Student Teach Test Review 2.1 - 2.5 By: Gianna Monaco and Nina Sponhiemer
2.1: Tangent Line and Differentiability Things to remember: Distance Time
Average Speed: Velocity over a given distance. Instantaneous Speed: Velocity at a specific point Secant Line: The slope of the line or average speed over the interval. Definition of a derivative: f(x+h) - f(x) h
2.1: Tangent Line Problems Definition of Differentiability: 1.) 2.) 3.)
Continuous The derivative exists The slope is equal going in from the left and out from the right.
NOT DIFFERENTIABLE AT CUSPS AND CORNERS
f(x+h) - f(x) h
Use the definition of the derivatives to find the derivatives for each function. f(x)=5x+7
2.1 Examples: Use the definition of the derivative to find the derivative of each function. s(t)=4t2+8t+16
2.2: Derivatives on the Calculator Math Print: From homescreen press “math” “8”.
f’(c)=d/dx (f(c))|x=c When graphing use:
Y1= d/dx (f(x))|x=x
2.3: Basic Differentiation Rules Part I Review: dy/dx is a _ _ _ _. Which means â€œthe derivative of __ with respect to __. d/dx is a _ _ _ _. Which means to â€œ _ _ _ _ the _ _ _ _ _ _ _ _ _ _. Three rules to go over in 2.3 are The Constant Rule, The Power Rule, and Sum and Difference Rule.
2.3: The Constant Rule The derivative of a constant is always _ _ _ _. Why is the constant always zero? Draw a graph at f(x)= -3 and draw a tangent line -3.
2.3 Examples: The Constant Rule Find the derivatives of each. y=7
g(x) = -2
f(x) = 415
y = q82 q is a constant.
2.3: The Power Rule When using the power rule when finding the derivative, you essentially bring the exponent down and multiply but the constant in front of the variable. Then you subtract one from the exponent.
f(x)= axn then d/dx axn=anxn-1
2.3 Examples: The Power Rule Find the derivatives of each.
2.3: Sum and Difference with Constants The derivative of the sum of _ _ _ functions is the sum of the derivatives _ _ _ _ _ _ _ _ _ _ _ _.
d/dx [f(x) + g(x)] = fâ€™(x) + gâ€™(x)
2.3 Examples: Sum and Difference with Constants f(x)= -9x3+3x2-4x+6
2.3 Part II: Differentiation Rules Part II Find the equation of the tangent line to the curve y = 5x2 at the point (3,45).
Step 1: take the derivative of the equation
Step 2: plug in the x value from the point into the derived equation (this is the slope)
Final Step: plug the point (x and y values) and slope into taylor form
2.3 Part II Examples Find the equation of the tangent line to y = x3 + x2 at (3, 36)
Find the equation of the tangent line to y = (2x + 5) / (x2 - 3) at x = 1
2.3 Part III The derivative of a function gives us two things: 1. The slope of the tangent line at that point. 2. The instantaneous rate of change at that point. Keep in mind: -position: x(t) -velocity: v(t) which is x ‘(t) -acceleration: a(t) which is x ‘’(t) or v ‘ (t)
2.3 Part III (con.) An iPhone is dropped from a height of 6 feet. Itâ€™s height G(n) feet at any time n milliseconds during itâ€™s agonizing fall is represented by G(n) = -16n2+ 6 -How many milliseconds was the book falling for? (graph in y= , then 2nd, then trace, then 2: zero)
-Derivative of function (tells of the velocity)
-Average velocity (displacement / # of terms)
2.3 Part III Examples Find the second derivative of the function x(t) = t3 - 11t2 . Explain what the second derivative shows.
Find the third derivative of the function x(t) = t3 - 11t2 . Explain what the third derivative shows.
2.4: Product and Quotient Rules Product Rule:
f’(x)g(x) + f(x)g’(x)
f’(x)g(x) - f(x)g’(x) (g(x)2 Derivative of g(x)
Derivative of f(x)
Derivative of g(x)
Derivative of f(x) g(x) is just squared.
2.4: Product Rule Find f’(x) if f(x)= (9x2+4x)(x3-5x2)
f’(x)g(x) + f(x)g’(x)
Find f’(x) if f(x)= (√x+4∛x)(x5-11x8)
fâ€™(x)g(x) - f(x)gâ€™(x) (g(x)2
2.4 Quotient Rules f(x)=(5x4+3x7) (x10-8x)
f(x)= _tan x_ sin x +1
2.5: Rates of Change and Particle Motion 1 IF f(x) describes the position of an object THEN f ‘(x) = dy/dx = change in DISPLACEMENT/ change in TIME Keep in mind: -position: x(t) -velocity: v(t) which is x ‘(t) -acceleration: a(t) which is x ‘’(t) or v ‘ (t) So, when finding the velocity of the position of the particle, you must take the derivative of that function before plugging in the variable of time. Also, when finding the acceleration of the position of the particle, you must take the second derivative of function before plugging in the variable of time.
2.5 Examples: Rates of Change and Particle Motion A boat is traveling down a river on a windy day. Suppose G(N) is the amount of time, in seconds, it takes to makes the boat accelerate if the boat has an initial speed of N. In complete sentences, along with units, explain the following: G ‘(30) = 50
G ‘(50) = 30
G(17) = 45
(G -1 ) ‘(18) = 36
G-1 (6) = 12
G(4) = 16
2.5: Understanding the graph A particle is at rest when it's position doesn't change overtime and shows a straight line across the distance axis of the graph. A particle changes direction when there is no definite tangent line. A particle is moving to the left when it has a negative slope. Velocity is the slope of the line. Speed is the absolute value of velocity
2.5 Examples: Understanding The Graph Between what intervals is the particle at rest? Between what intervals is the particle moving to the left? When is the particle changing direction? What is the speed at the following intervals: 0-2 2-4 6-8 8-10