MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Transcription

1 Calculus 1 Instructor: James Lee Practice Exam 3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. 1) 1) A) No absolute extrema. B) Absolute minimum only. C) Absolute maximum only. D) Absolute minimum and absolute maximum. 2) 2) A) Absolute minimum and absolute maximum. B) Absolute maximum only. C) Absolute minimum only. D) No absolute extrema. 1

2 Find the absolute extreme values of each function on the interval. 3) F(x) = - 3, 0.5 x 3 3) x2 A) Maximum = 3, ; minimum = 1 2, -12 B) Maximum = 1 2, ; minimum = (-3, -12) C) Maximum = 3, ; minimum = - 1 2, -12 D) Maximum = 1 2, 1 3 ; minimum = (3, -12) Find the extreme values of the function and where they occur. ) y = x x A) The maximum value is 0 at x = 0. B) The minimum value is 0 at x = 0. C) The minimum value is 0 at x = 1. The maximum value is 0 at x = -1. D) The minimum value is - 2 at x = -1. The maximum value is 2at x = 1. ) 5) y = 7 1-9x 2 5) A) The minimum is 0 at x = 1. B) The maximum is 7 at x = 2. C) The maximum is 7 at x = -2. D) The minimum is 7 at x = 0. 6) y = ln x x 2 6) A) Maximum value is 1 2e at x = e1/2 ; no minimum value. B) Maximum value is 1 2e at x = e1/2 ; minimum value is 0 at x = 1. C) Minimum value is 1 2e at x = e1/2 ; no maximum value. D) None Find the value or values of c that satisfy the equation f(b) - f(a) = f'(c) b - a in the conclusion of the Mean Value Theorem for the given function and interval. 7) f(x) = x + 80, [5, 16]. 7) x A) 5 B) 5, 16 C) - 5, 5 D) 0, 5 8) f(x) = x 2 + 2x + 2, [-1, 2]. 8) A) - 1 2, 1 2 B) 0, 1 2 C) 1 2 D) -1, 2 2

3 Find the function with the given derivative whose graph passes through the point P. 9) g (x) = 1 + 2x, P(-, ) 9) x2 A) g(x) = 1 x + x2-9 C) g(x) = 1 x + x2 + 9 B) g(x) = - 1 x + x2-9 D) g(x) = - 1 x - x2-9 10) r (t) = sec 2 t -, P(0, 0) 10) A) r(t) = sec t - t - 6 B) r(t) = sec t tan t - t -1 C) r(t) = sec t - t - D) r(t) = tan t - t Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. 11) f (x) = x 1/3 (x - 2) 11) A) Decreasing on (-, 0) (2, ); increasing on (0, 2) B) Increasing on (0, ) C) Decreasing on (0, 2); increasing on (2, ) D) Decreasing on (0, 2); increasing on (-, 0) (2, ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. For the given function: (a) Find the intervals on which the function is increasing and decreasing. (b) Then identify the function's local extreme values, if any, saying where they are taken on. (c) Which, if any, of the extreme values are absolute? 12) g(x) = 3x - 5x ) 13) f(x) = x 5 - x 2 13) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch the graph and show all local extrema and inflection points. 1) y = x x2 + 1) 3

4 A) Maximum: (0,1) No inflection point B) Local minimum: (-2,-1) Local maximum: (2,1) Inflection point: (0,0), (-2 3, -1 3),(2 3, 1 3) C) Local minimum: (2,-1) Local maximum: (-2,1) Inflection point: (0,0)

5 D) Local minimum: (-2,- 1 2 ) Local maximum: (2, 1 2 ) Inflection point: (0,0) 15) y = 2x3 + 9x2 + 12x 15) A) Local min: 1,10 No inflection point B) Local maximum: 0, 0 Local minimum: -6,216 Inflection point: -3,108 5

6 C) No extrema Inflection point: 0, 0 D) Local max: -2,-, min: -1,-5 Inflection point: - 3 2, ) y = x + cos 2x, 0 x 16) A) No local extrema. Inflection point: 2, 2 6

7 B) Local minimum:, -1 ; local maximum: 3, 3 Inflection point: 2, 1 C) Local minimum: 5 12, ; local maximum: 12, Inflection points:, and 3, 3 D) Local minimum: (1., -0.26); local maximum: (0.126, 1.031) Inflection points: (0.785, 0.393) and (2.356, 1.178) 7

8 For the given expression y (x), sketch the general shape of the graph of y = f(x). [Hint: it may be helpful to find y.] 17) y = x 2 (6 - x) 17) A) B) C) D) 8

9 Solve the problem. 18) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f. 18) x y Derivatives x < 2 y > 0,y < < x < 0 13 y = 0,y < 0 y < 0,y < < x < 2-3 y < 0,y = 0 y < 0,y > 0 2 x > 2-19 y = 0,y > 0 y > 0,y > 0 A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 19) Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. 19) y = x 3-15x 2 9

10 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 20) From a thin piece of cardboard 0 in. by 0 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary. A) 20 in. by 20 in. by 10 in.; 000 in. 3 B) 13.3 in. by 13.3 in. by 13.3 in.; in. 3 C) 26.7 in. by 26.7 in. by 13.3 in.; in. 3 D) 26.7 in. by 26.7 in. by 6.7 in.; 70.7 in. 3 20) Use l'hopital's Rule to evaluate the limit. 21) lim x 1 x 3-8x x - 1 A) 16 B) -13 C) 19 D) 11 21) 22) lim x 0 cos 7x - 1 x 2 22) A) 7 2 B) C) 0 D) ) lim x 11x 2 + 8x + 9 x 2 + 9x + 2 A) 1 B) - 11 C) 11 D) 11 23) 2) lim x x 2 + 7x + 15 x 3 + 2x A) 0 B) -1 C) D) 1 ^ Use l'hopital's rule to find the limit. 5-5cos 25) lim 0 sin 2 2) 25) A) 0 B) C) 5 2 D) 1 26) lim x x 2 + 2x - x 26) A) 1 B) 0 C) 2 D) ) lim x 0 + sin x x 6 27) A) 0 B) C) 1 D) - 10

11 Find the limit. 28) lim x 1 + x x A) B) 0 C) 1 D) 28) 29) lim x 0 + x -3/ln x 29) A) e 3 B) 1 e C) -3 D) 1 e 3 11

12 Answer Key Testname: PRACTICETEST3 1) A 2) B 3) A ) D 5) D 6) A 7) A 8) C 9) B 10) D 11) D 12) (a) increasing on , , ; decreasing on -, (b) local maximum at x = 0 (0, 1); local minima at x = - (c) no absolute extrema 13) (a) increasing at ; 0, 5 2, 5 2 ; decreasing on - 5, , 5 (b) local maxima x = - 5 (- 5, 0) and x = , and x = , 5 2 ; local minima at x = , , and x = 5 ( 5, 0) (c) no absolute extrema 1) B 15) D 16) C 17) D 18) C 19) The zeros of y = 0 and y = 0 are extrema and points of inflection, respectively. Inflection at x = 5, local maximum at x = 0, local minimum at x = ) D 21) B 22) B 23) C 2) A 12

13 Answer Key Testname: PRACTICETEST3 25) A 26) A 27) B 28) C 29) D 13