Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Calculus 3.1-3.4 Review Name:____________________________________ Mitchell o Show your work whenever possible and circle your final answers. 1) True or False: If f’’(c)=0, then (c,f(c)) is a point of inflection. Justify your answer. 2) True or False: If f(c) is a local maximum of a continuous function f on an open interval (a,b), then f ’(c)=0. Justify your answer. 3) True or False: If f’(c)=0 and f’’(c)<0, then f(c) is a local maximum. Justify your answer. 4) Find the critical values of f (x) = x − x − 2 . x2 5) Find the critical values of f (x) = . 5x + 4 6) How many critical values does the function 7) Which of the following values is the absolute 3 5 4 f (x) = (x − 2) (x + 3) have? (A) (B) (C) (D) (E) One Two Three Five Nine 2 maximum of the function f (x) = 4x − x 2 + 6 on the interval [0,4]? (A) (B) (C) (D) (E) 0 2 4 6 10 8) Find the extrema on [-1,3] for f where f (x) = 31 x 3 + 2x 2 − 5x + 1 . x+1 . x−1 Find all values of c that satisfy the conclusion of the mean value theorem on [ 23 ,5]. 10) Let f be the function given by f (x) = 3 2 9) Let f be the function given by f (x) = x 3 − 3x 2 + 2x . Find all values of c that satisfy the conclusion of the mean value theorem on [−1, 1]. 11) If f '(x) = x(x − 3)4 (x + 5)7 , at which values of x is f(x) a relative minimum value? x4 3 5 12) x = t3 − 3t 2 and y = t2 − 2t . Find the x and ycoordinates for each critical value on the curve and identify each point as having a vertical or horizontal tangent line. 13) If f (x) = at x= 14) Given the function defined by f (x) = 3x 5 − 20x 3 , find all values of x for which the graph of f is concave up. 15) The function f given by f (x) = 2x 3 − 3x 2 − 12x has a relative minimum at x= A. B. C. D. E. x>0 − 2 < x < 0 or x > 2 −2 < x < 0 or x > 2 x> 2 −2 < x < 2 − x5 , f ‘(x) attains a maximum value A. -1 B. 0 C. 1 D. 43 E. 53 A. -1 B. 0 C. 2 3 − 105 D. 4 3 + 105 E. 4 16) At what value of x does the graph of 1 1 y = 2 − 3 have a point of inflection? x x A. B. C. D. E. 17) Find the x-coordinates of all inflection points of x 8 9x 6 f if f (x) = . − 56 30 0 1 2 3 At no value of x 18) Given the function f defined by f (x) = cos x − cos 2 x for −π ≤ x ≤ π . 19) If y = 2x 3 + 4ax 2 + bx + 3 has an inflection point at (2,-3), find a and b. a) Find the x-intercepts of the graph of f. b) Find the x and y-coordinates of all relative maximum and minimum points of f. (justify your answers) c) Find the intervals on which the graph of f is increasing. 20) The graph of f, the derivative of f is shown below. On which of the following intervals is f decreasing? A. B. C. D. E. [2,4] only [3,5] only [0,1] and [3,5] [2,4] and [6,7] [0,2] and [4,6] y f’(x) x 7 21) The graph of f ’(x) is shown to the right. For what value of x does f(x) have a relative maximum? a) b) c) d) e) 1 2 3 4 5 y=f ‘(x) 0 1 2 3 4 (2,4) only (1,3) and (5, ∞ ) (- ∞ ,1) and (3,5) (0,2) and (4,6) (- ∞ ,0), (2,4) and (6, ∞ ) 0 1 2 4 6 -1 7 y=f ‘‘(x) -1 7 24) The graph of f ’(x) is shown to the right. f has an inflection point at which of the following x-values? a) b) c) d) e) y=f ‘(x) 23) The graph of f ’’(x) is shown to the right. Use the graph of f ’’(x) to approximate the intervals on which f(x) is concave up. a) b) c) d) e) 7 22) The graph of f ’(x) is shown to the right. For what value of x is f(x) concave down? a) b) c) d) e) -1 y=f ‘(x) -1 7 25) The function f and its derivatives have the properties indicated in the table below. x f(x) f’(x) f’’(x) − 2 < x < −1 + + - -1 2 0 0 0 1 UND UND −1 < x < 0 + - 1 0 0 0 0<x<1 + + 1<x<2 - 2 -1 UND UND 2<x<3 + - a) Find the x-coordinates of each point at which f attains an absolute maximum value or an absolute minimum value. For each x-coordinate you give, state whether f attains an absolute minimum or an absolute maximum. b) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer. c) Sketch a graph of a function with all of the given characteristics. 26) The figure below shows the graph of f’, the derivative of a function f. The domain of the function f is the set of all x such that −3 ≤ x ≤ 3 . a) For what values of x, −3 ≤ x ≤ 3 , does f have a relative maximum? A relative minimum? Justify your answers. b) For what values of x is the graph of f concave up? Justify your answer. c) Use the information found in parts a and b and the fact that f ( −3) = 0 to sketch a possible graph of f. y = f’(x) -3 7