Calculus AB and BC Exam 2020

Section I: Multiple-Choice PART A (AB OR BC) Graphing calculators are not permitted on this part of the exam. 1. 1 − cos 2 (2 x ) = x→0 (2 x )2 lim (A) 0 1 (B) 4 1 (C) 2 (D) 1 2 for x < −1 x  f (x ) =  2 x − 3 for − 1 ≤ x ≤ 2  4 x − 3 for x > 2 2. Let f be the function defined above. At what values of x, if any, is f not differentiable? (A) x = -1 only (B) x = 2 only (C) x = -1 and x = -2 (D) f is differentiable for all values of x. 2020 x f(x) fʹ(x) g(x) gʹ(x) 1 2 -4 -5 3 2 3. -3 1 8 4 The table above gives values of the differentiable functions f and g and their derivatives at selected values of x. If h is the function defined by h(x) = f(x)g(x) + 2g(x), then hʹ (1) = (A) 32 (B) 30 (C) -6 (D) -16 4. dy If x3 - 2xy + 3y2 = 7, then = dx 2 3x + 4 y (A) 2x (B) (C) (D) 5. 3x 2 − 2y 2x − 6 y 3x 2 2x − 6 y 3x 2 2 − 6y The radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius r and height h of the cylinder? (The volume V of a cylinder with radius r and height h is V = πr2h.) (A) -20πr (B) -2πrh (C) 4πrh - 5πr2 (D) 4πrh + 5πr2 2020 6. Which of the following is equivalent to the definite integral 6 ∫2 x dx? 4 4k n k =1 n n (A) lim ∑ n →∞ 6 6k n k =1 n n (B) lim ∑ n →∞ n (C) lim ∑ n →∞ k =1 4 4k 2+ n n 6 6k 2+ n k =1 n n (D) lim ∑ n →∞ O 1 2 3 4 5 6 7 8 Graph of g 7. The figure above shows the graph of the continuous function g on the interval x [0, 8]. Let h be the function defined by h(x) = ∫ g (t ) dt . On what intervals is h 3 increasing? (A) [2, 5]only (B) [1, 7] (C) [0, 1] and [3, 7] (D) [1, 3] and [7, 8] 8. ∫ x dx = 1 − 9x 2 1 (A) − 1 − 9 x 2 + C 9 1 (B) − ln 1 − 9 x 2 + C 18 1 (C) arcsin(3x ) + C 3 x (D) arcsin(3x ) + C 3 2020 y 3 2 1 –3 –2 –1 –1 1 2 3 x –2 –3 9. Shown above is a slope field for which of the following differential equations? (A) dy y − 2 = dx 2 (B) dy y 2 − 4 = dx 4 (C) dy x − 2 = dx 2 (D) dy x 2 − 4 = dx 4 10. Let R be the region bounded by the graph of x = ey, the vertical line x = 10, and the horizontal lines y = 1 and y = 2. Which of the following gives the area of R? (A) 2 ∫ 1e y dy e2 ∫ e ln x dx 2 (C) ∫ (10 − e y )dy 1 10 (D) ∫ (ln x − 1)dx e (B) 2020 PART B (AB OR BC) A graphing calculator is required on this part of the exam. y 4 3 2 1 –4 –3 –2 –1 O –1 1 2 3 4 x –2 –3 –4 Graph of f 11. The graph of the function f is shown in the figure above. The value of lim+ f ( x ) is x →1 (A) -2 (B) -1 (C) 2 (D) nonexistent 12. The velocity of a particle moving along a straight line is given by v(t) = 1.3tln (0.2t + 0.4) for time t ≥ 0. What is the acceleration of the particle at time t = 1.2? (A) -0.580 (B) -0.548 (C) -0.093 (D) 0.660 2020 x fʹ(x) 0 -1 11 2 9 4 8 5 5 2 13. Let f be a twice-differentiable function. Values of fʹ, the derivative of f, at selected values of x are given in the table above. Which of the following statements must be true? (A) f is increasing for -1 ≤ x ≤ 5. (B) The graph of f is concave down for -1 < x < 5. 3 (C) There exists c, where -1 < c < 5, such that f ′(c ) = − . 2 3 (D) There exists c, where -1 < c < 5, such that f ′′(c ) = − . 2 14. Let f be the function with derivative defined by f ′( x ) = 2 + (2 x − 8)sin( x + 3). How many points of inflection does the graph of f have on the interval 0 < x < 9 ? (A) One (B) Two (C) Three (D) Four 15. Honey is poured through a funnel at a rate of r(t) = 4e-0.35t ounces per minute, where t is measured in minutes. How many ounces of honey are poured through the funnel from time t = 0 to time t = 3? (A) 0.910 (B) 1.400 (C) 2.600 (D) 7.429 PART A (BC ONLY) Graphing calculators are not permitted on this part of the exam. x f(x) fʹ(x) 2 5 2 3 4 7 16. The table above gives values of the differentiable function f and its derivative fʹ at selected values of x. 5 5 If ∫ f ( x )dx = 14, what is the value of ∫ x ⋅ f ′( x ) dx ? 2 2 (A) 13 (B) 27 63 2 (D) 41 (C) 2020 17. The number of fish in a lake is modeled by the function F that satisfies the dF F   logistic differential equation = 0.04 F  1 −  , where t is the time in  dt 5000  months and F(0) = 2000. What is lim F (t ) ? t →∞ (A) 10,000 (B) 5000 (C) 2500 (D) 2000 18. A curve is defined by the parametric equations x(t) = t2 + 3 and y(t) = sin (t2). d2 y Which of the following is an expression for in terms of t ? dx 2 2 (A) -sin (t ) (B) -2tsin (t2) (C) cos (t2) - 2t2 sin (t2) (D) 2cos (t2) - 4t2 sin (t2) 19. Which of the following series is conditionally convergent? 5 3 k +1 k =1 ∞ 5 (B) ∑ ( −1)k k +1 k =1 ∞ 5k (C) ∑ ( −1)k k +1 k =1 ∞ 5k 2 (D) ∑ ( −1)k k +1 k =1 (A) ∞ ∑ (−1) k 20. Let f be the function defined by f(x) = e2x. Which of the following is the Maclaurin series for fʹ, the derivative of f ? x2 x3 xn + ++ + 2! 3! n! 2 3 2x 2x 2x n (B) 2 + 2 x + + ++ + 2! 3! n! (2 x )2 (2 x )3 (2 x )n (C) 1 + 2 x + + ++ + 2! 3! n! 2(2 x )2 2(2 x )3 2(2 x )n (D) 2 + 2(2 x ) + + ++ + 2! 3! n! (A) 1 + x + 2020 PART B (BC ONLY) A graphing calculator is required on this part of the exam. y O x 21. The figure above shows the graph of the polar curve r = 2 + 4sin θ. What is the area of the shaded region? (A) 2.174 (B) 2.739 (C) 13.660 (D) 37.699 22. The function f has derivatives of all orders for all real numbers. It is known that 12 3 f (4) ( x ) ≤ and f (5) ( x ) ≤ for 0 ≤ x ≤ 2 . Let P4 ( x ) be the fourth-degree 5 2 Taylor polynomial for f about x = 0. The Taylor series for f about x = 0 converges at x = 2. Of the following, which is the smallest value of k for which the Lagrange error bound guarantees that f (2) − P4 (2) ≤ k ? 25 3 ⋅ 5! 2 25 12 (B) ⋅ 5! 5 24 3 (C) ⋅ 4! 2 24 12 (D) ⋅ 4! 5 (A) 2020