Calculus Ap Ab Question
Free Response Questions 1969-2005
Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
1
AP Calculus Free-Response Questions
1969 AB 1 Consider the following functions defined for all x:
f1 ( x) = x f 2 ( x) = x cos x f3 ( x) = 3e2 x f 4 ( x) = x − x
Answer the following questions (a, b, c, and d) about each of these functions. Indicate your answer by writing either yes or no in the appropriate space in the given rectangular grid. No justification is required but each blank space will be scored as an incorrect answer. Questions f1 (a) (b) x? (c) (d) Does f (− x ) = − f ( x ) Does the inverse function exist for all Is the function periodic? Is the function continuous at x = 0? f2 Functions f3
f4
1969 AB 2 A particle moves along the x-axis in such a way that its position at time t is given by x = 3t 4 − 16t 3 + 24t 2 for − 5 ≤ t ≤ 5. a. Determine the velocity and acceleration of the particle at time t . b. At what values of t is the particle at rest? c. At what values of t does the particle change direction? d. What is the velocity when the acceleration is first zero? 1969 AB 3 Given f ( x) =
1 1 + ln x, defined only on the closed interval ≤ x < e. x e
a. Showing your reasoning, determine the value of x at which f has its (i) absolute maximum (ii) absolute minimum b. For what values of x is the curve concave up? c. On the coordinate axis provided, sketch the graph of f over the 2
interval
d. Given that the mean value (average ordinate) of f over the interval is 2 , state in words a geometrical interpretation of this number relative e −1 to the graph. 1969 AB 4 BC 4 The number of bacteria in a culture at time t is given approximately by
1 ≤ x < e. e
y = 1000(25 + te 20 ) for 0 ≤ t ≤ 100.
a. Find the largest number and the smallest number of bacteria in the culture during the interval. b. At what time during the interval is the rate of change in the number of bacteria a minimum? 1969 AB 5 Let
Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
1
AP Calculus Free-Response Questions
1969 AB 1 Consider the following functions defined for all x:
f1 ( x) = x f 2 ( x) = x cos x f3 ( x) = 3e2 x f 4 ( x) = x − x
Answer the following questions (a, b, c, and d) about each of these functions. Indicate your answer by writing either yes or no in the appropriate space in the given rectangular grid. No justification is required but each blank space will be scored as an incorrect answer. Questions f1 (a) (b) x? (c) (d) Does f (− x ) = − f ( x ) Does the inverse function exist for all Is the function periodic? Is the function continuous at x = 0? f2 Functions f3
f4
1969 AB 2 A particle moves along the x-axis in such a way that its position at time t is given by x = 3t 4 − 16t 3 + 24t 2 for − 5 ≤ t ≤ 5. a. Determine the velocity and acceleration of the particle at time t . b. At what values of t is the particle at rest? c. At what values of t does the particle change direction? d. What is the velocity when the acceleration is first zero? 1969 AB 3 Given f ( x) =
1 1 + ln x, defined only on the closed interval ≤ x < e. x e
a. Showing your reasoning, determine the value of x at which f has its (i) absolute maximum (ii) absolute minimum b. For what values of x is the curve concave up? c. On the coordinate axis provided, sketch the graph of f over the 2
interval
d. Given that the mean value (average ordinate) of f over the interval is 2 , state in words a geometrical interpretation of this number relative e −1 to the graph. 1969 AB 4 BC 4 The number of bacteria in a culture at time t is given approximately by
1 ≤ x < e. e
y = 1000(25 + te 20 ) for 0 ≤ t ≤ 100.
a. Find the largest number and the smallest number of bacteria in the culture during the interval. b. At what time during the interval is the rate of change in the number of bacteria a minimum? 1969 AB 5 Let