Preview A Level Mathematics for OCR A Student Book 2

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A Level Mathematics for OCR A Student Book 2 (Year 2) Vesna Kadelburg and Ben Woolley

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Contents

Contents 1

Section 1: A reminder of methods of proof.................. 1 Section 2: Proof by contradiction .................................. 3 Section 3: Criticising proofs ........................................... 5 2 Functions

10

Section 1: Mappings and functions ............................. 11 Section 2: Domain and range ...................................... 16 Section 3: Composite functions .................................. 20 Section 4: Inverse functions ......................................... 24 3 Further transformations of graphs

38

8 Further trigonometry

59

9 Calculus and exponential and trigonometric functions

178

Section 1: Differentiation............................................ 178 Section 2: Integration ................................................. 183

sa

4 Sequences and series

ft

Section 1: General sequences ..................................... 60 Section 2: General series and sigma notation ........... 64 Section 3: Arithmetic sequences ................................. 66 Section 4: Arithmetic series ......................................... 68 Section 5: Geometric sequences ................................ 71 Section 6: Geometric series ......................................... 75 Section 7: Infinite geometric series ............................. 77 Section 8: Using sequences and series to solve problems ............................................ 80

ra

5 Rational functions and partial fractions

87

D

Section 1: Review of the factor theorem ..................... 87 Section 2: Simplifying rational expressions................ 90 Section 3: Partial fractions with distinct factors........................................................... 93 Section 4: Partial fractions with a repeated factor ............................................................ 96 6 General binomial expansion

101

Section 1: General binomial expansion .................... 101 Section 2: Binomial expansions of compound expressions ................................................ 105 Focus on … Proof 1 ....................................................111 Focus on … Problem-solving 1 .................................113 Focus on … Modelling 1 ............................................115 Cross-topic review exercise 1 ....................................118

10 Further differentiation

190

Section 1: The chain rule ............................................ 191 Section 2: The product rule ....................................... 196 Section 3: The quotient rule....................................... 199 Section 4: Implicit differentiation .............................. 202 Section 5: Differentiating inverse functions ............. 207 11 Further integration techniques

212

Section 1: Reversing standard derivatives ................ 212 Section 2: Integration by substitution....................... 215 Section 3: Integration by parts .................................. 221 Section 4: Using trigonometric identities in integration ............................................. 225 Section 5: Integrating rational functions .................. 230 12 Further applications of calculus

237

Section 1: Properties of curves .................................. 238 Section 2: Parametric equations ................................ 244 Section 3: Related rates of change............................ 256 Section 4: More complicated areas........................... 259 13 Differential equations

274

Section 1: Introduction to differential equations .......275 Section 2: Separable differential equations ............. 277 Section 3: Modelling with differential equations ....... 281

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157

Section 1: Compound angle identities ..................... 157 Section 2: Double angle identities ............................ 161 Section 3: Functions of the form a sin x + b cos x ......................................... 166 Section 4: Reciprocal trigonometric functions......... 170

m

Section 1: Combined transformations ........................ 39 Section 2: Modulus function ........................................ 47 Section 3: Modulus equations and inequalities ......... 52

121

Section 1: Introducing radian measure ..................... 121 Section 2: Inverse trigonometric functions and solving trigonometric equations ............. 127 Section 3: Modelling with trigonometric functions .................................................... 132 Section 4: Arcs and sectors ........................................ 140 Section 5: Triangles and circles.................................. 144 Section 6: Small angle approximations .................... 148

e

1 Proof and mathematical communication

7 Radian measure

pl

Introduction ..................................................................... v How to use this book ..................................................... vi

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A Level Mathematics for OCR A Student Book 2

15 Numerical integration

320

Section 1: Integration as the limit of a sum .............. 321 Section 2: The trapezium rule .................................... 326 Focus on … Proofs 2 ................................................. 335 Focus on … Problem-solving 2 ................................ 337 Focus on … Modelling 2 ...........................................340 Cross-topic review exercise 2 ................................... 342 346

413

Section 1: Describing motion in two dimensions .... 414 Section 2: Constant acceleration equations ............ 419 Section 3: Calculus with vectors ................................ 422 Section 4: Vectors in three dimensions..................... 427 Section 5: Solving geometrical problems ................ 431 20 Projectiles

438

Section 1: Modelling projectile motion .................... 438 Section 2: The trajectory of a projectile ...................446 21 Forces in context

455

Section 1: Resolving forces ........................................ 455 Section 2: Coefficient of friction ................................ 461 Section 3: Motion on a slope ..................................... 469 22 Moments

486

Section 1: The turning effect of a force ....................486 Section 2: Equilibrium ................................................. 494 Section 3: Non-uniform rods ..................................... 499 Section 4: Further equilibrium problems.................. 501

m

16 Conditional probability

19 Applications of vectors

e

289

Section 1: Locating roots of a function .....................290 Section 2: The Newton–Raphson method ............... 295 Section 3: Limitations of the Newton–Raphson method ....................................................... 298 Section 4: Fixed-point iteration .................................304 Section 5: Limitations of fixed point iteration; alternative rearrangements......................309

pl

14 Numerical solution of equations

Section 1: Set notation and Venn diagrams ............. 347 Section 2: Two-way tables ..........................................354 Section 3: Tree diagrams ............................................ 357

Focus on … Proofs 4 ................................................. 510 Focus on … Problem-solving 4 .................................511

Section 1: Introduction to normal probabilities ....... 367 Section 2: Inverse normal distribution ...................... 374 Section 3: Finding unknown μ or σ............................ 376 Section 4: Modelling with the normal distribution .......379

Focus on … Modelling 4 ........................................... 512

sa 366

17 The normal distribution

387

ft

18 Further hypothesis testing

ra

Section 1: Distribution of the sample mean .............388 Section 2: Hypothesis tests for a mean ....................390 Section 3: Hypothesis tests for correlation coefficients................................................. 397 Focus on … Proofs 3 .................................................404 Focus on … Problem-solving 3 ................................405

Cross-topic review exercise 4 ................................... 513

Paper 1 practice questions ........................................ 517 Paper 2 practice questions ........................................ 519 Paper 3 practice questions ........................................ 523 Formulae ...................................................................... 527 Statistical tables .......................................................... 529 Answers ........................................................................530 Glossary........................................................................ 591 Index ............................................................................. 594 Acknowledgements .................................................... 595

D

Focus on … Modelling 3 ........................................... 407 Cross-topic review exercise 3 ...................................409

iv

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9 Calculus of exponential and trigonometric functions

• learn how to differentiate ex, ln x, sin x, cos x and tan x • learn how to integrate ex, 1 , sin x, and cos x x

m

pl

• review applications of differentiation to find tangents, normal and stationary points • review applications of integration to find equation of a curve and areas.

e

In this chapter you will:

Before you start…

You should be able to use rules of indices and logarithms.

1 Write ln(3x4) in the form A + B ln x.

Student Book 1, chapter 13

You should be able to differentiate xn.

2 Differentiate y

Student Book 1, chapter 14

You should be able to integrate xn for n ≠ −1.

3 Find the exact value of

Chapters 6 and 7

You should be able to use compound angle formulae and small angle approximations.

4 Use small angle approximations to find the approximate value of cos π + π .

( x − x ))(( x + x ) . ∫

2

1

x 3 + 3 dx .

2x 2

(3

100

)

ft

sa

Student Book 1, chapters 2 and 7

ra

Extending differentiation and integration

D

You have already seen many situations that can be modelled using trigonometric, exponential and logarithm functions. For example, in chapter 7 of Student Book 1 you studied exponential models for population growth, and in chapters 7 and 8 of this book you saw an example of modelling water waves using sine and cosine functions.

We are often interested in the rate of change of quantities in such models; for example, at what rate is the population increasing after three years? Finding rates of change involves differentiation. Integration reverses this process – given the rate of change you can find the equation for the original quantity.

So far you have learnt how to differentiate and integrate functions of the form axn. In this section we will extend rules of differentiation and integration to include a wider variety of functions.

Section 1: Differentiation You already know from Student Book 1, chapter 7 that the rate of growth of an exponential function is proportional to the value of the function.

178

Rewind You learnt about inverse functions and their graphs in section 4 of chapter 4.

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9 Calculus of exponential and trigonometric functions In particular, for y = ex the rate of growth, which is the same as the derivative, equals the y value.

The derivative of the natural logarithm function is perhaps surprising, but you will see below how it follows from the fact that y = ln x is the inverse function of y = ex.

Key point 9.1 dy = ex dx

•

If y = ln x then

dy 1 = dx x

e

If y = ex then

pl

•

PROOF 9

Let point B be the point on the graph of y = In x which is the reflection of A in the line y = x.

sa

The reflection swaps x and y coordinates, so B has x-coordinate a. y y = ex

y=x

y = ln x

A

a

ft

B

x

a

ra

O

We know how to differentiate y = ex, and how the graphs of y = ex and y = ln x are related. So we will use the gradient of the first graph to find the gradient of the second graph.

m

Let point A on the graph of y = ex have y-coordinate a.

The gradient of y = ex at A is a.

We know that the gradient of y = ex at A equals the y value.

The gradient of the reflected line:

To see what happens to the gradient on reflection in y = x, consider reflecting a triangle.

D

gradient a1 = a 1

a

A

B

1

a gradient 1 a

This shows that, if a line with gradient m is reflected in y = x, the gradient of the reflected line is 1 . m

The gradient of the graph y = In x at B is a1 .

This is 1 divided by the x-coordinate of B, so the 1 gradient of y = In x is x . © Cambridge University Press 2017 The third party copyright material that appears in this sample may still be pending clearance and may be subject to change.

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A Level Mathematics for OCR A Student Book 2 The rules for differentiating sums and constant multiples of expressions still apply. Sometimes an expression needs to be simplified before it can be differentiated.

WORKED EXAMPLE 9.1 Differentiate y = ex+3−3 ln (4x2). y = e3ex − 3 (In 4 + 2 In x)

dy 3 x 6 =e e − dx x

e x is multiplied by a constant (e3).

3 ln 4 is a constant, so its derivative is 0.

The derivative of 6 ln x is 6 × 1 .

pl

e

= e3ex − 3 In 4 − 6 In x

We can only differentiate ex and ln x, so we need to simplify the expression using rules of indices and logarithms.

m

x

sa

To differentiate sinx and cos x you need to remember differentiation from first principles (see Book 1, section 13.2). You also need to use compound angle formulae from chapter 8, section 1 and small angle approximations from chapter 7, section 6.

WORKED EXAMPLE 9.2

Use differentiation from first principles to prove that the derivative of sin x is cos x, where x is measured in radians.

ft

Let f(x) = sin x ⎧⎪ sin( +

f ′( x ) = hlim ⎨ →0

⎬ ⎭

ra

⎩

h

) − sin x ⎫⎪

180

−

}

(

Use small angle approximations:

)

D = cos x

+

Use sin (A + B) = sin A cos B + sin B cos A

⎧ + ( )cos x − sin x ⎫⎪ ⎪ sin x − ≈ hlim ⎨ ⎬ →0 h ⎪ ⎪ ⎩ ⎭ ⎧ − 1h2 sin x + h co x ⎫ ⎪ 2 ⎪ = hlim ⎨ ⎬ →0 h ⎪ ⎪ ⎩ ⎭

{

{

⎧ sin x cos h + sinh cos x − sin x ⎫ ⎬ h ⎩ ⎭

= hlim ⎨ →0

= hlim − h →0

f ′( ) = lim h→0

x+

x

cosh

1 h2 2

sinh ≈ h

}

Let h tend to 0.

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9 Calculus of exponential and trigonometric functions Differentiating cos x follows the same method. The result for tan x can also be proved in this way, as well as using the quotient rule which you will meet in chapter 10, section 3.

Tip

Key point 9.2 dy = cos x dx

•

If y = sin x then

•

If y = cos x then dy = − sin x

These formulae only apply if x is in radians.

pl

•

e

dx dy If y = tan x then = sec2 x dx

m

The following two examples should help you remember two applications of differentiation: finding equations of tangents and normals and finding stationary points.

WORKED EXAMPLE 9.3

f′(x) = −sin x + ex

We need the gradient, which is f ′ (0 ) To find the equation of a straight line we also need coordinates of one point.

ra

ft

∴ f′(0) = −sin 0 + e0 = 1 When x = 0, y = f (0) = cos 0 + e0 = 1 + 1 = 2

sa

Find the equations of the tangent and the normal to the graph of the function f (x) = cos x + ex at the point x = 0. Give your answer in the form ax + by + c = 0, where a, b, c are integers.

The tangent passes through the point on the graph where x = 0. Its y-coordinate is f(0).

Tangent:

y y1 = m ( x − x1 )

)

D

y 2 1( x − y = x +2 y x −2 = 0

Put all the information into the equation of a line.

Normal:

y 2 y

1( x − x +2

)

The gradient of the normal is − 1 and it passes through m the same point.

y + x −2 = 0

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A Level Mathematics for OCR A Student Book 2

WORKED EXAMPLE 9.4 Find the coordinates of the stationary point on the graph of y = l x + 12 and determine its nature. x

dy 1 = − 2 x −3 dx x

We need to find where the gradient equals 0.

Write 12 as x −2. x

1 2 − =0 x x3

x > 0, o x = 2

ln x is only defined for x > 0.

y = ln 2 + 1

(

The stationary point is 2 , 1 l 2 + 1 2

2

)

d2 y = − 12 + 6 x −4 dx 2 x

( ) ( )

Use rules of logs to rewrite the y-coordinate.

To determine the nature of the stationary point we need to evaluate the second derivative at x = 2 .

m

=− 1 2+ 6

pl

2

4

= − 1 + 6 = 1> 0 4

sa

2

e

x 2 −2 2 0

The stationary point is a minimum.

1

ft

EXERCISE 9A

Differentiate the following: = 3e x

a i

ii

y

c i

y ln x 3 x + 4e x 5

ii

d i

y 3 sin x

ii

y 2c

e i

y 2 x −5 cos x

ii

y t n x+5

f

y = sin x + 2 c x

ii

y 1 ta x −1 sin x

D 2

i

5

ii

x

2

3

Differentiate after simplifying fi rst: a i

y ln x 3

ii

y 2 ln x 5

b i

y = 3 l 2x

ii

y l 5x

c i

y = e x +3

ii

y = e x −3

d i

y = e2ln x

ii

y = e3l

e i

y = 3sin x

ii

y = 2 cos x + 4 sin x

f 182

x

5 y 1 ln x 3 x y 4 e + 3 ln x 2

b i

ra

2l x

y = 2e

i

cos x 2x e − ex y= ex

x

ii

x 2 x+

cos x x 4 e − e2 x y= 2e x

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9 Calculus of exponential and trigonometric functions

Find the exact value of the gradient of the graph of f ( ) 1 e x 7l x at the point x = ln 4 2 Find the exact value of the gradient of the graph f ( ) e x − ln x when x = ln 3 2

3 4 5

Find the value of x where the gradient of f ( )

6

Find the value of x where the gradient of g ( x ) x 2 −12ln x is 2.

7

Find the rate of change of f ( x )

8

Find the rate of change of g ( x )

9

Find the equation of the tangent to the curve y

e x is –6.

i x + x 2 at the point x = π .

2 1 tan x − 3 cos x x 3 at the point x = π .

4 6

+ x which is parallel to y 3 x .

e

x

10 Find the equation of the tangent to the curve y 5sin x which is parallel to 2 x y 6 . in x + cos x , 0 - x < 2 π, fi nd the values of x for which h ′ ( x ) = 0.

Given that h ( x )

12

Find the equations on the tangent and the normal to the graph of y 3 the coefficients in an exact form.

13

Given that y = tan x +

14

Find and classify the stationary points on the curve y

15

Show that the function f ( x ) = l x + 1k has a stationary point with y-coordinate lnkk +1 .

16

Find the range of the function f

17

Find and classify the stationary points of:

1

4

x

2

fo 0

dy 2 π , solve the equation dx = 1 − 23 .

x

x

x

b y 2

x

5x

i x + 4 cos x in the interval 0 x 2 π.

ex 4 x + 2 .

sa

a y ln x − x

x −2 2 sin x at x = π4 . Give all

m

1

pl

11

18 The volume of water in millions of litres (V) in a tidal lake is modelled by V time in days after being switched on.

t + 100 where t is the

ft

a What is the smallest volume of the lake?

ra

b A hydroelectric plant produces an amount of electricity proportional to the rate of flow of water through a tidal dam. Assuming all flow is through the dam, find the time, in the first 6 days, when the plant is producing maximum electricity.

Section 2: Integration

D

We can reverse all the differentiation results from the previous section to integrate several new functions.

Key point 9.3

•

∫e

•

∫ 1x dx = l

•

∫ sin

d = − cos x + c

•

∫ cos

d = si x + c

x

d dx = e x + c x +c

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A Level Mathematics for OCR A Student Book 2

Note the modulus sign in ∫ 1x dx = l x + c. This is because the x in ln x cannot be negative, but the x in 1 can. Taking the modulus means that x we can find the area between the x-axis and the negative part of the curve y = 1 (it can be seen that this is the same as the corresponding x region on the positive x-axis). y

–b

–a a

x

b

m

pl

O

e

y= 1 x

WORKED EXAMPLE 9.5 3 Find ∫ 5 x 32 x dx .

2x

5 − 3x 3 5 3 dx = ∫ − x dx 2x 2 2x 2 5 3 = ln x − x 2 + c 2 4

We don’t know how to integrate a quotient, but we can split the fraction and integrate each term separately.

ra

ft

∫

sa

You can combine these facts with rules of integration you already know.

Sometimes an expression needs to be rewritten in a different form before integrating.

WORK IT OUT 9.1

D

Which of the following statements is correct? Identify the mistake in the other three.

Statement 1

Statement 2

y= 1

∫x

x

⇒

dy = ln x dx

1 dx = l 2

( )+ c

Statement 3

Statement 4

y = ln (3 x )

y = ln (3 x )

⇒

dy = 1 dx 3 x

⇒

dy 1 = dx x

Remember that integration is the reverse of differentiation, so if you know the derivative of a function you can use integration to find the function itself.

184

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9 Calculus of exponential and trigonometric functions

WORKED EXAMPLE 9.6 A curve passes through the point ( −1, 3) and its gradient is given by dx = 3e x − 1. Find the equation of the curve. dy

y = ∫3e x − 1 dx

Integrate dy to get y. dx

= 3e x − x + c

e

Using x

Don’t forget the constant of integration.

1 y = 3:

Use given values of x and y to find the constant of integration.

m

pl

3 = 3e −1 − ( − ) + c 3 ⇒ c = 2− e 3 x ∴y 3 x +2− e

WORKED EXAMPLE 9.7

sa

You can also use integration to find the area between the curve and the x-axis. This involves evaluating definite integrals. It is always a good idea to draw a diagram to make sure you are finding the correct area.

Rewind You know from Student book 1 chapter 14 that when a curve is below the x-axis the integral is negative.

ft

Find the exact area enclosed between the x-axis, the curve y si x and the lines x = 0 and x = π3 . y

ra

y = sin x

O

x

D

π 3

π 3

π

A = ∫ sinxx dx = [ −

x ]03

π⎞ ⎛ = − cos − ( − ⎝ 3⎠

)

Sketch the graph and identify the required area.

Integrate and write in square brackets.

0

1 ⎛ 1⎞ = − − (− ) = ⎝ 2⎠ 2

Evaluate the integrated expression at the upper and lower limits and subtract the lower from the upper.

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A Level Mathematics for OCR A Student Book 2

WORK IT OUT 9.2 Three students are considering the following question: 6

∫1

Evaluate x dx . −2

Which of the following solutions is correct? Identify the mistake in the other three.

What is the value of the area shaded in the second solution?

1

6

x ⎤⎦ −2

−2

Solution 3

This is the area between the graph and the x-axis:

We are trying to integrate between −2 and 6, but this includes x = 0, which is not in the domain of ln x.

So there is no answer.

y

= ln 6 − l −2 = ln 3 –2

6

m

x

O

e

6

∫ x dx = ⎡⎣l

Solution 2

pl

Solution 1

0

1 ∫−2 x But 0

1

sa

We need to calculate the two parts separately and add them up: 6

1

∫ x dx 0

∫ x dx = ln 0 − l

−2

ft

−2

ra

And ln 0 is not defined, so there is no answer.

EXERCISE 9B 1

Find the following integrals.

∫ 5e

x

dx

D

a i b i c i d i e i f

2

i

∫ 5 dx ( + ) ∫ 2 dx ∫ 3cos x dx

ii

∫ sin x −22cos x dx ∫ x + si x dx

ii

2e x

ii ii

ii

∫ 9e dx 7e ∫ 11 dx x

x

+

3

∫ 5 dx ∫ 4sin x dx (

)

∫ 2cos x3− sisin x dx ∫ cos x − 1x dx

Find the following integrals a i

186

ii

∫ 2x dx

ii

∫ x3 dx

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9 Calculus of exponential and trigonometric functions

∫ 2x1 dx ∫ 2x5 dx ∫ x x− 1 dx ∫ 3xx+ 2 dx 2 x + 3x ∫ x x dx

b i c i

e i

3

ii

2

d i

f

ii

ii ii

2

i

ii

∫ 3x1 dx ∫ 3x2 dx ∫ x x+ 5 dx ∫ 3x −x 5x dx x −4 x ∫ x x dx 3

2

3

2

Find the exact value of these defi nite integrals: 3

dx

ii

∫

ii

ln 2

∫ (3

)

ii

2 dx

ii

3

)dx

∫ 34x dx 2

1

e3

∫ 2x dx e

∫ 23x dx

π 2

2π

ii

sa

1

∫ cos x dx

ii

∫ 3si

∫ sin x dx

π 2π

0 π

g i

∫ (4

m

∫

i

dx

5

3 dx 2x

e2

f

x

1

3

e i

∫ 2e

0 ln 3

1

d i

dx

ln 5

e x dx

0

c i

x

1

0 ln 3

b i

∫ 2e

pl

∫ 3e

a i

x

e

2

x − 4 cos x dx

−π

ii

in x dx ∫ 2 cos x − ssin 0

Find the exact value of the area enclosed by the curve y = 2 , the

4

3x

ft

x-axis and the lines x = 2 and x = 6. Give your answer in the form a ln b.

5

ra

Find the area enclosed by the curve y = 3 sin x, the x-axis, and the line x = π . π 3

6

3

Find the exact value of

∫ (sin x + 2 cos x )dx.

0

e5

∫3

Find the exact value of x dx . −3 1 2 a Evaluate x dx.

7

D

∫

8

−9

b State the value of the area between the graph of y = 1 , the x-axis x and the lines x = –9 and x = –3. dy

9

Find the equation of the curve given that = cos x + sin x and the dx curve passes through the point ( π,1).

10

Find sin x + cos x dx .

11

The derivative of the function f(x) is 1 .

∫

Elevate See Support sheet 9 for a further example of finding the equation of a curve and for more practice questions.

2 cos x

2x

a Find an expression for all possible functions f(x). b If the curve y = f(x)passes through the point (2,7), find the equation of the curve.

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A Level Mathematics for OCR A Student Book 2 The diagram shows part of the curve y e x 5 si x .

12

y

1

Find the shaded area, enclosed between the curve, the x-axis and the lines x = 0 and x = 1.

The diagram shows a part of the curve with equation y = 4 + x − 5.

13

x

y

e

x

O

Fast forward

O

6

m

pl

Areas of regions that are partly above and partly below the x-axis are discussed in more detail in chapter 12, section 4.

x

sa

a Show that the curve crosses the x-axis at x = 1 and x = 4.

b Find the exact value of the shaded area. Give your answer in the form p − 4ln q, where p and q are rational numbers.

2k

Show that the value of the integral ∫ 1x dx is independent of k.

14

k

)

ra

(

ft

15 The gradient of the normal to a curve at any point is equal to the x-coordinate at that point. If the curve passes through the point e2 , 3 , fi nd the equation of the curve in the form y ln g ( x ) where g(x) is a rational function.

Checklist of learning and understanding

D

• You can now differentiate basic trigonometric, exponential and logarithm functions: d d d d x d 1 2 x , (cos x ) sin x , (tan x ) x, (sin x ) e e x , (ln x ) = dx dx dx dx dx x • The results for sin x and cos x can be derived using differentiation from first principles.

• You can also integrate some new functions: 1 x d = e x + c , ∫ dx = l x + c ∫ ∫ cos x dx = sin x + c , ∫e dx x • You can combine these with the rules of differentiation and integration you already know to find: • Rates of change

( )

• Equations of tangents and normal • Stationary points • Equation of a curve with a given gradient • Area between a curve and the x-axis

188

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9 Calculus of exponential and trigonometric functions

Mixed practice 9 1 2 3

x Find the equation of the tangent to the curve y = e + 2 sin x at π the point where x = 2 .

()

If f ′ ( x ) = si x and f π3 = 0 fi nd f(x). a Find the exact value of the gradient of the graph of f ( x ) 1 e x 7l x at the point x = ln 4.

2

b Find the exact value of the gradient of the graph f ( x ) = e x − ln x when x = ln3.

2

e

5

π

x Find the exact value of the integral ∫e + i x + 1 dx .

0

The diagram shows the curve with equation y = x − 7 + 10 , x which crosses the x-axis at 2 and 5.

Find the exact value of the shaded area.

6

Find the indefi nite integral ∫ 1 + x

7

Find and classify the stationary points on the curve y = sin x + 4 c x in the interval 0 < < 2 π.

2

x dx .

m

x

y

pl

4

Find and classify the stationary points on the curve y = tan x − 4 x for −π < < π . Give only the x-coordinates, and 3 leave your answers in terms of π .

9

Find the equation of the normal to the curve y = 3ex at the point x = ln 3. Give your answer in the form x + ky = p + ln q where k, p and q are integers.

10

The population of bacteria (P) in thousands at a time t in hours is modelled by P et t , t . 0 .

sa

8

x

O

ii

At what time does the number of bacteria reach 14 million?

Find ddPt .

ra

b i

Find the initial population of bacteria.

ft

a i

ii

Find the time at which the bacteria are growing at a rate of 6 million per hour.

y

2

Find d P2 and explain the physical significance of this dt quantity.

D

c i

iv Find the minimum number of bacteria, justifying that it is a minimum.

11 The diagram shows the graphs of y si x and y Find the exact value of the shaded area.

12 Find

2x dx . ∫ coscos x − ssin in x

x

O

3c

x.

Elevate See Extension sheet 9 for a selection of more challenging problems.

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Chapter 09.indd 189

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