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AP Calculus AB Name: ________________ Chapter Three test 1) Determine the function whose graph has vertical asymptotes at x = +2 and a horizontal asymptote at y = 0. x a) f(x) = (x 2)2 x b) f(x) = 2 x +4 c) f(x) = d) f(x) = e) x3 x2 4 3x x2 4 none of these 2) Find the values of x that give relative extrema for the function f(x) = 3x5 – 5x3. a) relative maximum: x = 0; relative minimum x = 5 3 b) relative maximum: x = -1; relative minimum x = 1 c) relative maximum: x = +1; relative minimum x = 0 d) relative maximum: x = 0; relative minimum x = +1 e) none of these 3) Decide whether Rolle’s Theorem can be applied to f(x) = x2 – 2x on the interval [0, 2]. If Rolle’s Theorem can be applied, find all values of c in the interval such that f’(c) = 0. If Rolle’s Theorem cannot be applied, state why. a) Rolle’s can be applied; c = 1 b) Rolle’s cannot be applied: f(x) differentiable on (0, 2) c) Rolle’s cannot be applied: f(0) = f(2) d) Rolle’s cannot be applied; f(x) is not continuous on [0, 2] e) none of these 4) Find all critical numbers for the function f(x) = a) b) c) d) e) -1 –1, -3 –3 f(x) has no critical numbers none of these x +1 x +3 5) The product of two positive numbers is 972. Minimize the sum of the first and three times the second. a) 30 and 1625 b) both numbers are 18 3 c) 18 and 54 d) impossible, one number will be negative e) none of these 6) Find all the intervals on which the graph of the function is concave upward: f(x) = 8x3 – 2x4 a) (- , 3) b) (- , 0) (2, ) c) (3, ) d) (0, 2) e) none of these 7) Find all horizontal asymptotes for f(x) = a) b) c) d) e) y=+1 y=4 y=+4 y=0 none of these 4x x2 + 9 8) Find all points of inflection of the graph of f(x) = x4 – 6x3 a) (0, 0) b) (0, 0) and ( 9 2 , - 218716 ) c) (3, -81) d) (0, 0) and (3, -81) e) none of these 9) Find all of the open intervals on which f(x) is increasing or decreasing: f(x) = 1 2 x a) increasing ( ); decreasing ( ) b) decreasing ( ); increasing ( ) c) strictly increasing d) strictly decreasing e) none of these 10) Determine whether the Mean Value Theorem applies to f(x) = 3x – x2 on the interval [2, 3]. If the “MVT” does apply, find all values of c in f(b) f( a) [2, 3] such that f’(c) = . If it does not apply, state why. b a a) MVT applies; c = 2 3 c) MVT applies; c = 5 2 MVT does not apply; f(2) = f(3) d) MVT does not apply; f(x) is not differentiable on [2, 3] e) none of these b) 11) Determine from the graph whether f possesses extrema on the interval (a, b) a) maximum at x = c, minimum at x = b b) maximum at x = c, no minimum c) no maximum, minimum at x = b d) no extrema 12) 13) 14) Find the limit: lim 8 + a) 81 b) c) d) d) -8 0 x sin x 3x 3 none of these A farmer has 160 feet of fencing to enclose 2 adjacent rectangular pig pens. What dimensions should be used so that the enclosed area will be a maximum? 80 a) 40 ft. by ft. 3 8 b) 4 15 ft. by 15 ft. 5 c) 40 ft. by 40 ft. d) 20 ft. by e) none of these 80 ft. 3 The side of a cube is measured to be 3.0 inches. If the measurement is correct to within 0.1 inch, use differentials to estimate the propagated error in the volume of the cube. a) +0.000001 in3. b) +0.06 in3. c) +2.7 in3. d) +0.27 in3. e) none of these 15) Which of the following is the correct sketch of the graph of the function y = -x3 – 12x + 20? 16) Find the differential of y = tan3x a) sec23xdx b) 3sec3xtan3xdx c) 3sec23xdx d) 3sec3xtan3xdx e) none of these