1988 FRQs AP Calculus
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This document contains 6 multi-part calculus problems involving functions, derivatives, integrals, areas, volumes and series. The problems cover topics like finding derivatives and integrals of functions, analyzing graphs of functions, calculating areas and volumes of revolved regions, determining convergence of series, and relating functions to their derivatives.
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1988 FRQs AP Calculus
- 1. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB1) Let f be the function given by f(x) = x4 - 16x2 . (a) Find the ___domain of f. (b) Describe the symmetry, if any, of the graph of f. (c) Find f ’(x). (d) Find the slope of the line normal to the graph of f at x = 5.
- 2. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB2) A particle moves along the x-axis so that its velocity at any time t ≥≥≥≥ 0 is given by v(t) = 1 - sin(2ππππt). (a) Find the acceleration a(t) of the particle at any time t. (b) Find all values of t, 0 ≤ t ≤ 2, for which the particle is at rest. (c) Find the position x(t) of the particle at any time t if x(0) = 0.
- 3. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB3) Let R be the region in the first quadrant enclosed by the hyperbola x2 - y2 = 9, the x-axis, and the line x = 5. (a) Find the volume of the solid generated by revolving R about the x-axis. (b) Set up, but do not integrate, an integral expression in terms of a single variable for the volume of the solid generated when R is revolved about the line x = -1.
- 4. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB4) Let f be the function defined by f(x) = 2xe-x for all real numbers x. (a) Write an equation of the horizontal asymptote for the graph of f. (b) Find the x-coordinate of each critical point of f. For each such x, determine whether f(x) is a relative maximum, a relative minimum, or neither. (c) For what values of x is the graph of f concave down? (d) Using the results found in parts (a), (b), and (c), sketch the graph of y = f(x) in the xy-plane provided below.
- 5. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB5) Let R be the region in the first quadrant under the graph of y = x x2 + 2 for 0 ≤≤≤≤ x ≤≤≤≤ 6 . (a) Find the area of R. (b) If the line x = k divides R into two regions of equal area, what is the value of k? (c) What is the average value of y = x x2 + 2 on the interval 0 ≤ x ≤ 6 .
- 6. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (AB6) Let f be a differentiable function, defined for all real numbers x, with the following properties. (i) f ’(x) = ax2 + bx (ii) f ’(1) = 6 and f ’’(1) = 18 (iii) ⌡⌠ 1 2 f(x)dx = 18 Find f(x). Show your work.
- 7. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (BC1) Let f be the function defined by f(x) = (x2 - 3)ex for all real numbers x. (a) For what values of x is f increasing? (b) Find the x-coordinate of each point of inflection of f. (c) Find the x- and y-coordinates of the point, if any, where f(x) attains its absolute minimum.
- 8. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] 0.5 1 1.5 2 2.5 3 x 1 2 3 4 y (BC2) Let R be the shaded region between the graphs of y = 3 x and y = 3x x2 + 1 from x = 1 to x = 3 , as shown in the figure above. (a) Find the area of R. (b) Find the volume of the solid generated by revolving R about the y-axis.
- 9. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] B x y A 100 C (BC3) The figure above represents an observer at point A watching balloon B as it rises from point C. The balloon is rising at a constant rate of 3 meters per second and the observer is 100 meters from point C. (a) Find the rate of change in x at the instant when y = 50. (b) Find the rate of change in the area of the right triangle BCA at the instant when y = 50. (c) Find the rate of change in θ at the instant when y = 50.
- 10. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (BC4) Determine all values of x for which the series ∑ k = 0 ∞∞∞∞ 2kxk ln(k + 2) converges. Justify your answer.
- 11. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] 0.25 0.5 0.75 1 1.25 1.5 x 0.2 0.4 0.6 0.8 1 y = 1 - sin x (BC5) The base of a solid S is the shaded region in the xy-plane enclosed by the x-axis, the y-axis, and the graph of y = 1 - sin x, as shown in the figure above. For each x, the cross section of S perpendicular to the x-axis at the point (x, 0) is an isosceles right triangle whose hypotenuse lies in the xy- plane. (a) Find the area of the triangle as a function of x. (b) Find the volume of S.
- 12. Calculus WorkBook 88 Name:_______________ Format © 1997 – 1999 MNA Consulting [email protected] (BC6) Let f be a differentiable function defined for all x ≥≥≥≥ 0 such that f(0) = 5 and f(3) = -1. Suppose that for any number b > 0 the average value of f(x) on the interval 0 ≤≤≤≤ x ≤≤≤≤ b is f(0) + f(b) 2 . (a) Find ⌡⌠ 0 3 f(x)dx. (b) Prove that f ’(x) = f(x) - 5 x for all x > 0. (c) Using part (b), find f(x).