Practice Set 30 Instantaneous Rate of Change

Transcription

1 Practice Set 0 Instantaneous Rate of Change No Calculator Objectives Find the average rate of change of a function on an interval. Find the instantaneous rate of change of a function at a point. Find the equation of the line tangent to a function at a point. Notes f x h f x Average Rate of Change Slope of a secant line msec h f x h f x Instantaneous Rate of Change Slope of a tangent line mtan lim h0 h 1. (ACT/SAT) A polynomial function M is defined as 10 real number a, then what is the value of a? M x x x x. Ma for some If 0. (ACT/SAT) Multiple choice A polynomial function P is defined as following could be the graph of the function y Px in the xy-plane? A. B. C. D. P x. x x. Which of the For problems -4, if f x x and g x x find:. (14) f gx 4. (14) gf x For problems 5-6, if the graph of f(x) contains the point (,), find the point guaranteed to be on the graph of: 5. (15) fx 6. (15) fx Derivative Investigations 161

2 For problems 7-14, find the average rate of change on the interval given.. 7. f x x x on 0, 1 8. gx x on, 9. hx 4x 5x on 1, f x x x on 1, 11. gx x 4 on, 5 1. hx x on, 1. f x x 1x on 0, 14. y x x 4 on 1, 1 For problems 15-, find the equation of the line tangent to the curve at the given value of x. Leave your answer in point-slope form. 15. f x x x at x g x x at x 16 Derivative Investigations

3 17. hx 4x 5x at x f x x x at x 19. gx x 4 at x 0 0. h x x at x 1. f x x 1x at x 0. y x x 4 at x 1 For problems -8, identify f(x). You need NOT evaluate the given limit.. x h x h x x lim h0 h 4. x a x a x x lim a0 a 5. e lim h0 xh x e h 6. sin x b sin x lim b0 b lim x h 1 x 1 h 0 h 8. cos x c cos x lim c0 c Derivative Investigations 16

4 Practice Set 1 Polynomial, Product, Quotient Rule No Calculator Objectives Find the derivative of: o a polynomial function o a function consisting of the product of two polynomials o a function consisting of the quotient of two polynomials Find the second derivative of a polynomial function. Notes Average Rate of Change: f b f a b a Instantaneous Rate of Change: f ' x slope of secant line f x h f x lim slope of tangent line x h f x x x x 4x f ' x 11x 9x 1x 16x Polynomial Rule of Differentiation: Product Rule of Differentiation h x f x g x h' x f ' x g x g' x f x o o f x F S f ' x F' S S' F Quotient Rule of Differentiation f x f ' x g x g' x f x o hx h' x gx gx o N N' D f x f ' x D' N D D (ACT/SAT) Multiple choice A third degree polynomial function y gx g0 5. Which of the following must be a factor of g? is defined so that g 0 and A. x 5 B. x + 5 C. x D. x +. (ACT/SAT) Multiple choice A polynomial function P is defined as Px x 1 x x. Which of the following could be the graph of the function y Px in the xy-plane? A. B. C. D. For problems -4, if f x 5x 1and gx x 1, find:. (14) f gx 4. (14) gf x 164 Derivative Investigations

5 For problems 5-6, if the graph of f(x) contains the point (,), find the point guaranteed to be on the graph of: 5. (15) f x (15) f x For problems 7-14, find f (x). You need NOT simplify your answer. 7. f x 4x x 7 8. f x x 5 x x x 5 9. f x x 1 x 7x fx x x fx x x 1 x f x x x x 1. f x x7 5x x 14. fx x 5x 6 x x Derivative Investigations 165

6 For problems 15-, find. You need NOT simplify your answer y x x 16. x x y x y x x x 6x y x 4x 19. y x x 1 x 1 0. y x x 1 x x 1 1. y 4 x 5x 4 x. y x 6 4x x 4 For problems -4, find f " x. 4. f x x x x 4x 4. f x 5x x 7 For problems 5-6, find. 5. y x 6. 4 y x x x Derivative Investigations

7 Practice Set Applications of the Derivative No Calculator Objectives Find the equation of the line tangent to f(x) at x = a. Find the equation of the line normal to f(x) at x = a. Apply the derivative to position, velocity, and acceleration. Notes Equation of tangent line to f(x) at x = a: y f a f ' ax a Slope of a normal line to f(x) at x = a: Position/Velocity/Acceleration Updated 1 f ' a slope/derivative POSITION VELOCITY ACCELERATION s(t) v(t) a(t) area 1. (ACT/SAT) Multiple choice The polynomial What is the remaining factor? has x 5 and x 10 x 5x 50x 1000 as factors. A. x + B. x C. x + 0 D. x 0. (ACT/SAT) Multiple choice A polynomial has zeros at 5,, and 9. Which of the following could be the 11 factorization of the polynomial? A. x 5x 11x 9 B. x 5x 11x 9 C. x 5x 9x 11 D. x 5x 9x 11 For problems -4, if f x x x 1and g x x, find:. (14) f gx 4. (14) gf x For problems 5-6, if the graph of f(x) contains the point ( 4,), find the point guaranteed to be on the graph of: 5. (15) f x 6. (15) f x Derivative Investigations 167

8 For problems 7-9, find the equation of the line tangent to f(x) at x = f x x x 1 8. f x 7 4x x x x 1 9. fx x x 1 For problems 10-1, find the equation of the line normal to f(x) at x = f x 4x 4x 11. f x x 1 x 4x 1. fx x 5 x Problems 1-15 use the following scenario: Let ht 16t 40t 5 represent the height of an object at a particular time t. 1. Find the velocity of the object at t =. 14. Find the acceleration of the object at t = Find the value of t for which the velocity is zero (the object is at rest). 168 Derivative Investigations

9 Problems use the following scenario: A particle travels along the x-axis so that its position at any time t 0 s t 1t 6t 1 is given by 16. For what values of t > 0 is the particle at rest? 17. Find the value of t for which the acceleration of the particle is 1. Problems 18-0 use the following scenario: A particle moves along the x-axis so that its position at any time t 0 s t t 7t 14t 8 is given by 18. Find the instantaneous velocity function in terms of t. 19. Find the acceleration of the particle at t = Find the values of t for which the particle s acceleration is zero. Problems 1- use the following scenario: A ball is thrown straight down from the top of a 0-foot building with s t 16t v t s an initial velocity of feet per second. Use the position function for free-falling objects: 1. Determine the position function for the ball Determine the velocity at t =.. Determine the acceleration at t =. Derivative Investigations 169

10 Problems 4-7 use the following scenario: A coin is dropped from the top of a building that is 16 feet tall. Use s t 16t v t s the position function for free-falling objects: 4. Determine the position function for the coin Find the instantaneous velocities when t = 1 and t =. 6. (calculator required) Find the time required for the coin to reach ground level. Round your answer to the nearest tenth. 7. (calculator required) Find the velocity of the coin at the time it impacts the ground. Round your answer to the nearest tenth. For problems 8-1, use st t t 1t t 1 8. Find the initial position of the object.. 9. Find the initial velocity of the object. 0. Compute v(1) v(0). 1. Find the average acceleration of the object over the interval [0, 1]. 170 Derivative Investigations

11 For problems -5, use st t 4t t.. Find the initial position of the object.. Find the initial velocity of the object. 4. Compute v() v(0). 5. Find the average acceleration of the object over the interval [0, ]. Derivative Investigations 171

12 Practice Set Assessment 8 Review No Calculator 1. (ACT/SAT) Multiple choice Given the polynomial function Px x 18x, which are its zeros? A. 9, 6,, B. 9, 0, C., D., 0,. (ACT/SAT) Multiple choice Given ht t 8 t 4 t t 1 have? 1 4, how many distinct zeros does h Section 14 Composition of Functions ( pts) Compute the value of the composition of two functions (linear, quadratic, and/or exponential). Create a new function as the composition of two other functions (linear, quadratic, polynomial, and/or exponential). Given the composition of two functions, identify the inner function and outer function. For problems -4, if f x 6x 7 and g x x, find:. (14) f gx 4. (14) gf x Section 15 Transformations of Functions ( pts) Understand how a function f(x) is transformed by multiplying or adding a value in different parts of the function. Describe the transformation from one function to another. Create the equation of a function transformed from another. Identify graphs of transformations [f(x) + k, f(x + k), kf(x), f(kx)] given the graph of f(x). For problems 5-6, if the graph of f(x) contains the point ( 4,6), find the point guaranteed to be on the graph of: 5. (15) f x f x 1 6. (15) Section 0 Instantaneous Rate of Change (8 pts) Find the average rate of change of a function on an interval. Find the instantaneous rate of change of a function at a point. Find the equation of the line tangent to a function at a point. For problems 7-8, find the average rate of change on the interval given. 7. f x x x 1 on 0, 8. f x x 4x 1 on 1, 7 17 Derivative Investigations

13 For problems 9-1, using the limit definition of the first derivative, find the equation of the line tangent to the curve at the given value of x. Leave your answer in point-slope form. 9. f x x x 4 at x f x x x 8 at x 11. f x x 8x at x 1. f x x 6x at x For problems 1-16, identify f(x). You need NOT evaluate the limit. x h x h 5 x x 5 1. lim 14. cos x h cos x lim h0 h h0 h 15. lim h0 x h 1 x 1 h xh x 4 e x h e x lim h0 h Section 1 Polynomial, Product, Quotient Rule (8 pts) Find the derivative of: o a polynomial function o a function consisting of the product of two polynomials o a function consisting of the quotient of two polynomials Find the second derivative of a polynomial function. For problems 17-, find f (x). You need NOT simplify your answer f x x x x fx x 4 x 1 f x 5x x x f x x x x fx x x x. f x x x 7 x Derivative Investigations 17

14 For problems -8, find. You need NOT simplify your answer.. x 6 y x y x x x 5 5. y x x 9x 4 6. y x 4 x y x 4x x 5 8. y x x 1 x For problems 9-0, find f " x f x x x f x x 7x For problems 1-, find. 1. y 7x x 9. y x 4x 8x Derivative Investigations

15 Section Applications of Derivatives (8 pts) Find the equation of the line tangent to f(x) at x = a. Find the equation of the line normal to f(x) at x = a. Apply the derivative to position, velocity, and acceleration. For problems -4, find the equation of the line tangent to the curve at the given value of x.. f x x x 1at x 1 4. y x 4 at x For problems 5-6, find the equation of the line normal to the curve at the given value of x. 5. f x 4x 6x at x 1 6. y x 6x at x Problems 7-9 use the following scenario: A particle s position is represented by st t 4t Find the velocity of the particle at t =. 8. Find the average velocity on the interval 1,. 9. Find the acceleration at t = 1. Problems 40-4 use the following scenario: The velocity of a particle is represented by v t 7t 1t Find the velocity of the particle at t = Find the acceleration at t = Find the average acceleration on the interval [0, ]. Derivative Investigations 175

16 Practice Set 4 The Chain Rule No Calculator Objectives Find the derivative of the composition of two functions f gx Notes Chain Rule Method of Differentiation h x f g x, h' x f ' g x g' x o If then o hx f gx, then h' x Derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function 1. (ACT/SAT) Multiple choice Which of the following is the value of tan0? A. 1 B. 0 C. 1 D. undefined. (ACT/SAT) Multiple choice Which of the following is the measure of the angle below in radians? A. 5 B. 7 4 C D. 1. (6) Multiple choice Mary babysits (b) for $4 per hour. She also works as a tutor (t) for $7 per hour. She is only allowed to work 1 hours per week. She wants to make at least $65. Which of the following systems of inequalities best models the situation described above? A. b t 1 4b 7t 65 B. b t 1 4t 7b 65 C. b t 1 4b 7t 65 D. b t 1 4t 7b (6) Multiple choice The liquid portion of a diet is to provide at least 00 calories, 6 units of vitamin A, and 90 units of vitamin C daily. A cup of dietary drink X (x) provides 60 calories, 1 units of vitamin A, and 10 units of vitamin C. A cup of dietary drink Y (y) provides 60 calories, 6 units of vitamin A, and 0 units of vitamin C. Which of the following systems of inequalities best models the meeting of the minimum daily requirements? A. C. 60x 60y 00 1x 6y 6 10x 0y 90 60x 1x 10x 46 60y 6y 0y 46 B. D. 60x 1x 10x 46 60y 6y 0y 46 60x 60y 00 1x 6y 6 10x 0y Derivative Investigations

17 5. (1) If x 1 f x 4, evaluate lim f x. x x 6. (1) Find the vertical intercept of f x () Rewrite f x x 4 as a simplified piecewise function. 8. () Given the graph of f(x) below, graph g x f x 1 For problems 9-4, find the derivative of each function. 9. y x gx 4 9x f x 9 x 4 1. v t 1 t 5 1. y x x y x 1 x 1 4 Derivative Investigations 177

18 15. gx 1 x 16. st 1 t 17. y 1 x f x x x f x x 1 x 0. y x x 1 1. gx x 5 x. vt 1 t 1 t. y 5 x 4. f x 4x 5. If st t t 8, find v. 6. If f x x 1 x, find f ' 1. x 7. If y, find x 0. x 1 t 1 If v t, find a. t Derivative Investigations

19 For problems 9-0, find the equation of the line tangent to f(x) at the given value of x. f x x 1 at x 1 9. f x x at x 0. 5 For problems 1-, find f x x f " x.. fx 1 x For problems -4, solve f ' x 0 for x f x x 1 x 4. f x x 7 x 1 4 Derivative Investigations 179

20 Practice Set 5 Local Linearization No Calculator Objectives Find the local linearization of f(x) at x = a. Use the local linearization of f(x) to approximate values of the function near x = a. Notes Use the linearization of f x x x x 1 at x to approximate f 1 f ' x x 6x f ' y 1 x y 1.1 f f (ACT/SAT) Multiple choice What is the value of the tangent of the angle shown below? A. 1 B. C. D.. (ACT/SAT) Multiple choice Which of the following is the value of sin? A. 1 B. 0 C. D. 1. (6) Multiple choice You can work a total of no more than 10 hours each week at your two jobs. Housecleaning (x) pays $5 per hour and your sales job (y) pays $8 per hour. You need to earn at least $56 each week to pay your bills. Which of the following systems of inequalities best models the number of hours you can work at each job? A. x y 10 5x 8y 56 B. x y 56 5x 8y 10 C. x y 10 5x 8y 56 D. x y 10 5x 8y (6) Multiple choice The area of a parking lot is 600 square meters. A car (c) requires 6 square meters. A bus (b) requires 0 square meters. The attendant can handle only 60 vehicles. Which of the following systems of inequalities best models the situation described above? A. c b 600 6c 0b 60 B. c b 60 6c 0b 600 C. c b 60 6c 0b 600 D. c b 600 6c 0b 60 x 5. (1) Find the equation of the horizontal asymptote of f x e 180 Derivative Investigations

21 f x. x1 6. (1) Find the zeros of 7. () Rewrite f x x 4 1 as a simplified piecewise function. 8. () Given the graph of f(x) below, graph gx f x 1 For problems 9-18, use local linearization of f(x) to approximate the value of the function f x x ; f f x x x; f x 11. f x ; f.1 1. f x x x; f 1.1 Derivative Investigations 181

22 1. f x x 4x ; f f x x x ; f 1.9 x x f x x; f f x ; f f x x ; f. 17. f x x 77 x x ; f Derivative Investigations

23 Practice Set 6 Implicit Differentiation No Calculator Objectives Find Find implicitly in terms of x and y. implicitly in terms of x and y. Notes x xy y 1 x 1 y x y 0 x y y x y x x y x y x y 5 x y 0 y 1 y x x 1 y x d y y y x y y 1. (ACT/SAT) Multiple choice Which of the following is the value of cos? A. 1 B. 0 C. 1 D. undefined. (ACT/SAT) Multiple choice If 5, what is the value of in radians? A. 4 B. 5 4 C. 5 D (6) Multiple choice Kelly can work for her dad (d) and make $6 per hour, or she can work for Jana s Mowing Service (j) and make $14 per hour. She needs to make at least $84, and can only work 10 hours total. She can work at most 5 hours for Jana s Mowing Service. Which of the following systems of inequalities best models the situation described above? A. d j 10 14d 6j 84 j 5 B. d j 10 6d 14j 84 j 5 C. d j 10 14d 6j 84 d 5 D. d j 10 6d 14j 84 d 5 4. (6) The B&W Leather company wants to add handmade belts (b) and wallets (w) to its product line. Both belts and wallets require cutting and sewing. Belts require hours of cutting time and 6 hours of sewing time. Wallets require hours of cutting time and hours of sewing time. If the cutting machine is available 1 hours a week and the sewing machine is available 18 hours per week, which of the following systems of inequalities best models the situation described above? A. b w 1 6b w 18 B. b w 18 6b w 1 C. b w 1 6b w 18 D. b w 18 6b w 1 Derivative Investigations 18

24 f x 4 4. x 5. (1) Find the vertical intercept of 6. (1) Select all that apply Which of the following represents an exponential decay function? x 1 x x 1 x A. f x e B. f x e 1 C. f x e 4 D. f x e 4 7. () Rewrite f x x 4 as a simplified piecewise function. 8. () Given the graph of f(x) below, graph gx f x For problems 9-16, find in terms of x and y. 9. x y x 1 y x x xy 1. x y xy Derivative Investigations

25 1. y x 14. x y x x y 15. xy y x y 16. x y x y For problems use the equation x y Show that x 18. Show that y 1 y For problems 19-1, find in terms of x and y. 19. x y x y 1 Derivative Investigations 185

26 1. xy y 1 (OMIT) For problems -, find the equation of the line tangent to the curve at the given point... at, 1 x xy y 1 at (, ) x y x y For problems 4-5, find the equation of the line normal to the curve at the given point. 4. x y 5 at, 4 5. xy x 5y at, 186 Derivative Investigations

27 Practice Set 7 Assessment 9 Review 75 Points No Calculator 1. (ACT/SAT) What is the measure of the angle below in degrees?. (ACT/SAT) What is the value of cos 60? Section 6 Modeling with Systems of Inequalities ( pts) Create an inequality or system of inequalities given a model. Select an appropriate graph which represents a systems of inequalities model. Select an appropriate solution from the graph of a systems of inequalities model. Set up a system of inequalities given a model, and select an appropriate solution which fits the model.. (6) Multiple choice Katie is buying plants (p) and soil (s) for her garden. The soil costs $4 per bag, and the plants cost $10 each. She wants to buy at least 5 plants. She cannot spend more than $100. Which of the following systems of inequalities best models the situation described above? A. 10p 4s 100 p 5 B. 10p 4s 100 s 5 C. 10p 4s 100 p 5 D. 10p 4s 100 p 5 4. (6) Multiple choice LW Band members are selling pizzas to raise money for their upcoming trip. Cheese pizza (c) cost $8 and pepperoni pizza (p) cost $9. A band member must sell at least two of each kind of pizza, and must sell at least $180 worth of pizza. Which of the following systems of inequalities best models the situation described above? A. 8c 9p 180 c p B. 8c 9p 180 c p 4 C. 8c 9p 180 c p D. 8c 9p 180 c p Section 1 Exponential Functions ( pts) Evaluate an exponential function (given the function, a table of values, or a graph) by hand and with the use of a calculator, rounding where appropriate. Create a data table for an exponential function using the graphing calculator s table function. Identify various qualities of exponential functions (i.e. horizontal intercept, vertical intercept, increasing, decreasing, zeros). Determine the y-intercept of an exponential equation (without a horizontal shift) as well as the equation of the horizontal asymptote of an exponential equation. Determine the end behaviors of an exponential function. 5. Select all that apply Which of the functions below is decreasing? 1 f x 7 A. x x x4 x B. f x 4 C. f x e D. f x e 4 6. Select all that apply Which of the functions below has a zero? x x1 x x A. f x e B. f x e 4 C. f x e 5 D. f x e Derivative Investigations 187

28 Section The Absolute Value Function ( pts) Understand that the absolute value function is a piecewise function. Graph transformations of an absolute value function. Graph absolute value functions. Solve absolute value equations. 7. Rewrite f x x 4 5 as a simplified piecewise function. 8. Given the graph of f(x) below, graph gx f x 1 Section 0 Instantaneous Rate of Change (6 pts) Find the average rate of change of a function on an interval. Find the instantaneous rate of change of a function at a point. Find the equation of the line tangent to a function at a point. For problems 9-10, find the average rate of change on the given interval. 9. f x x x 4 on, f x x x on 1, 1 For problems 11-1, given the limit, identify f(x). 11. lim a0 x a x a x x a x h x 1. lim h 0 h 188 Derivative Investigations

29 Section 1 Polynomial, Product, Quotient Rule (6 pts) Find the derivative of: o a polynomial function o a function consisting of the product of two polynomials o a function consisting of the quotient of two polynomials Find the second derivative of a polynomial function. For problems 1-16, find the derivative. You need not simplify your answer. 1. y x x 9x 4x 14. f x x x 5 x x 15. gx x x 9 5x y x x 4 Section Applications of the Derivative (8 pts) Find the equation of the line tangent to f(x) at x = a. Find the equation of the line normal to f(x) at x = a. Apply the derivative to position, velocity, and acceleration. 17. Find the equation of the line tangent to f x x x at x =. 18. Find the equation of the normal line to f x x x 1 at x = If st t t 7, find a( 1). 0. If st t 4t 7t 9, find v(0). Derivative Investigations 189

30 Section 4 The Chain Rule (1 pts) Find the derivative of the composition of two functions f gx y x 1, then 1. Multiple choice If 4 A. 4x 1 B. 4x x 1 C. 16 x 1 D. 16x x 1. Multiple choice If f x, then f ' x x x 1 A. x x 1 x B. x 1 x C. x 1 x D. x 1 For problems -6, find the derivative of each function.. gx 1 x x y x x 5. f x x 1 6. vt 4 1 t For problems 7-8, find f " x. 7. f x x fx 4 x 190 Derivative Investigations

31 For problems 9-0, given f ' x, solve f ' x f ' x 6 x 1 x 4 x x f ' x 6 x 4 x x x 4 4 For problems 1-, use f x x 1 x. 1. Find f ' x.. Find f ' 1. For problems -4, use gx x 4 x. Find g' x. 4. Find g' 1. Derivative Investigations 191

32 Section 5 Local Linearization (10 pts) Find the local linearization of f(x) at x = a. Use the local linearization of f(x) to approximate values of the function near x = a. 5. Multiple choice Which of the following is the local linearization of f x x x 6x at x 1? A. y 5 7 x 1 B. y 7 5 x 1 C. y 7 5 x 1 D. y 5 7x 1 6. Multiple choice Which of the following is the local linearization of f x 4x 5x at x 1? A. y 1 x 1 B. y 1 x 1 C. y 1 x 1 D. y 1 x 1 For problems 7-8, find the local linearization of f(x) at the given value of x. x 4 x 7. f x x x 4 at, f x at, 1 For problems 9-40, use the local linearization of f(x) to approximate the function at the given value of x. 9. f x x x x ; f f x x x 1x ; f 1.1 Section 6 Implicit Differentiation (1 pts) Find implicitly in terms of x and y. Find 41. Multiple choice If implicitly in terms of x and y. x y xy, then A. y 1 x 1 B. 1 y x 1 C. y x D. xy y 19 Derivative Investigations

33 4. Multiple choice If y xy 1, then at the point (, ) is A. 6 5 B. 5 C. 8 D. 5 For problems 4-44, find. 4. x y x x y 1 For problems 45-46, find the equation of the tangent line at the given point. 45. xy 6 at, xy 8 at, 1 Derivative Investigations 19

34 Answers to Selected Exercises Practice Set 0 Instantaneous Rate of Change P A. 4x 1x 9 4. x 5. 0, 6., y 1 x y 1 4 x 17. y 1 x y 8 7 x 19. y y 8 8 x 1. y 0 1 x 0. y x 1. f x x x 4. x 5. f x e 6. f x sin x 7. fx 1 x 1 8. f x cos x f x x x Practice Set 1 Polynomial, Product, Quotient Rule P C. A. 5x x 10x 5., 6. 1, 7. f ' x 1x 4x f ' x 10x 4x 8 4x x 4 6x x x 9 9. f ' x 5x 4 x 7x 1 x 5 1 4x 7 x 1 1 x x x 1 1x x f ' x x f ' x 1. f ' x 17 5x x 5 x x f ' x x 4 5x x 16. x 1 x x x x 1 x 1 6xx x 1 8 x x x 1 f ' x 10x x x 5x x x x 5x 6 x x 17. x x 4 6x 1 4x 6 x x 0. x 1 x x 1 x 1 x x 1 x x 1 15x. 5 6x 5 x x 4 x 1 x f '' x f " x 1x 1x. 1x 6x 6 Practice Set Applications of the Derivative P C. A. 4x 14x x x 1 5., , 8 7. y 1 0x 1 8. y 6 14 x 1 9. y 1 1x y 0 x y x y 4 x st 16t t v t t 14t 14 s t 16t v 1 ; v Derivative Investigations

35 Practice Set Assessment 8 Review P D. 4. 1x x , , y 7 5 x y 6 1 x 11. y 4x 1. y 7 0x 1. f x x x 5 x f x cos x 15. f x x f x e x 17. f ' x 10x 1x x 1 1x f ' x 19. f ' x x 1 x 1 x x 0. x 1 1. f ' x. x x x x x x x x x 6 x x x 4 6x x x 9 xx 4 xx 4 x 4 f ' x 0x 6x x. f ' x x 4x x x x x x 16x 1x f " x 6x 1x 0. f " x 40x x 1x 8. 6x x x x y 4 1x 5. y x 1 6. y 1 x x 8. y 1 0x Practice Set 4 The Chain Rule P B. A. C 4. D fx x 4 x x 4 x 8. 6 x g' x x 1 f ' x 8x 9 x 1. at 1 t 4 1 f ' x x x 4x x x x x x 1 4 x 14. 8x 1 x 1 6 x 1x g' x x f ' x 1 x 16. vt x 1 x 0. t x x 1 1 t 1 t 1 1 t. a t 1 t. 1 t y 5 x f " x.,, 1 4. x x 1 15 t y 1 10x ,, x x x 5 x 5 g' x x x 4. f ' x 4 4x f " x 1x 1 48x x 1 Derivative Investigations 195

36 Practice Set 5 Local Linearization P B. D. D 4. C 5. f x fx x 11 x 4 x 1 x Practice Set 6 Implicit Differentiation P C. B. B 4. A A, C 7. fx 9. x 10. y 14. x xy 1 x 1 1 y x y x y x x 4x 10 x x y xy y 1. x x xy x xy x y y x 1. y 5 1. OMIT. y x. y 1 x 4. y 4 x 1 5. y x y x y 5 Practice Set 7 Assessment 9 Review P A 4. C 5. A, D 6. A, C 7. fx x 1 x 4 x 9 x f x x x 1. fx f ' x x x x 5x x x g' x 5 x x 4 4x 1 5x 1 x x D. B. g' x 1 x 1 x 9 x 1. x x 4x x 4x x 9 x y 1 1x 18. y 11 x 4 x x x x 1 x x 6. at 7. f " x 48x 1 8. f " x 1 t 4 4 x f ' x 8 x 1 x x x 1. f ' 1 5. g' x 6 x 4 x 6 x x y 4 4x 8. y 1 x 4. x 44. y ,, 8 7 f ' x g' A 6. C B 4. D 4xy 4x y y 1 x 46. y 1 x x x 1 10,, Derivative Investigations