Review Sheet Chapter 3

Transcription

1 Review Sheet Chapter 3 1. Find the value of the derivative (if it exists) of the function at the extremum point (0,0). A) 0 B) 1 C) -1 D) E) 2. Find the value of the derivative (if it exists) of the function at the extremum point (0,7). A) 0 D) 7 B) Does not exist E) None of the above C) 7 Page 1

2 3. Find any critical numbers of the function, t < 4. A) 0 D) Both A and B B) E) Both A and C C) 4. Find any critical numbers of of the function,. A) D) B) E) Both A and B C) F) Both A and C 5. Locate the absolute extrema of the function on the closed interval. A) No absolute max; Absolute min: f( 6) = 148 B) Absolute max: f(3) = 14 ; Absolute min: f( 6) = 148 C) Absolute max: f( 6) = 148 ; No absolute min D) Absolute max: f( 6) = 148 ; Absolute min: f(3) = 14 E) No absolute max or min 6. Locate the absolute extrema of the given function on the closed interval [ 10,10]. A) Absolute max: f(1) = 5 D) No absolute min B) Absolute min: f(-1) = 5 E) Both A and D C) No absolute max F) Both A and B 7. Locate the absolute extrema of the function on the closed interval. A) Absolute max: = ; Absolute min: f(0) = 1. B) Absolute max: f(0) = 1 ; Absolute min: =. C) Absolute max: = ; No absolute min. D) No absolute max; Absolute min: f(0) = 1. E) None of the above. Page 2

3 8. Determine whether Rolle's Theorem can be applied to the function on the closed interval [ 4,2]. If Rolle s Theorem can be applied, find all values of c in the open interval ( 4,2) such that. A) Rolle's Theorem applies; c = 1 D) Rolle's Theorem applies; c = 1 B) Rolle's Theorem applies; c = 2.5 E) Both A and D C) Rolle's Theorem does not apply 9. Determine whether Rolle's Theorem can be applied to the function on the closed interval such that.. If Rolle's Theorem can be applied, find all numbers c in the open interval A) Rolle's Theorem applies; D) Rolle's Theorem does not apply B) Rolle's Theorem applies; E) Both A and B C) Rolle's Theorem applies; F) Both C and D 10. Determine whether Rolle's Theorem can be applied to the function on the closed interval. If Rolle's Theorem can be applied, find all numbers c in the open interval such that. A) Rolle's Theorem applies; 0 D) Rolle's Theorem applies: 0.5 B) Rolle's Theorem applies; 1 E) Rolle's Theorem does not apply C) Rolle's Theorem applies; Determine whether the Mean Value Theorem can be applied to the function on the closed interval [6,12]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (6,12) such that. A) MVT applies; 8 D) MVT applies; 11 B) MVT applies; 10 E) MVT applies; 7 C) MVT applies; 9 Page 3

4 12. Determine whether the Mean Value Theorem can be applied to the function on the closed interval. If the Mean Value Theorem can be applied, find all numbers c in the open interval such that. A) MVT applies; D) MVT applies; B) MVT applies; E) MVT does not apply C) MVT applies; 13. Identify the open intervals where the function is increasing or decreasing. A) Decreasing: ; Increasing: B) Increasing: ; Decreasing: C) Increasing on D) Decreasing on E) None of the above 14. Identify the open intervals where the function is increasing or decreasing. A) Decreasing: ; Increasing: B) Increasing: ; Decreasing: C) Increasing: ; Decreasing: D) Increasing: ; Decreasing: E) Decreasing for all x Page 4

5 15. For the function : (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results. A) (a) x = 0, 8 (b) Increasing: ; Decreasing: (c) Relative max: B) (a) x = 0, 8 (b) Decreasing: (c) Relative min: C) (a) x = 0, 1 (b) Increasing: (c) Relative max: D) (a) x = 0, 1 (b) Decreasing: (c) Relative min: E) (a) x = 0, 1 (b) Increasing: (c) Relative max: ; Relative min: ; Increasing: ; Relative max: ; Decreasing: ; Relative min: ; Increasing: ; Relative max: ; Decreasing: ; No relative min. Page 5

6 16. Consider the function on the interval. (a) Find the critical points of the function; (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Use a graphing utility to confirm your results. A) (a) (b) Decreasing: ; Increasing: B) (a) (c) Relative min at (b) Increasing: (c) Relative min at ; Decreasing: ; Relative max at C) (a) (b) Decreasing: (c) Relative max at ; Increasing: ; Relative min at D) (a) (b) Decreasing: ; Increasing: E) (a) (c) Relative min at (b) Increasing: ; Decreasing: (c) Relative max at Page 6

7 17. The graph of f is shown in the figure. Sketch a graph of the derivative of f. A) B) C) Page 7

8 D) E) Page 8

9 18. The graph of f is shown in the figure. Sketch a graph of the derivative of f. A) B) C) Page 9

10 D) E) Page 10

11 19. The function describes the motion of a particle moving along a line. (a) Find the velocity function of the particle at any time t; (b) Identify the time intervals when the particle is moving in a positive direction; (c) Identify the time intervals when the particle is moving in a negative direction; and (d) Identify the times when the particle changes its direction. A) (a) ; (b) ; (c) ; (d) t = 2 and t = 10. B) (a) ; (b) ; C) (a) D) (a) (c) ; (d) t = 2 and t = 10. (b) ; (c) Particle is never moving in a negative direction; (d) Particle never changes direction. (b) Particle is never moving in a positive direction; (c) ; (d) Particle never changes direction. E) None of the above. 20. Determine the open intervals on which the graph of is concave downward or concave upward. A) Concave downward on B) Concave downward on ; concave upward on C) Concave upward on ; concave downward on D) Concave downward on ; concave upward on E) Concave upward on ; concave downward on Page 11

12 21. Find the points of inflection and discuss the concavity of the function. A) Inflection point at ; concave upward on ; concave downward on B) Inflection point at ; concave downward on ; concave upward on C) Inflection point at ; concave upward on ; concave downward on D) Inflection point at ; concave downward on ; concave upward on E) None of the above 22. Find the points of inflection and discuss the concavity of the function. A) Inflection point at x = 7. Concave down on B) No inflection points. Concave down on C) Inflection point at x = 7. Concave up on D) No inflection points. Concave up on E) Inflection point at x = 0. Concave up on ; concave down on 23. Find the points of inflection and discuss the concavity of the function on the interval. A) Concave downward on ; concave upward on. Inflection point at B) Concave upward on ; concave downward on. Inflection point at C) No inflection points. Concave up on D) No inflection points. Concave down on E) None of the above 24. Find all relative extrema of the function Use the Second Derivative Test where applicable. A) Relative max: ; no relative min B) Relative max: ; no relative min C) No relative max or min D) Relative min: ; no relative max E) Relative min: ; no relative max Page 12

13 25. Find all relative extrema of the function. Use the Second Derivative Test where applicable. A) Relative max: D) No relative max B) Relative min: E) Both A and C C) No relative min F) Both B and D Page 13

14 26. The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes. A) B) C) Page 14

15 D) E) None of the above Page 15

16 27. With out a calculator, match the function with one of the following graphs. A) B) C) D) E) Page 16

17 28. Find the limit. A) D) 0 B) E) Does not exist C) Find the limit. A) B) 1 C) 0 D) E) 30. Find the limit. A) 6 B) C) D) 1 E) 31. Find the limit. A) B) C) 8 D) 1 E) Page 17

18 32. The graph of a function f is is shown below: Which of the following graphs is the graph of its derivative A) '? B) C) Page 18

19 D) E) Page 19

20 33. The graph of a function f is is shown below: Which of the following graphs is the graph of its derivative? A) B) C) Page 20

21 D) E) 34. Determine the slant asymptote of the graph of. A) No slant asymptotes D) B) E) C) 35. Find the length and width of a rectangle that has perimeter meters and a maximum area. A) 2, 6 B) 1, 7 C) 4, 4 D) 5, 3 E) 8, 1 Page 21

22 36. Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 meters. A) Square base side ; height D) Square base side ; height B) Square base side ; height E) Square base side ; height C) Square base side ; height 37. A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used. A) B) C) D) E) 38. Find the differential dy of the function. A) D) B) E) C) 39. Find the differential dy of the function. A) D) B) E) C) 40. The measurements of the base and altitude of a triangle are found to be 46 and 42 centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places. A) 11 B) 9.9 C) 12.1 D) E) The measurement of the edge of a cube is found to be 13 inches, with a possible error of 0.08 inch. Use differentials to estimate the propagated error in computing (a) the volume of the cube and (b) the surface area of the cube. Give your answers to two decimal places. A) 48.67, B) 32.45, C) 36.50, D) 32.45, 9.98 E) 40.56, Page 22

23 Answer Key - chapter 3 review 1. A 2. B 3. C 4. E 5. D 6. F 7. B 8. D 9. E 10. A 11. C 12. A 13. A 14. B 15. A 16. C 17. D 18. C 19. A 20. E 21. A 22. D 23. E 24. E 25. E 26. B 27. D 28. B 29. C 30. C 31. A 32. B 33. D 34. E 35. C 36. D 37. A 38. E 39. C 40. A 41. E Page 23