Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
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
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
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 R denote the region enclosed between the graph of y = x 2 and the graph of y = 2 x. a. Find the area of region R. b. Find the volume of the solid obtained by revolving the region R about the y-axis.
1969 AB 6 An arched window with base width 2b and height h is set into a wall. The arch is to be either an arc of a parabola or a half-cycle of a cosine curve. a. If the arch is an arc of a parabola, write an equation for the parabola relative to the coordinate system shown in the figure. (x-intercepts are (−b,0) and (b,0). y-intercept is (0, h). ) b. If the arch is a half-cycle of a cosine curve, write an equation for the cosine curve relative to the coordinate system shown in the figure.
c. Of these two window designs, which has the greater area? Justify your answer. 1969 AB 7
e x + e− x . 2 b. Let R be a point on the curve and let the x-coordinate of R be r (r ≠ 0). The tangent line to the curve at R crosses the x-axis at a point Q. Find the coordinates of Q. c. If P is the point (r , 0), find the length of PQ as a function of r and the limiting value of this length as r increases without bound. a. On the coordinate axes provided, sketch the graph of y = 1970 AB 1 BC 1 Given the parabola y = x 2 − 2 x + 3: a. Find an equation for the line L , which contains the point (2, 3) and is perpendicular to the line tangent to the parabola at (2, 3). b. Find the area of that part of the first quadrant which lies below both the line L and the parabola. 1970 AB 2 A function f is defined on the closed interval from -3 to 3 and has the graph shown below.
a. Sketch the entire graph of y = f ( x) . b. Sketch the entire graph of y = f ( x ). c. Sketch the entire graph of y = f (− x). 1 d. Sketch the entire graph of y = f x . 2 e. Sketch the entire graph of y = f ( x − 1).
1970 AB 3 BC 2 Consider the function f given by f ( x) =
4 x3 1 + 4x3.
a. Find the coordinates of all points at which the tangent to the curve is a horizontal line....