rectangular prism, that is 12 m long, 6 m wide and always 2 m deep, what is the rate at which the height of the water is changing when the water is 0.5m deep? Would the rate change when the height is at 1.5m? Explain.

2m

12 m

6m

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Example 2: Determine the Rate of Change in Volume Problems Involving Cones and Prisms.

A conical vat in Charlie’s Chocolate Factory is being filled with liquid chocolate at a constant rate of 6 ft3/min. If the vat stands point down with a height of 8 ft and a radius of 4 ft, find the rate at which the level of chocolate is rising when the depth is 2 ft? What about when it is at 4 ft?

How to Do It

What to Think About What mathematical model can be used to represent this situation?

Which quantities are changing, and which are not?

Which quantities rates of change are unknown?

Why do we need to write the radius in terms of the height?

Could we solve the given problem by substituting for height in terms of the radius?

Why or why not?

Does it make sense that the rate that the height increases is slowing down as the cone fills up?

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Your Turn A trough is 20 feet long and 4 feet across the top. Its ends are isosceles triangles with heights of 3 feet. Water is running into the trough at a rate of 2.1 ft3/min. What is rate at which the height of the water in the trough is increasing when the water is 1.5 ft deep?

Would the rate change when the height is at 2 ft deep? Explain.

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Practice 1.

As a spherical balloon is being inflated, its radius (cm) after t minutes is given by , where . What is the rate of change with respect to t of each of the following at t = 8. a. r(t) b. The volume of the balloon c. The surface area

2.

A stone is dropped into the lake, causing water waves to create concentric circles. If, after t seconds, the radius of the waves is 35t cm, find the rate of change with respect to t of the area of the circle caused by the wave at: a. t = 1 sec b. t = 2 sec c. t =3 sec

3.

A circular metal pancake griddle is being heated, its radius changes at a rate of 0.02cm/min. When the radius is 10 cm at what rate is the area changing?

341

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4.

A balloon is being filled up with gas at a rate of 4ft3/min. Find the rate at which the radius is changing when the diameter is 16 in.

5.

Salt leaks out of a hole in a container and forms a conical pile whose height is always double the radius. If the height of the pile is increasing at a rate of 5 cm/min, what is the rate at which the sand is leaking out of the container when the height is 8cm.

6.

A spherical snowball is melting, and the radius is decreasing at a constant rate, changing from 11 inches to 7 inches in 45 minutes. How fast was the volume changing when the radius was 9 inches?

7.

The ends of a water trough 8ft long are equilateral triangles whose sides are 3 feet long. If water is being pumped into the trough at a rate of 4ft 3/min, find the rate at which the water level is rising when the depth is 8 inches.

8.

A point P(x, y) moves on the graph of the equation , the xcoordinate is changing at a rate of 3 units/second. How fast is the y coordinate changing at the point (1,7).

9.

A spherical balloon is deflating at a rate of 9 ft3/hr. At what rate is the radius changing when the volume is 375 ft3?

10. A stone is dropped into the pond and it creates circular waves whose radii increases at a rate of 0.4 m/sec. At what rate is the circumference of a wave changing when the radius is 5 m?

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7.2 Related Rates Involving Motion Warm Up Consider a 13-foot ladder leaning against a vertical wall. If the bottom of the ladder begins to slide along the ground away from the wall at a rate of 0.5 ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 5 feet from the ground?

Draw a diagram that represents this situation.

Label the quantities.

Which quantities are changing? Which quantities are constant?

Can you create an equation that models this situation?

What would the derivative of this equation be if this situation is changing over time?

Can you answer the question posed?

What are the units of your solution?

Why is your solution negative?

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Example 1: Solving Related Rates Problems When One Quantity is Constant A taxi leaves Walmart and begins travelling east for 3 Km at a speed of 60 Km per hour.

The driver turns north at an intersection and travels at a rate of 90 Km per hour. When the taxi is 4 km away from the intersection, how fast is the distance between the taxi and Walmart increasing?

How to Do It

What to Think About Which quantities are changing, and which are constant?

How is the distance the taxi has travelled east changing when driving north?

What mathematical model can be used to represent how the distance between the taxi and Walmart changes over time?

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Your Turn a) An armadillo is walking away from a 20 m tall tree at a rate of 3 m/min. At what rate is the distance from the top of the tree changing when it is 3 m from the base of the tree?

b) A boat is being pulled into a boat launch that is 2 m above the level of the surface of the lake.

Assume the rope is attached to the bow of the boat and is 50 cm above the water. If the rope is pulled in at a rate 0.2 m/sec, how fast is the boat approaching the launch when it is 10 m from the dock?

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Example 2: Solving Related Rates Problems When All Quantities are Changing In a high-speed chase, a police cruiser approaches a right-angled intersection from the north. The officers are chasing an escaped fugitive on a speeding motorcycle that has turned the corner and is now heading east. When the cruiser is 0.6 miles north of the intersection and the motorcycle is 0.8 miles east, the police officers determine with radar that the distance between them and the motorcycle is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the fugitive?

How to Do It

What to Think About Which quantities are changing in this situation?

At the instant of measurement, which quantities are known, and which others can be determined?

Why is the rate of change of the distance between the intersection and the police cruiser considered negative?

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Your Turn You are the captain of a small sailboat drifting at sea at midday and spot a ferry 15 km west of you. You are travelling at 6 km/h north and the ferry is travelling east at 8 km/h.

At 2 pm, how fast is the distance between the boats changing?

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Practice 1.

A ladder 25 feet long leans against a vertical building. If the top of the ladders slides down the wall at a rate of 4ft/sec, how fast is the ladder sliding away from the wall when the top of the ladder is 7 feet from the ground?

2.

Sebastian starts at point A and runs east at a rate of 12ft/sec. One minute later Orla runs north at a rate of 10 ft/sec. At what rate is the distance between them changing 1 minute after Orla starts.

3.

A man 6 feet tall walks at a rate of 6ft/sec away from a lamp that is 18 feet above the ground. When he is 10 feet from the base of the lamp at what rate is his shadow changing?

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4.

Jake is driving east at 60km/h and Samantha is driving north at 75 km/h. Both cars are approaching the intersection of the two roads. At what rate is the distance changing between them when Jake’s car is 0.5km from the intersection and Samantha’s car is 0.4 km from the intersection?

5.

A point P(x, y) moves along the graph Find

6.

350

such that

, where t is time.

at the point (2,4).

Sebi is flying a kite at cattle point and holds the string 4ft above ground level and lets the string out at a rate of 2ft/sec as the kite moves horizontally at height of 100 ft. Assuming the string is taut, find the rate at which the kite is moving when 150 ft of string has been let out.

7.

A hot air balloon is rising vertically at a rate of 2 ft/sec. An observer waiting for their turn is seating 100yards from a point directly below the balloon. At what rate is the distance between the balloon and the observer changing when the balloon is 500 feet high.

8.

When two electrical resistors R1 and R2 are connected in parallel the total resistance R is given by

. If R1 and R2 are increasing at rates of 0.02

ohm/sec and 0.01 ohm/sec, respectively, at what rate is R changing at the instant when R1= 90 ohms and R2= 30 ohms?

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7.3 Linearization and Newtons Method of Approximation Warm Up Let .

Show that the line tangent to the graph of

Set see.

and

at the point

is

. Zoom in on the two graphs at

Consider and . Find How good of an approximation is for

and ?

. Sketch what you

.

What happens to the accuracy of the approximations as you move further away from the point of tangency?

Is

352

an underestimation or an overestimation? Explain.

.

Definition: for a function f that is differentiable at  The linearization is a linear approximation model used to estimate a past or future value of an unknown function. The linearization uses a specific point on the function and the rate of change information at that point; thus, it is equivalent to the tangent line at the specific point.  In practice, we often find such that This is generally a reasonable approximation provided

is

sufficiently small.  Newton’s Method -if is a first approximation to a root of an equation, , then successive approximations are given by:

Example 1: Determining the linearization and using it to approximate a function. a) Find the linear approximation

(or linearization) of

b) How accurate is the approximation

at for values of

. near ?

c) Sketch what the linearization looks like at values close to 1.

How to Do It

What to Think About What is the linearization equivalent to?

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How well does the linearization approximate the curve near the point of tangency?

What happens to the approximation at ?

354

? At

Your Turn a) What is the linearization to the graph of

b) How good is the approximation for

at

?

?

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Example 2: Using Newton’s Method to Estimate the Root of an Equation Use Newton’s method to find the third approximation starting with .

How to Do It

What to Think About How accurate is your estimation?

Your Turn Find the third approximation to the root of the equation Newton’s Method starting at

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to the root of the equation

using

Example 2: Using Newton’s Method to Estimate the Value of a Function Use Newton’s method to find How to Do It

correct to four decimal places.

What to Think About How accurate is your estimation?

Your Turn Use Newton’s method to find

correct to four decimal places.

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Practice 1.

Find the linear approximation to the function the approximation for ?

2.

Find the linearization to the graph of approximation for

3.

at x = 0? How good is

at

? How good is the

?

Find the linear approximation

(or linearization) of

a. How accurate is the approximation

at for values of

b. Sketch what the linearization looks like at values close to 1.

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. near ?

4.

a. Find the linear approximation

(or linearization) of

b. How accurate is the approximation

at

for values of

.

near ?

5. Use Newton’s method with as the initial approximation for the root of correct to six decimal places

6. Use Newton’s Method to approximate the root of the equation given the interval correct to six decimal places. a.

b.

7. Use Newton’s Method to approximate the followings numbers a. b.

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7.4 Chapter Review 1. Water is being emptied out of a spherical fishbowl of radius 10 cm. If the depth of the water in the tank is 6 cm and is decreasing at a rate of 3cm/sec, at what rate is the radius of the top of the surface of the water decreasing?

2. A water tank has the shape of a right circular cone of altitude 12 ft and base radius 4 ft, with the vertex at the bottom of the tank. If water is being taken out of the tank at a rate of 10ft3/min, for fast is the water level falling when the depth is 5 ft?

3. At what rate is the area of an equilateral triangle increasing if its base is 12 cm long and is increasing at a rate of 0.4 cm/sec?

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4. A balloon is being filled at a rate of 121π cm3/sec. At what rate is the balloon increasing when its radius is 12 cm.

5. The base of a rectangle is increasing at 4cm/s while the height is decreasing at 3 cm/s. At what rate is the area changing when its base is 20 cm and its height is 12 cm?

6. A square is expanding. When each edge is 16 cm its area is increasing at a rate of 80cm2/sec. At what rate is the length of each ledge changing?

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7. Two roads intersect at right angles. At 9 am a car passes through the intersection headed due west at 60 km/h. at 10 am a truck heading south at 75 km/h passes through the same intersection. Assume they maintain their speeds and directions. At what rate are they separating at 1 pm?

8. A circle is inscribed in a square as shown. The circumference of the circle is increasing at a constant rate of 8 cm/sec. As the circle expands the square expands to maintain its tangency.

a. Find the rate at which the perimeter of the square is increasing.

b. At the instant when the area of the circle is 36π square inches, find the rate of increase of the area between the circle and the square.

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9. Water is draining from a conical tank with a height of 10 feet and diameter 8 feet into a cylindrical tank. The depth of the water in the conical tank is changing at a rate of (h – 8) feet per minute.

a. Volume of a cone

,write an expression for the volume of water in

the conical tank in terms of h.

b. At what rate is the volume of the water in the conical tank changing when h = 4?

c. At what rate is the height of the water in the cylinder changing when h = 4?

363

10. The radius of a basketball is increasing a rate of 0.01 cm/sec. a. When the basketball has a radius of 10 cm, what is the rate of increase of its volume.

b. At the time when the volume of the basketball is 48π cubic centimeters, what is the rate of increase of the area cross section through the center of the basketball.

c. When the radius and the basketball are increasing at the same rate, what will the radius be?

11. Use Newtons Method with initial approximation to find the third approximation to

the root of the equation

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Unit 3 Project: Shrinking Lollipop This project compares the predicted rate of change of the radius of a lollipop to the actual rate of change.

Test the hypothesis that the rate of change of the volume of a lollipop is proportional to its surface area. It seems reasonable that the rate of change of the volume of a lollipop would be proportional to its surface area, since the candy dissolves from the surface as you suck the lollipop. In mathematical terms, we predict that

where k is a constant of proportionality. But is this prediction true? What can we learn, by mathematical deduction, about k, and how can we test this prediction experimentally?

1. Prepare a table with time in the first column, radius in the second column.

2. Measure (mm) and record the radius (diameter/2) of a lollipop (Tootsie Roll Pops work well).

3. Suck on the lollipop for one minute and measure the radius. Record your data.

4. Repeat step 3 until the lollipop is gone or you reach candy of a different consistency (e.g. the inner nougat of the Tootsie Roll Pop). Note: it is important to suck on the lollipop the same from one trial to the next. If you suck especially vigorously on one trial but only lightly on the next, you won’t be able to interpret your results.

5. Graph your results: radius (y-axis) vs time (x-axis). What kind of function do you see: linear? parabolic? ? ?

6. Use the regression function on your calculator or Desmos to determine an equation.

7. Now consider the prediction. Use a chain rule decomposition of k. (Hint:

to help find

)

8. Do the experimental results support your prediction?

Questions: 1. What happens to the shape of a square lollipop (or other shape with edges and corners) as it dissolves? Why? 2. How does the theoretical compare to the calculus you have performed?

365

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Unit 4: Integral Calculus This unit focuses on the following Big Idea from the British Columbia provincial Calculus 12 curriculum: 

Derivatives and integrals are inversely related.

By the end of this unit, you should be able to:  Recognize antiderivatives of basic functions.  Interpret the definite integral as the limit of a Riemann sum in integral notation.  Express the limit of a Reimann sum in integral notation.  Approximate a definite integral.  Calculate a definite integral using areas and properties of definite integrals.  Analyze functions defined by an integral.  Calculate antiderivatives.  Evaluate definite integrals.  Interpret the meaning of a definite integral within a problem.  Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume  Analyze differential equations to obtain general and specific solutions.  Interpret, create, and solve differential equations from problems in context.

By the end of this unit, you should know the meaning of these key terms:     

Antidifferentiation Area under a curve Definite integral Differential equation Indefinite integral

?

   

Integral Integration by parts Rotational volume Riemann sum

?

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Chapter 8: Antidifferentiation Differentiation allows us to determine how a function is changing. Most of the time, in real-world experiments, we collect data that measures change. We then try to determine a function, which is a derivative that represents the data. Once the derivative is known, we can use antidifferentiation to find the original function.

Integration is like filling a tank from a tap.

The input (before integration) is the flow rate from the tap.

Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank.

Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap).

As the flow rate increases, the tank fills up faster and faster.

With a flow rate of fills up at .

, the tank

We have integrated the flow to get the volume.

Example: with the flow rate in litres per minute, and the tank starting at t = 0.

After 3 minutes

:

 the flow rate has reached  the volume has reached

And after 5 minutes

liters

:

 the flow rate has reached  the volume has reached

368

liters/min,

liters/min, liters

8.1 The Antiderivative – Working Backwards Warm up: Work backwards to find the original Function.

What to Think About

What pattern do you observe?

What happens when the coefficient does not fit the pattern?

Generalized pattern

Are there any other functions that you could differentiate to get

?

Can you determine a general formula to follow?

Finding the Integral (the antiderivative) using the “Reverse Power Rule” If Integral notation,

. , asks you to find the antiderivative.

The constant of integration C must always be added when anti-differentiating because the derivative of a constant is zero. Without more information about the original function we cannot know the value of the constant term.

369

Example 1: Finding the Antiderivative Using the Reverse Power Rule.

Find the antiderivative of the following How to Do it a)

What to Think About How do you apply the reverse power rule?

Why must you include a constant? b) Can you write the derivative as a power term?

Your Turn a)

370

b)

Example 2: Determining the Antiderivative of a Function Find the antiderivative of the following How to Do it

What to Think About How can you write the function with an exponent?

a)

b)

Your Turn a)

b)

371

Derivative Notation

Differential Notation

Practice

372

1.

4.

2.

5.

3.

6.

Integral Notation

7.

10.

8.

11.

9.

12.

373

8.2 Antiderivatives of Trigonometric Functions

Warm up Consider all the primary Trigonometric differentiation formulas:

These differentiation formulas lead to the following antiderivatives:

374

Example 1: Find the Antiderivative by Reversing Trigonometric Differentiation.

How to do it a)

What to think about What would you differentiate to get ?

b) If you remember the derivatives can you find the antiderivatives? c)

Your Turn a)

b)

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Example 2: Using Trigonometric Identities Before Finding the Antiderivative.

How to do it a)

What to think about Can you rewrite the function using a trig identity?

b)

Why does using a trigonometric identity help you to antidifferentiate?

Your Turn a)

376

b)

Practice 1.

7.

2.

8.

3.

9.

4.

5.

10.

6.

377

8.3 Antidifferentiation Using Substitution

Warm up Take the derivative of the following functions:

After reviewing the derivatives using chain rule how can you use this idea to find the antiderivative?

How else can be written?

378

“Reverse chain rule”

Example 1: Reversing the Chain Rule Using “u/du” Substitution How to do it a)

What to Think About Are you trying to find the antiderivative of a composite function?

Is the derivative of the embedded function present?

b)

Why do you need to expand this function first?

Which rule did you use to find the antiderivative?

379

Your Turn a)

b)

380

Example 2: Finding the Antiderivative Using “u/du” Substitution How to do it a)

What to Think About Can you simplify the expression using a u/du substitution?

Why is a u/du substitution not possible for this function? b)

What would we need to do to find the antiderivative?

Have you ever heard of the binomial expansion theorem?

381

Your Turn a)

b)

c)

Pascal’s Triangle and the Binomial Expansion Theorem

The binomial expansion theorem is a quick way to expand a binomial raised to a large integer exponent. Pascal’s triangle provides a quick way to determine the coefficients.

382

Practice Evaluate each integral 1.

2.

3.

4.

5.

6.

383

384

7.

8.

9.

10.

11.

12.

8.4 Advanced Antidifferentiation Techniques Warm Up Differentiate each expression

Trigonometric Integration Formulas

385

Example 1: Evaluate Integrals Involving Trigonometric Functions and u/du Substitution.

Evaluate the following integrals How to Do It What to Think About Why is the u/du helpful for this a) question?

Is a u/du always necessary?

Sometimes, it is best to ask the question “what would I differentiate to get this function?”

b)

c)

Your Turn a)

386

b)

Example 2: Evaluate the Indefinite Integral How to Do It a)

What to Think About What would you need to differentiate to get ?

How does the constant p effect the final answer?

b)

Why do you not have to use a reverse of the quotient rule?

Your Turn Determine each indefinite integral:

a)

b)

387

Practice 1.

388

4.

2.

5.

3.

6.

7.

10.

8.

11.

9.

389

8.5 Antidifferentiation Involving Exponential and Logarithmic Functions Warm Up Integrate the following:

What did you notice? What about the last one?

Did it follow the rule?

The antiderivative of the reciprocal function:

390

Example 1: Removing Constants Before Finding the Antiderivative.

How to Do It Evaluate each integral

What to Think About It is often beneficial to remove a constant before finding the antiderivative.

a)

Why do we have to include the absolute value? b)

Your Turn

Evaluate each integral a)

b)

391

Example 2: Finding the Antiderivative of a Reciprocal Function.

How to Do It

What to Think About What would you differentiate to

Evaluate each integral

get a)

1 ? x +1

Why may we omit the absolute value in this case?

b)

Your Turn Evaluate each integral a)

392

b)

Example 3: Evaluate the Indefinite Integral How to Do It

What to Think About Is there a constant to be factored out?

Evaluate each integral a)

What would you differentiate to get ?

What would you differentiate to get the exponential component of this function?

b)

Why must you first break this up into three separate fractions first? c)

Your Turn a)

b)

393

Practice

394

1.

5.

2.

6.

3.

7.

4.

8.

9.

13.

10.

14.

11.

15.

12.

395

8.6 Integration by Parts Warm Up: Use the product rule for differentiation to find the derivative of the following function.

What is the antiderivative of the integrand of

?

Simplify the above expression to show

Can you rearrange the last equation to determine an expression for

396

?

Definition Integration by parts is an integration technique that needs to be used to find the antiderivative for some functions. This technique reverses the product rule of differentiation.

There is an additional technique known as tabular integration that is a condensed version of the integration by parts process. Tabular integration or the Tic Tac Toe method works well for certain functions.

Did You Know The Tic Tac Toe method was showcased in the movie 1988 film Stand and Deliver. The movie is based on the true story of Jamie Escalante and his James A. Garfield high school students in East Los Angeles. Under the teaching of Escalante, Mexican American students proved that social and economic factors could not hold them back from success at the AP level in mathematics. There are many scenes involving calculus in the course and the film highlights perseverance and self-improvement through sustained effort.

The film is preserved in the National Film Registry in the United States Library of Congress for its cultural and historical significance.

The origination of the tabular integration is credited to Vedula N. Murty, Professor Emeritus of Mathematics and Statistics at Penn State University.

397

Example 1: Determine the Integral Using Integration by Parts a) How to Do It

b) What to Think About

a)

What integration technique should be applied?

b)

Which function should be u and which should be dv?

What happens if you are unable to integrate the new function after using integration by parts?

Your Turn Evaluate

398

Example 2: Recursion with Integration by Parts

How to Do It

What to Think About What technique should be used to integrate this product?

What can you differentiate to get the integrand?

If a recursion happens, in other words we get back to where we started, what can be done to solve for the integral?

Why must a constant of integration be included?

399

Your Turn

Practice

400

1.

3.

2.

4.

5.

6.

Additional Practice 7.

9.

8.

10.

401

402

11.

15.

12.  ecosx sin xdx

16.

13.

17.

14.

18.

19.

22.

20.

23.

21.

24.

403

Other Useful Formulas:

404

8.7 Chapter Review Evaluate each indefinite integral.

1.

4.

2.

5.

3.

6.

405

7.

10.

8.

11.

9.

406

12.

13.

14.

15.

407

408

Chapter 9: Solving Differential Equations There are many ways to model real-world situations mathematically. Differential equations describe how phenomena change. A differential equation is a mathematical model used to measure the change of a quantity that is determined by another quantity.

There are a variety of solution techniques for these types of equations depending on the complexity of the equations involved. Differential equations are composed of several terms, just as conventional algebraic equations are.

Consider the example of Yellow fever in the body.

In vaccinology, scientists are interested in the changing of cells, molecules and virus concentrations with respect to time. The following equation is one that represents a model of the virus (V) for Yellow Fever.

dV = π v ( V )H*-cv V -k v VA t dt Each of the variables represents a specific quantity that is being measured related to the Yellow Fever antibody.

409

9.1

Solving Differential Equations Analytically

Warm Up Consider: What does

mean?

Multiply both sides by

How can you solve for y?

Draw a possible graph of y.

Definition An equation involving a derivative is called a differential equation.

410

There is one differential equation that you probably know. It is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us.

To see that this is in fact a differential equation we need to rewrite it. First, remember that acceleration, a, can be written in one of two ways. or Where v is the velocity of the object and s is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position. Now Newton’s Second Law can be written as a differential equation in terms of either the velocity, v, or the position, s, of the object as follows. or Example 1: Evaluate a Differential Equation by Separating the Variables How to Do It

What to Think About How can you separate the variables so that all x terms are on one side of the equation?

How can you solve for y?

Why is there no constant of integration on the left side of the equation?

What does the constant of integration determine about the behavior of y?

What do you need to know to find C?

411

Your Turn Evaluate the differential equation

Example 2: Evaluate Differential Equations Implicitly to Solve for “y” How to Do It a)

What to Think About How can you separate the variables so that all x terms are on one side and all y terms are on the other?

What do you need to do to remove the differentials from the equation?

Why is there no constant of integration on the left side of the equation?

What does C1 equal?

412

What to Think About How can you separate the variables so that all x terms are on one side and all y terms are on the other?

b)

Why do we not need to use the absolute value sign any more?

Your Turn a)

b)

413

Example 3: Interpret the Solution to a Differential Equation Graphically If we go back to the differential equation graphically?

How to Do It

, what does the solution mean What to Think About What does the represent What does the C represent? T

How can you know which is the correct graph?

Your Turn Graph the solutions to the differential equations and create at least two different curves. a)

414

b)

Example 4: Find the Particular Solution Given an Initial Value for the Function: Suppose the slope of the tangent line to each point on the curve is x and the curve passes through the point How to Do It

. Find the specific curve.

What to Think About How can you separate the variables? .

How does the initial value help us find the exact function y?

Your Turn Given that

and the solution curve passes through the point

, what is y?

415

Practice

416

1.

4.

2.

5.

3.

6.

7. A particle moves along a line with an acceleration of at a time t. When , its velocity is 3 and its position . When , what is the position s.

8. An automobile accelerates from a standing start with a constant acceleration of . How far does it travel in the first 10s?

9. Suppose that the acceleration a(t) of a particle at time t, is given by a(t) = 6t  3 and where s(t) is the position function. Find v(t) and s(t).

417

10. A tree farm sells a bush after 7 years of growth. The growth rate of the tree during the first seven years is approximated by

, where t is the time in

years and h is the height in cm. The seedlings are 10 cm tall when planted t = 0. a. Find the height after t years

b. How tall are the bushes when they are sold?

418

9.2 Solving Differential Equations Graphically Warm up Given that

, find the slope at each point Use each point

At each of the points

.

to calculate

draw a short line through the point with slope

.

y

x

Can you sketch a graph of y through the point

?

What does

tell us

about the function y?

419

Slope Field A slope field uses the idea of local linearity; that is, if a function is differentiable at a point, then the tangent line approximates the function close to that point.

By drawing a slope field, we can graphically “see” the family of functions or the general solution to the given differential equation. The slope field is like the wind blowing through a field of grass. You can see the change behavior that the differential equation is describing. Then by choosing a starting point, an initial value, one can “determine” the particular solution. Some differential equations can only be solved by considering the slope field.

Example 1: Sketching a Slope Field Consider the differential equation

through the point

. What does

the slope field look like? What curve passes through the initial value?

How to Do It

What to Think About What does

tell us about

the slope at each point?

How can we use the slope field to draw a curve through the initial point?

420

Your Turn Sketch a graph of the slope field for

Example 2: Determine a Specific Solution to a Differential Equation.

Determine the slope field for

. Sketch a solution function that passes through

?

Confirm algebraically the solution to the differential equation.

How to Do It

What to Think About What does

tell us about the

slope at each point?

How can we use the slope field to draw a curve through the initial point?

Why does the solution show only “half” of the curve?

421

Why must we choose between the positive and negative versions of the y function?

How do we decide which is the solution?

Your Turn a)

422

What is the slope field for the differential equation

?

b)

Draw the solution to this differential equation that passes through the point , then solve the differential equation algebraically given this initial condition.

423

Practice 1. Draw slope fields for the following differential equations a)

b)

2. a) Draw the slope field for the differential equation b) Draw the unique curve that passes through the point (2, 1).

424

c) Solve the differential equation with this initial condition.

3. Match the slope field (a – d) to one of the differential equations (i – iv). a) b)

c)

i.

d)

ii.

iii.

iv.

425

4. Draw the solution of each differential equation. Then, determine the corresponding differential equation below each function.

426

a.

b.

c.

d.

9.3 Approximating Area Under a Curve Using Riemann Sums Warm up Consider a car travelling with the following velocity-time information. a.

How far has the car travelled between

b.

How far has the car travelled between

Can you determine the exact distance travelled when the function is a curve? a

Definition The process of evaluating a product in which one factor varies is called finding the definite integral. The definite integral will give you the area under the curve. We can approximate the definite integral in several ways.

427

Example 1: Estimating the Area Under a Curve Geometrically How to Do It What to Think About Can you determine an exact answer in this case?

Is there another way other than counting we can find the area under the curve?

What shapes can we use to determine the area?

What other shape can be used to represent the area?

428

Your Turn Graph

and calculate the area under the graph from

to

429

Example 2: Rectangular Approximation Methods to Determine the Area Under a Curve.

Graph the function and make a table of values. Approximate the area under the curve from by using 4 rectangles of equal width.

How to Do It

What to Think About How do you determine the width of each sub-interval?

Can you use the entire interval to determine the width of each sub-interval?

What does i represent?

Method 1: Left Rectangular Approximation

What does n represent?

What height should we choose for each rectangle?

What does the ai represent in the area equation?

How do you determine for each rectangle?

Does the left rectangular approximation method overestimate or underestimate the area in this example?

430

Method 2: Right Rectangular Approximation

What height should we choose for each rectangle?

How do you determine for each rectangle?

Does the right rectangular approximation method overestimate or underestimate the area in this example?

Does a right rectangular approximation always underestimate?

Can you provide any counterexamples?

Method 3: Trapezoidal Approximation

What do

and

represent?

Is the trapezoidal approximation an overestimation or an underestimation for this function?

431

Method 4: Midpoint Rectangular Approximation

Compared to the trapezoidal approximation, what is different about the average used in the midpoint approximation?

Which is the most accurate approximation, LRAM, RRAM, Trapezoidal, or MRAM?

Rule Left Endpoint

Right Endpoint

Midpoint

Trapezoidal

432

Formula NOTE:

These formulas work only for n equal sub-intervals.

Your Turn A particle starts at

and moves along the x-axis with velocity

. Where is the particle at rectangles of equal width.

for time

? Approximate the area under the curve using four

a.

Calculate the left-hand area (LRAM):

b.

Calculate the right-hand area (RRAM):

c.

Calculate the trapezoid area:

433

Example 3: Solve a Contextual Problem Using Trapezoidal Approximation A machine fills a milk carton with milk at a constant rate. The rates (in cases per hour) are recorded at hourly intervals during a 11-hour period, from 6:00 am to 5:00 pm Use the trapezoidal approximation method with n = 11 to determine approximately how many cases of milk are filled by the machine over the 11-hour period.

How to Do It

What to Think About What is the interval width?

Why is 5:00 pm converted to 17 o’clock

434

Your Turn The economy is continuously changing, we can analyze it with certain measurements. The following table records the annual inflation rate as measured each month for 13 consecutive months. Use the trapezoidal rule with n = 12 to find the overall inflation rate for the year.

435

Practice 1. The function f is continuous on the interval [2, 8] and has values that are given in the table. Using the subintervals [2, 5], [5, 7] and [7, 8], what is the trapezoidal approximation of

?

x 2 5 7 8 f(x) 10 30 40 20 2. The following table shows the speedometer readings of a truck, taken at ten-minute intervals during one hour of the trip. Use the table and the midpoint rule to estimate the distance that the truck traveled in the hour. Watch your units!

Time (min)

0

10 20 30 40 50 60

Speed (km/h) 40 45 50 60 70 65 60

3. Use the Trapezoidal Rule with n = 2 to approximate the integral

436

.

4. The temperature, in degrees Celsius

, of the water in a pond is a differentiable

function W of time t. t days 0

20

3

31

6

28

9

24

12

22

15

21

The table above shows the water temperature as recorded every 3 days over a 15-day period. a)

Use the data from the table to find an approximation for

. Show the

computations that lead to your answer. Indicate units of measure.

b)

Determine the LRAM over the time interval 0 ≤ 𝑡 ≤ 15 days with 5 subintervals .

c)

Determine the RRAM over the time interval 0 ≤ 𝑡 ≤ 15 days with 5 subintervals.

437

9.4 Calculating Area Under Functions Graphically Warm up Graph the function

. Find the area under the curve from

.

What happens when part of the graph is below the x-axis?

What is the difference between net area, and total area?

438

Example 1: Determining the Area Under Piece-wise Functions Geometrically Evaluate the following integral given the graph of

below

How to Do It

What to Think About Where should you start?

Do we need to find the equation of each linear section?

What happens when some of the area is below the ?

Does the integral ask you to determine the net area or the total area?

What is the difference between the net area and the total area?

439

Your Turn Given the piece-wise function below, write an integral statement and evaluate.

440

a.

Find the net area on [0,7].

b.

Find the total area on [0,7].

Example 2: Finding the Area Under a Curve of Known Shape Evaluate How to Do It

What to Think About What shape is the curve of the function?

What formula can be used to determine the area under the curve over the interval [-3,3]?

Your Turn Using the graph calculate the following integral.

441

Example 3: Solving Real-World Integration Problems Using Geometry.

A car moves along the highway at a constant rate of 65 miles per hour from 6:00 am to 8:00 am. Express the total distance travelled as an integral and evaluate.

What to Think About Under which type of function will the shape under the curve be a rectangle?

Consider the units that determine the area function, why is the result in miles?

Your Turn Find the output from a pump producing 30 gallons per minute during the first 2 hours of operation. Express your answer using correct units.

442

Example 4: Evaluating Definite Integrals Using a TI Calculator.

Evaluate How to Do It

What to Think About Which buttons on your calculator should you press to evaluate a definite integral?

Your Turn Use your calculator to evaluate the following integral.

443

Example 5: Solving Real-World Problems Using a Calculator The rate of consumption of oil in Canada during the 1990’s (in billions of barrels per year) is modeled by the function , where t is the number of years after January 1, 1990. Find the Total consumption of oil in Canada from January 1, 1990 to January 1, 2000.

How to Do It

What to Think About Why are the constants of integration 0 and 10?

Could you use different constants to get the same result?

What are the units obtained after integrating?

Your Turn The rate at which our homes consume electricity is measured in kilowatts. Most homes consume electricity at a rate of 1 kilowatt for 1 hour. Suppose that the average

æ πt ö , where è 12 ÷ø

consumption rate of your home is modeled by the function C ( t ) = 3.6-2.4sinç

is measured in kilowatts and t is measured in hours past midnight. Find the average daily consumption for your home, measured in kilowatt hours.

444

Practice 1. Determine the area under each graph a. b.

2. Evaluate the integral using geometrical shapes. a.

=

c.

=

b.

=

d.

=

445

3. Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. a.

b.

446

=

=

c.

=

d.

=

4. a.

The graph of f below consists of a line segment and a semicircle. Evaluate each definite integral using geometric formulas =

b.

c.

d.

e.

f.

447

9.5 Chapter Review 1.

Consider the curve given by

a) Show that

b) Find all points on the curve whose y-coordinate is 1. Write an equation for the line tangent to the curve at each of the points.

2.

Consider the differential equation equation

448

. Find the solution to the differential

3.

A function is differentiable for all real numbers. The point and the slope at each point is given by

is on the graph of .

a. Find the second derivative and evaluate it at the point

b. Find

by solving

.

using the initial condition

.

449

4.

For the following use the LRAM, RRAM, Trapezoidal, and MRAM to approximate the definite integral for the stated value of n. a.

LRAM

RRAM

Trapezoid

450

b.

LRAM

RRAM

Trapezoid

451

c.

LRAM

RRAM

Trapezoid

452

5.

Evaluate the definite integral using geometrical shapes. a.

b.

6.

Given

c.

=

d.

=

and

=

=

find:

a.

b.

c.

453

7.

The graph of

is below. Evaluate each definite integral by using geometric

formulas.

a.

d.

b.

e.

c.

454

8.

The graph was created by recording data on an instrument used to measure quantities. Estimate the y-value and approximate the area of the shaded region using n = 6 and the trapezoidal Rule.

455

Unit 4 Project: Newton’s Law of Cooling Objective 

To translate a law of physics into a differential equation.



To solve a separable differential equation.



To compare experimental results with theoretical predictions.

Definitions 

Temperature is how hot or cold something is.



Energy transferred from one body to another due to a temperature difference is called heat.

Background The transfer of heat from a body with high temperature to a body with lower temperature is what runs your car’s engine and keeps food cold in your refrigerator. Let’s investigate how heat moves in time.

Newton’s Law of Cooling The rate at which a body cools is directly proportional to the difference between the temperature of the body and that of the surrounding medium.

Materials      

456

Thermometer 50 ml beaker of room temperature water Graphing Calculator or computer with spreadsheet Microwave Stopwatch/timer (can be used on phone) Bucket of ice water

Part 1: The Experiment First measure the temperature of the ice water and record the temperature on the sheet provided. Take your beaker to the nearest sink/fountain and fill up 400ml of water. Use a microwave to warm the water for 5 minutes. Carefully take out the beaker and put in the thermometer with the red point down. (Have your stopwatch ready) record the temperature then slowly place the beaker in the ice water and record the temperature change every 10 seconds for 3 minutes in a table, graphing calculator list, or spreadsheet.

Part 2: The Calculus 1) Use the regression feature of your calculator or spreadsheet to determine the function of best fit for the temperature column.

2) Plot your data points and the regression equation on the same set of axes on the graph provided at the end of this project.

Let: T = Temperature of an object at time t (in minutes) = Ambient temperature (of ice water) Given: The rate of change of the temperature

, is (by Newton’s Law of Cooling) proportional

to the difference between the temperature of the beaker T and the ambient temperature . This means that:

Here a bit of care is needed: Clearly if the water in the beaker is hotter than the ice water , then the beaker is cooling down which means that the derivative

should

be negative. (Remember the connection between a decreasing function and the sign of the derivative?)

457

The equation we need has to have the following sign pattern:

3) Solve the differential equation above for a general equation. (Plug in the ambience temp.)

4) Find the particular solution by using 2 of the values taken from your table.

5) Use your calculator to record your new table of values from your particular solution on the third column of your data sheet or spreadsheet.

6) Sketch a graph of your calculus equation on the same axes as the data and regression equation. Use the blank graph on the next page making sure to use different colors.

458

Analysis Plot your results on the graph below. Write a key at the end to decipher which graph belongs to which data.

7) Compare your experimental and theoretical results: How are they alike and how do they differ?

459

460

Unit 5: Applications of Integration This unit focuses on the following Big Idea from the British Columbia provincial Calculus 12 curriculum: 

Integral calculus develops the concept of determining a product involving a continuously changing quantity over an interval.

By the end of this unit, you should be able to: 

Analyze functions defined by an integral.



Evaluate definite integrals.



Interpret the meaning of a definite integral within a problem.



Apply definite integrals to solve problems involving the average value of a function, motion, area, and volume.

By the end of this unit, you should know the meaning of these key terms: 

Area under a curve





Area between two curves



Average value of a function



Net change



Definite integral



Volume by revolution

Fundamental Theorem of Calculus

461

Chapter 10: Integrals and the Fundamental Theorem of Calculus

462

n=5

n = 15

n = 50

n=

10.1 Integrating Using the Fundamental Theorem Warm up Approximate

using a left rectangular approximation method with:

a) 3 equal subintervals x 1 2 3

b) 4 equal subintervals

4 5 6 7

c) 6 equal subintervals

8 9 10

d) 12 equal subintervals

11 12 13

e) Use your calculator to determine the exact value of

What happens to the approximation as we increase the number of rectangles?

463

Example 1: Making the Connection Between the Area Under a Curve and the Antiderivative. a) Write the area

in terms of

and

given

, then

calculate the value of the integral. b) Determine How to Do It

What to Think About How can you split up the area under the curve from 0 to 8 at x = 2 into two known shapes?

What is the connection between ?

464

and

Your Turn Write the area

in terms of

and

given

then

calculate the value of the integral over the interval [2, 4].

Theorem: The Fundamental Theorem of Calculus If a function f(x) is continuous on a closed interval [a, b] and F(x) is the antiderivative of f(x) on the interval [a, b], then

Note: It is not necessary to include the constant of integration C in the antiderivative because

465

Definition: The Definite Integral (more formally) Let f be a continuous function defined on a closed interval . Let the interval be subdivided into n intervals of equal

length

. As we increase the

number of subdivisions, , and , we can use an infinite number of infinitely thin rectangular slices to calculate a more accurate approximation for the area under a curve by using a limit.

The Riemann sum of all these rectangles calculates the area under the curve as follows,

where is an x-value chosen arbitrarily in the chose new notation using the differential, of f over as:

subinterval. Mathematicians , to write the definite integral

If

, then the area under the

is integrable over a closed interval

curve

from a to b is the integral of f from a to b,

The above statement is read “integral of f from a to b”. a lower limit of integration b upper limit of integration is the function of the integrand.

the integral sign

is the variable of integration.

See: https://www.desmos.com/calculator/tgyr42ezjq for an exploration of this concept.

466

Example 2: Using the Fundamental Theorem of Calculus Evaluate the following: a)

b)

How to Do It

What to Think About How do you use the antiderivative and the fundamental of theorem of calculus to solve?

Why is the answer negative?

How does this relate to the graph?

What conclusion can you make?

How do you know before you start if the answer will be negative or positive?

Your Turn Evaluate the following: a.

b.

467

Example 3: Connecting the FTC to the Net Area Under the Curve Find

How to Do It

graphically and confirm your answer using the FTC.

What to Think About What do you notice about the shape?

Is there any symmetry?

Can you guess what the net area will be?

Compare the areas on either side of the x-axis, what conclusion can you make?

Does your answer make sense?

How do you know before you start if the answer will be negative or positive or zero?

468

Definition: If f is a function whose domain contains x whenever it contains x, then a. f is even if for all x in the domain of f

b.

f is odd if

for all x in the domain of f

Your Turn Solve the following graphically then confirm using the Fundamental Theorem.

a.

b.

469

SUMMARY – Graphical connections to the Fundamental Theorem of Calculus

1.

The observations made can be summarized as follows: If is above the x-axis between a and b then .

2.

If

a

b

a

b

is below the x-axis between a and b then .

3.

If then

is above and below the x-axis between a and b which is the net area.

A3

A1 a

A2

b

4.

5.

6.

a

b

Note: when b > a, and we calculate the integral from b to a, dx < 0.

470

c

Example 4: Changing the Limits of Integration When Using the Substitution Method Solve How to Do It

by substitution.

What to Think About How would you integrate using substitution if this was an indefinite integral?

What values will u take, when integrating over

?

Why must you change the constants of integration when using substitution?

Your Turn Solve

471

Practice Evaluate each integral:

472

1.

5.

2.

6.

3.

7.

4.

8.

9. Given:

. Let

. Change the limits of integration and write

the new integral in terms of u, then solve.

10. Evaluate the integral

11. Suppose

and leave the answer in exact form

, then evaluate

473

12. Given

,

, and

a.

b.

c.

13.

Given

a.

b.

474

and

, find

find

10.2 Derivative of a Definite Integral Warm Up Today we are going to analyze the Fundamental Theorem of Calculus more closely.

Consider what the theorem states:

What is the relationship between

For the integral

identify

and

?

and

Can you state the Fundamental Theorem of Calculus an alternative way?

475

The Fundamental Theorem of Calculus can be stated another way

This represents the notion that the integral of a rate of change function gives us the net change of the integral function over the interval of integration.

Example 1: Evaluate the derivative of a definite integral a) How to Do It

b) What to Think About What does

ask you to

do?

What does

ask you to do?

How does your answer relate to the original integrand?

What does

ask you to

do?

What does

ask you to do?

How does your answer relate to the original integrand??

476

There is one other form of the FTC often called the Second Fundamental Theorem of Calculus:

If the function g is defined as

a  x  b then

for a < x < b.

In other words, g is an antiderivative of f and therefore f is the derivative of g.

The second fundamental theorem of calculus is sometimes referred to as the Newton-Leibniz axiom

Your Turn What does each expression represent? a)

b)

477

Example 2: Evaluate Using the Second Fundamental Theorem with the Chain Rule a) How to Do It

b)

What to Think About What is different about this expression?

When differentiating which rule must you follow when there is a function in the place of a constant of integration?

What part of the expression indicates that the chain rule must be used?

Theorem: Extension of the Second Fundamental Theorem of Calculus

478

Your Turn What does each expression represent? a)

b)

Example 3: Connecting the FTC to the area under the curve Consider the Area Function Evaluate the following: a) b) c) d) e) f) How to Do It

What to Think About What do the constants of integration mean graphically?

What does graphically?

represent

479

Why is the answer negative?

What does the derivative of the area function represent graphically?

How is the value of this expression represented graphically?

Why is this solution not negative?

480

Your Turn Let

where f(t) is the function graphed here. a) Evaluate

y

.

3 2 1 -3

b) Estimate

.

-2

-1

1

2

3

x

-1 -2 -3

c) On what interval is g(x) increasing?

d) Where does g have its maximum value?

481

Practice Evaluate each expression:

1.

6.

For

, find

.

7.

For

8.

For

, find

.

9.

For

, find

.

2.

, find

.

3.

4.

5.

482

10.

for

.

a) Find the value of x where F attains its minimum value.

b) Find open intervals over which F is only increasing or only decreasing.

c)

Find open intervals over which F is only concave up or only concave down.

483

11.

Let

where

is the function graphed to the right.

a) At what values of x do the local maximum and minimum of g occur?

4 3 2 1 1 -1 -2 -3 -4

b) Where does g attain its absolute maximum?

c) On what intervals is g concave downward?

484

2

3

4

5

6

7

8

9

10

10.3 The Average Value of a Function Warm Up The area of the two shaded regions are equal. a) Find the area under the curve g(x) on the interval [1,3].

b) Find the value of g(c), such that the area of the rectangle on the interval [1,3] is equal to the area under the curve of on the interval [1,3]

c) How is g(c) related to the area of g(x)?

c

d) Is the c the midpoint of the interval?

The area under any curve,

, can

be represented as a rectangle with height g(c) over the interval [a,b].

Can you write this relationship as an equation?

485

Definition: The Average Value of a Function If f is integrable on the interval [a,b] then

is the average

value of the function on the interval .

Example 1: Finding the average value of a function on an interval a) Find the average value of the function on the interval b) Find c such that How to Do It

is the average value of

.

on the interval [0,2].

What to Think About Graphically, what does the average value represent?

Why does dividing the integral by the width of the interval gives us the average value?

486

What value will the function have at c?

Why do we eliminate one of the solutions?

Your Turn a) What is the average value of the function

on the interval

b) Find c such that

on the interval [0,4].

is the average value of

?

487

Theorem: Mean Value Theorem for Integrals If f is a continuous function on such that:

, then there exists a number c in

The area under the graph of f from a to b is equal to the product of the interval length and the average value . The average value is the height of the area represented as a rectangle.

The average value (mean value) of the rate of change function can be written and shown in two ways:

You will recognize the first as the average value of

on

, otherwise

known as the average rate of change of f on . The second, you will recognize from the mean value theorem as the average rate of change of f on .

Using the fundamental theorem of calculus, we can see how these two expressions are equivalent.

Thus, if we are looking for the average rate of change for a function and we are given the derivative function, we use the mean value theorem for

integrals

.

If we are looking for the average rate of change for a function and we are given the original function, we use the mean value theorem for derivatives

488

Example 2: Finding the average rate of change Find the average velocity for given each situation below a) The position function b) The velocity function How to Do It

What to Think About What formula should you use to calculate the average value?

Why do you not use an integral?

What formula should you use to calculate the average value?

What do you get when you integrate a rate of change?

How do you determine the average value given the net change over an interval?

489

Your Turn Show that the average velocity of a car over a time interval [a,b] is the same as the average of its velocities during the trip.

Practice 1. Let f be a continuous function on [0, 3]. If, value of

then the greatest possible

is

2. Find the average value of the function

3. Let

be a continuous function on

and smallest possible values of

490

on the interval

. If ?

, what are the greatest

4. Find the average value of the function on the given interval a.

b.

5. Given the graph of f a. Evaluate

b. Determine the average value of f on the interval [0,8]

c.

Determine the answer to a, b is the graph if translated 3 units upwards

491

10.4 The Integral as Net Change Warm Up Given the graph of the velocity of a particle moving on the x-axis. The particle starts at x = 2 when t = 0.

a) What is the net distance traveled by the particle during the trip?

b) Find where the particle is at the end of the trip.

c) Find the total distance traveled by the particle.

492

Example 1: Motion Along a Line The velocity of a particle is given by

cm/sec. If the particle is

moving along a horizontal x-axis for , a) Draw the graph of the velocity function and describe the motion of the particle. b) Find the net distance traveled by the particle in the first 4 seconds. c) Suppose the initial position of the particle is . What is the particle’s position at , . d) Use your calculator to find the total distance traveled by the particle in the first 4 seconds How to Do It

What to Think About How do we know from the graph when it is moving left or right?

What is the point where the direction changes?

When you integrate the rate of change function what do you get?

493

What does

represent?

Why must we consider this when determining the particle’s position at a different time?

Which buttons do you need to press to do this on your calculator?

Your Turn The velocity of a particle is given by

cm/sec. If the particle is

moving along a horizontal x-axis for a) What is the net distance traveled by the particle in the first 6 seconds?

494

b) Suppose the initial position of the particle is position at

,

. What is the particle’s

.

c) What is the total distance traveled by the particle in the first 4 seconds?

Making Predictions using the FTC

Since the integral of a rate of change function gives us the net change of the original function, we use the facts that

and

to determine future or past values of the function. We can rearrange the fundamental theorem of calculus to create the following formula given the velocity function and one value, , to find a future value . In other words, the value of x at b is the value of x at a plus the net change of x from a to b.

or given

, to find a past value .

495

Did You Know Three hundred years before Newton and Leibniz first developed calculus Nicole Oresme (1323 – 1382), in his Treatise on the Configuration of Qualities and Motions proved geometrically that under uniform acceleration, the distance traveled is equal to the distance traveled at constant average velocity.

Example 2: Using your calculator to solve real-world applications using net change.

People enter a line for a turnstile at a rate modeled by the function

given by

where is measured in people per second and t is measured in seconds. As people go through the turnstile the exit the line at a constant rate of 0.85 person per second. There are 22 people in line at time . a) How many people enter the line for the turnstile during the first 5 minutes? b) During the first 5 minutes, there are always people in line. How many people are in line at ? c) After 5 minutes, what is the first time that there are no people in line?

How to Do It

What to Think About How do you determine the net change of people entering the line in the first 5 minutes?

496

How many people are in line to begin with?

How many people enter the line in the first 5 minutes?

How many people exit the line in the first 5 minutes?

What equation can be used to model this situation?

Are there other strategies to get this answer?

Can we determine this by starting at t = 0? Explain.

497

Your Turn Oil is pumped into an underground tank at a constant rate of 5 liters per second. Oil leaks out of the tank at a rate of

liters per second, for

hours. At t = 0, the

tank contains 115 liters of oil. a) How many liters of oil leak out of the tank in the first 40 seconds?

b) How many liters of oil are in the tank at t = 40 seconds?

c) Write an expression for the total amount of oil in the tank at any time t

498

Practice 1. A particle moves along the x-axis so that its velocity at time t is oven by

. At time t = 0, the particle is at a position x = 2. a) Find the acceleration of the particle at time t = 1. Is the speed of the particle increasing at t = 1? Explain your answer.

b) Find all times in the open interval (0,5) when the particle changes direction.

Justify your answer.

c) What is the total distance traveled by the particle from t = 0 to t = 5?

d) What is the particles position at t = 4?

499

2. A 100-gallon aquarium in a dentist’s office contains a rare collection of fish at t = 0.

During a 12-hour period from 8:00 am t = 0 to 8:00pm t = 12 the doctor is cleaning the tank by refilling the tank at a rate of

and during the same time

interval there is water being removed from the tank at a rate of

a) How many gallons of water is being pumped into the tank during the time interval hours?

b) Is the amount of water in the tank rising or falling at time t = 8 hours? Give a reason for your solution.

c) How much water is in the tank at t = 12 hours?

500

10.5 Chapter Review For question 1 to 3 find the average value over each interval.

3.

1.

2.

4. Show that the average value of interval [a,b]

is equal to the average rate of change of f over the

For question 5 to 11 evaluate the definite integrals 5.

6.

501

7.

10.

8.

11.

9.

For questions 12 to 15 evaluate the derivative.

502

12.

14.

13.

15.

For questions 16 to 19 change the limits of integration and then evaluate the function.

16.

18.

17.

19.

503

504

Chapter 11: Area and Volume Every day you see examples of how integration is used in a wide variety of contexts.

For example: 

Consumer Testing – A car magazine tests two kinds of engine. One engine had acceleration modeled by , t seconds after starting from rest. The acceleration of a turbo-charged model could be approximated by ,t seconds after starting from rest.

How much faster is the turbocharged model at the end of a 10 second test run. 

Population Predictions – A government estimates that the population of the country (in 1000 people per year) will grow at the rate of

. If an education

program is instituted, they believe the population growth will change to over a 5-year period.

How many fewer people will be in the country if the education program is implemented and is successful?

505

11.1 Area Between Two Curves Warm Up Find the area between the x-axis and the curve y = x 2 from to .

Find the area under the curve y  from

to

.

What would you do to find the area between the line, y = – 1 and the functions above?

506

1 x

Example 1: Area Between Two Functions Over a Closed Interval.

Write an expression that represents the area over the interval [a, b]: a) Between and the x-axis b) Between

and the x-axis

c) Between

and

How to Do It

a

b

What to Think About What integral can you use to calculate the area under the curve?

How can you combine the first two areas to determine the area between these two curves?

How can you combine the difference of the areas into one integral expression?

507

Area of the Region Between Two Curves To determine the area between two curves, partition (slice) the interval into n equal sub-intervals of . Use a rectangular area for each sub-interval with as the width and as the height of the ith sub-interval. By taking the limit as , we can find the exact area using the definite integral. Note, the region will be bounded on an interval between two xvalues where .

Your Turn Find the area bounded by each region.

1.

and the x-axis

2.

between

508

Example 2: Finding the Area Between Two Curves Find the area of the region between and How to Do It

.

What to Think About What do the graphs look like?

What interval contains the region?

How do you determine the intersection points?

Sketch a rectangular slice with width . What is the area of this rectangular partition?

Does it matter that one curve is below the x-axis?

What are the limits of integration for this region?

509

Your Turn 1. Find the area between

and

2. Determine the area of the region enclosed by

510

.

Example 3:

Finding the Area Between Two Curves that Intersect at More than Two Points Find the area of the region between and .

How to Do It

What to Think About At which x-values do the graphs intersect?

Sketch a slice of the region on the interval [3, 1]. What is the area of this rectangular partition with width ?

Sketch a slice of the region on the interval [1,1]. What is the area of this rectangular partition?

What integrals can be used to find the area of each region?

511

Your Turn 1. Find the area between

512

and

.

Practice Sketch the region bounded by the graphs of the given equations, show a typical vertical slice and find the area of the region.

1. and

from 2 to 4

2.

, from -1 to 1

513

514

3.

and

4.

and

5.

and

6.

and

7.

and

8.

and

from

from

515

11.2 Volumes of Solids Warm Up You are given a roll of sushi and want to determine the volume of the entire roll

. a. Cut the sushi up into slices of width

. Sketch the shape of each slice.

b. Write an expression for the volume of one slice with width .

and radius

c. Imagine the top of the sushi roll can be represented by the line

. The

3-D sushi roll is a cylinder that is created by rotating around the x-axis so that the axis goes through the center of the roll. Sketch the roll over the interval [a, b] and label one slice with , .

How can you determine the volume using infinitely thin compact discs (cylinders)? d. Create an equation to find the volume of the entire roll.

516

Definition: Let be a continuous function on . The volume V of the solid of revolution generated by revolving the region bounded by the graphs of and the x-axis around the x-axis is

Note: is the radius R of the solid of revolution from the x-axis to the outermost function.

Example 1: Calculating the Volume of a Solid of Revolution About the x-axis.

If , find the volume of the solid generated by revolving the region under the graph of f from x = -1 to x = 1 about the x-axis.

How to Do It

What to Think About What does this shape look like?

What is the shape a slice that is perpendicular to the x-axis?

What is the width of the slice?

What other dimension do I need to determine the volume of the slice?

What is the expression for the volume of a slice?

517

Your Turn: Find the volume of the solid generated by revolving the region under the graph of f about the x-axis. a)

b)

518

Example 2: Finding the Volume of a Solid with a Hole in the Middle – the Washer Method.

Find the volume of the solid generated by the region bounded by the graphs of , the line How to Do It

,

,

rotated around the x-axis.

What to Think About What does this shape look like?

What shape best describes a slice?

What is the volume of the whole slice if there was no hole in the middle?

What is the volume of the hole in the middle of each slice?

What is the expression for the volume of each slice?

519

Your Turn Find the volume of the solid generated by the region bounded by the graphs of the line rotated about the x – axis.

Definition: The volume V of the solid of revolution generated by revolving the region bounded by the graphs of two functions and around the x-axis is

Note: There are two radii that must be considered in this situation. The larger radius, R, is determined as the difference between the outermost function and the axis of rotation. The smaller radius, r, is determined as the difference between the innermost function and the axis of rotation.

520

Practice Find the volume of each function, revolved around the x-axis.

[0,1] and the x – axis.

1.

[1,4] and the x – axis

2.

3.

The region in the first quadrant enclosed by the y-axis and the regions of and

is revolved about the x-axis to form a solid. Find the

volume.

521

For questions 4 and 5 calculate the volume of the solid and include sketches of the area to be revolved about the x-axis and the resulting solid.

4.

What is the volume of the solid that is generated when the region between and from 1 to 2 is rotated about the x-axis?

5.

Determine the volume of the solid that is generated when the region bounded by

522

and

is rotated about the x-axis.

11.3 Finding Volume Using Cross-Sectional Area Warm Up So far, we have been considering solids created by rotating a region around a line. Imagine taking a cross-sectional slice of one of these solids, what does one of the slices look like? Write an expression for its volume.

Imagine another type of solid that is formed with a known cross-section. How can we find the volume of one of these solids? For example, consider a solid with a flat circular base, that has cross-sections perpendicular to the x-axis, that are squares of side length equal to the distance across the circular base at that point.

Draw or build a solid that has these features.

What would this shape look like if the crosssectional area was a semi-circle, or an isosceles right triangle, or some irregular area?

See http://mathdemos.org/mathdemos/sectionmethod/sectiongallery.html https://www.geogebra.org/m/bwRfChS9

Definition:

The volume V of the solid with cross sectional area A(x) taken perpendicular to the x -axis,

523

Example 1: Finding the Volume of a Solid with Known Cross-section Perpendicular to the x-axis.

Find the volume of a solid with a circular base of radius 3, that has crosssectional areas perpendicular to the x-axis that are isosceles right triangles with one leg in the plane of the base.

How to Do It

What to Think About What is the equation of the base region?

What is the area expression for the cross-sectional area of this shape?

Why does the integrand have the constant

?

Why does the integrand have the constant 2?

Why does the integrand have an exponent of 2?

524

Your Turn Find the volume of a solid with a circular base of radius 3, that has cross-sectional areas perpendicular to the x-axis that are squares.

Definition: The volume V of the solid with cross sectional area A(x) taken perpendicular to the y -axis,

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Example 2: Finding the Volume of a Solid of Known Cross Section Between Two Curves.

Find the volume of a solid bounded above by and below by with cross-sectional areas perpendicular to the x-axis that are squares where the sidelength is equal to the distance between the two boundary curves over the interval .

How to Do It

What to Think About What is the area expression for the cross-sectional area of this shape?

Why is the function not divided by 2?

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Your Turn Find the volume of a solid bounded above by and below by with crosssectional areas perpendicular to the x-axis that are semi-circles where the side-length is equal to the distance between the two boundary curves over the interval .

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Practice 1.

The base of a solid is the circular region bounded by the graph of Find the volume of the solid: a) If every cross-section perpendicular to the x-axis is a square.

.

b) If every cross-section perpendicular to the x – axis is an isosceles triangle and the height equal to the base.

c) If every cross-section perpendicular to the x-axis is a semicircle.

d) If every cross-section perpendicular to the x-axis is an equilateral triangle.

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2.

A solid has as its base the region bounded by and . Find the volume of the solid if every cross-section is perpendicular to the x-axis. a) Is an isosceles right triangle with hypotenuse in the xy – plane

b) Is a square

3.

The base of the solid is bounded by the graphs of and . Find the volume of the solid if every cross-section perpendicular to the y-axis is a semicircle with diameter in the xy – plane.

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4.

The base of the solid is bounded by the graphs of

and

and the y-axis. Find the volume of the solid if every cross-section perpendicular to the x-axis is an isosceles right triangle with one leg in the xyplane.

5.

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The base of the solid is bounded by the graphs of and .

Find the volume of the solid if every cross-section perpendicular to the x-axis is a square with side extending from

11.4 Chapter Review 1.

Let R be the region in the first quadrant enclosed by the graphs of

and

R a. Find the area of R

b. The region R is rotated about the x-axis. Find the volume of the solid.

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2.

Let h and k be the functions given by

and

. Let

R be the region in the first quadrant enclosed by the y-axis and the graphs of h and k, and S be the region in the first quadrant enclosed bythe graphs h and k.

S

R a. Find the area of R

b. Find the area of S

c. Find the volume of the solid when S is revolved about the x-axis.

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3.

Let R be the shaded region in the first quadrant enclosed by the graphs of and

and the y-axis.

R

a. Find the area of the region R

b. The volume of the solid generated when the region R is revolved about the x -axis.

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4.

Let R be the region bounded by the x-axis, the graphs of

and the

line

a. Find the area of region R

b. Find the volume of the solid generated when R is evolved about the xaxis.

R

c. The region R is the base of a solid. For the solid, each cross-section perpendicular to the x-axis is an isosceles triangle with one of its legs bound by the two graphs. Find the volume of the solid.

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Unit 5 Project: Solid of Revolution 1. Find a real object that is a 3-D object generated by revolving a 2-D region about a straight axis (i.e., solid of revolution). a. Your object must not be a shape with a simple volume formula (i.e., no spheres, cylinders, cones, etc.)

Doorknob

Polynomial rotated about the x-axis

b. You may choose an object whose volume can be calculated by the disk method or by the washer method.

2. Identify the equation(s) that represent the boundaries of the 2-D region to be revolved about the x-axis or y-axis. a. Use graphs and transformations to get an equation that is a reasonably good fit for the curved boundary(ies). b. Your equation(s) can be determined using technology (e.g., Desmos, GeoGebra, etc.) to create curves of best fit. c. If necessary, write a piecewise equation for the curved boundary(ies).

3. Determine the integral that represents the exact volume of your object. Identify which axis the 2-D region is being revolved about.

4. Use a variety of calculus methods to approximate the object’s volume.

5. Your report must include the following: a. A neatly drawn sketch of the two-dimensional region being revolved. b. An integral expression representing the volume of your object. c. Calculations showing how you approximated the volume of the object.

Include correct units!

d. Separate calculations for each of the following volume

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approximations: i. Left Riemann Sum approximation with at least 4 subintervals. ii. Right Riemann Sum approximation with at least 4 subintervals. iii. Midpoint Riemann Sum approximation with at least 4 subintervals. iv. Trapezoidal Sum approximation with at least 4 subintervals. e. A volume approximation of your object calculated the “integrate” function of your graphing calculator. f. Representation of your solid (e.g., drawing, photograph, etc.)

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