AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle moving along the -ais is given by a third-degree polynomial P(t). The roots of P(t) are all in the open interval < t < a. Which of the following statements must be true? I. The velocity of the particle will be zero at least once and at most three times for < t < a. II. In the interval < t < a, the particle moves both left and right. III. The total distance traveled by the particle from t = to t = a is given by P(t) dt. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III a 3. What is the area of the shaded region in the figure below? y y = 1 2 sin (A) 1 (B) π (C) 2 (D) π 1 (E) 2π O
4. Find the antiderivatives for the following: (a) ( 12 3 8 3) d (b) 3 3 t 2 dt ( ) (c) cosθ sinθ + 2 dθ (d) 2 d (e) 5 d 5. Solve the following differential equations for the given initial conditions. (a) f () = 2 3 5 f (2) = 1 (b) f () = cos f (π ) = 1 f (π ) = π (c) f () = 4 1/3 and f () contains the point (8, 2)
6. Given f () = 1 2 + 2. (a) Find the average value of the function on the interval [1, 4]. (b) Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals. 7. Let F() = f (t) dt where f is the function graphs to the right. NOTE: The graph of f is made up of straight lines and a semicircle. (a) Find F( 4) (b) Find F(2) (c) Find F (3) (d) On what intervals is F concave up? Justify your answer. (e) Which is larger: F( 3) or F(3)? Justify your answer. (f) What is the maimum value of F on the interval [ 4, 4]? Eplain.
< < 1 1 1 < < 2 2 2 < < 3 3 3 < < 4 f () 1 2 f!() 4 DNE 3 f!!() 2 DNE 8. Let f be the function that is continuous on the interval,4). The function f is twice differentiable ecept at = 2. The function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not eist at = 2. (a) For < < 4, find all values of at which f has a relative etremum. Determine whether f has a relative maimum or a relative minimum at each of these values. Justify your answer. (b) On the aes provided, sketch the graph of a function that has all the characteristics of f. (c) Let g be the function defined by g() = f (t) dt on the open interval (, 4). For < < 4, find all 1 values of at which g has a relative etremum. Determine whether g has a relative maimum or a relative minimum at each of these values. Justify your answer. (d) For the function g defined in part (c), find all values of, for < < 4, at which the graph of g has a point of inflection. Justify your answer. y O
A GRAPHING CALCULATOR IS REQUIRED FOR SOME OF THE FOLLOWING PROBLEMS. 1. The function y = sin + cos is a solution of which differential equation? (A) I only (B) II only (C) III only (D) I and II (E) II and III I. y + dy d = 2sin II. III. y + dy d = 2cos dy d y = 2sin 2. The average value of the function f () = e sin on the closed interval 1,π is (A).129 (B).145 (C).155 (D).276 (E).31 3. The present price of a new car is $14,5. The price of a new car is changing at a rate of 12 + 18 t dollars per year. How much will a new car cost 5 years from now? (A) $15,2 (B) $15,3 (C) $16,44 (D) $18,12 (E) $22,6
4. Given the function f () = 2/3 on the interval [1, 5]. (a) Sketch the graph of this function on the given interval. (b) Approimate the area using a right endpoint Riemann sum with 4 subintervals of equal length. Show the set-up required for this approimation. (c) Does this approimation overestimate or underestimate the actual area? Eplain (d) Approimate the area using a midpoint Riemann sum with 4 subintervals of equal length. Show the set-up required for this approimation. 5. Find F () for F() = 1 t 2 + 5 dt. 2 6. Find F () for F() = 2 1 t 2 + 5 dt. 2 1 7. Find F () for F() = 2 t 2 + 5 dt. 2
8. The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by R(t) = 2 + 5sin( 4πt 25 ) A pumping station adds sand to the beach at a rate modeled by the function S, given by S(t) = 15t 1+ 3t Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for t 6. At time t =, the beach contains 25 cubic yards of sand. (a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. (b) Write an epression for Y(t), the total number of cubic yards of sand on the beach at time t. (c) Find the rate at which the total amount of sand on the beach is changing at time t = 4. (d) For t 6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.