AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E) 5 5 5 ( 4) d (4 ) d ( 4) d ( + 4) d (4 ) d y=+ 7 6 5 4 3 y y=5-3 - - 3. d t 4 t d dt = (A) t 4 t (B) 4 (C) 4 4 (D) (E) 4
3. π 4 sin d + π 4 cos d = (A) (B) (C) (D) (E) 4. If 3 f() d = A and 5 f() d = B, then 5 3 f() d = (A) A + B (B) A B (C) (D) B A (E) 5. The average value of the function f() = ( ) on the interval from = to = 5 is (A) 6 (B) 6 (C) 64 (D) 66 (E) 56 3 3 3 3 3 6. The average value of the function f() = ln on the interval [, 4] is (A).4 (B).4 (C).59 (D).48 (E) 8.636 7. d d 3 cos t dt = (A) sin 3 (B) 3 sin 3 (C) cos 3 (D) 3 sin 3 (E) 3 cos 3
8. If the definite integral 9. 3 the Trapezoid Rule with n = 4, the error is (A) (B) 7 (C) (D) 65 3 6 ln d = ln (A) + C ln (B) + C (C) ln + C (D) ln + C (E) ln + C ( + ) d is approimated by using (E) 97 3. 3 d = (A) 3 5 5 + C (B) 5 3 5 + C (C) 5 3 + C (D) 3 + C (E) 5 3 3 + C. The average value of f() = e 4 on the interval [ 4, 4 ] is (A).7 (B).545 (C).9 (D).8 (E) 4.36
. Find the distance traveled (to three decimal places) in the first four seconds, for a particle whose velocity is given by v(t) = 7e t, where t stands for time. (A).976 (B) 6.4 (C) 6.359 (D).7 (E) 7. 3. Approimate to three decimal places. sin d using the Trapezoid Rule with n = 4 (A).77 (B).7 (C).555 (D).9 (E).9 4. Use the Trapezoid Rule with n = 4 to approimate the area between the curve y = 3 and the -ais over the interval [3, 4]. (A) 35.66 (B) 7.766 (C) 63.6 (D) 3.56 (E) 5.5 5. The graph of the function f shown below consists of a semicircle and a straight line segment. Then 3 y=f() 4 y 4 f() d = -5-4 -3 - - 3 4 5 - - (A) π + (B) (π ) (C) π + (D) (π + ) (E) π +
6. d 5 cos t dt = d (A) 5 cos 5 cos (B) 5 sin 5 sin (C) cos 5 cos (D) sin 5 sin (E) 5 cos 5 sin 7. d d 5 cos t dt = (A) cos 5 5 cos (B) 5 cos cos 5 (C) cos 5 cos 5 (D) 5 cos 5 cos (E) 5 cos 5 + cos 8. Given that the function f is continuous on the interval [, ), and that f(t) dt =, then f (t) dt = (A) (B) /4 (C) 4 ln (D) 4 (ln ) (E) Can t be determined 9. Given that the function f is continuous on the interval [, ), and that f(t) dt =, then f (t) dt = (A) (B) 6 (C) 4 (D) (E)
Part II. Free-Response Questions. On the graph below, shade in the appropriate area indicated by the integral 3 f()d y y=f(). In the graph to the right, indicate the rectangles that would used in computing a RRAM (right rectangular approimation) to 3 f()d using five rectangles, each of width. (Don t try to compute anything; just draw the relevant picture.) Does it appear the result gives an overestimate or an underestimate of the true area? y y=f()
4 y y=f() - 4 -. The graph of the function f, consisting of three line segments, is given above. Let g() = (a) Compute g(4) and g( ). f(t) dt. (b) Find the instantaneous rate of change of g, with respect to, at =. (c) Find the minimum value of g on the closed interval [, 4]. Justify your answer.
3. Let f be the function depicted in the graph to the right. Arrange the following numbers in increasing order: y y=f() f() d, 3 f() d, 3 f () d. 4. Assume that the average value of a function f over the interval [, 3] is 3.5. Compute 3 f()d. 5. Epress the area of a circle of radius r as a definite integral. 6. Find an antiderivative for each function below: (a) f() = + (b) g() = e (c) v(t) = gt + v (here, v is just a constant) (d) h() = /, where < (This is not hard, you just need to be careful!) (e) k() = ( + )/ (f) θ() = /( + )
7. For each definite integral below, sketch the area represented. (Don t try to compute anything, just draw the relevant picture.) (a) (b) (c) (d) e π π 3 ln d cos d t(3 t) dt f() d, where f() = { if if >. 8. Compute the definite integrals (a) (c) (e) 3 3 4 (9 )d (b) e d (d) d + (f) π sin d d d + 9. Solve for a in each of the below: (a) (b) (c) a a a ( )d = d + = π 4 d =
3. Suppose that f is an even function and that where a >. Compute a a f()d. a f()d =, 3. Suppose that f is an odd function and that a >. Compute a a f()d. 3. Solve the differential equations: (a) y = 3, y() =. (b) y = e, y() =. (c) ( + )y =, y() =. (d) y = y, y() =. (e) y = y, y() =. 33. Suppose that you know that f () =. What can you say about f()? Write down the most general form that f() could have. 34. Suppose that the velocity of a particle is given by v(t) = + sin t, t, where v is given in units of cm/sec. How far has this particle traveled after sec? 35. Assume that a water pump is pumping water into a large tank at a variable rate: after t hours, water is being pumped at a rate of v(t) = 5t + t gallons/hour.
(a) Write down the definite integral that epresses how much water has been pumped into the vessel after 4 hours. (Don t compute this.) (b) Compute how much water is in the tank after 4 hours, assuming that the tank was empty to begin with and that F (t) = 5 (t ln( + t)) is an antiderivative for v(t).
36. The rate at which people enter an amusement park on a given day is modeled by the function E defined by E(t) = 56 t 4t + 5. the rate at which people leave the same amusement park on the same day is modeled by the function L defined by L(t) = 989 t 38t + 37. Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 t 3, the hours during which the park is open. At time t = 9, there are no people in the park. (a) How many people have entered the park by 5: P.M. (t = 7)? Round your answer to the nearest whole number. (b) The price of admission to the park is $5 until 5: P.M. (t = 7). After 5: P.M., the price of admission to the park is $. How many dollars are collected from admission to the park on the given day? Round your answer to the nearest whole number. (c) Let H(t) = t 9 (E() L()) d for 9 t 3. The value of H(7) to the nearest whole number is 375. Find the value of H (7) and eplain the meaning of H(7) and H (7) in the contet of the amusement park. (d) At what time, t, for 9 t 3, does the model predict that the number of people in the park is a maimum?
3 3 (, 3) Graph of f (, 3) 37. The graph of the function f shown above consists of two line segments. Let g be the funtion given by g() = (a) Find g( ), g ( ), and g ( ). f(t) dt. (b) Over which interval(s) within (, ) is g increasing? Eplain your reasoning. (c) Over which interval(s) within (, ) is the graph of g concave down? Eplain your reasoning. (d) On the aes provided, sketch the graph of g on the closed interval [, ]. 3 3
t (minutes) R(t) (gallons per minute) 3 3 4 4 5 55 7 65 9 7 38. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of t. A table of selected values of R(t), for the time interval t 9 minutes, is shown above. (a) Use data from the table to find an approimation for R (45). Show the computations that lead to your answer. Indicate units of measure. (b) The rate of fuel consumption is increasing fastest at time t = 45. What is the value of R (45)? Eplain your reasoning. (c) Approimate the value of 9 R(t) dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approimation less than the value of 9 R(t) dt? Eplain your reasoning. (d) For < b 9 minutes, eplain the meaning of in terms of fuel consumption for the plane. b b R(t) dt Eplain the meaning of R(t) dt in terms of fuel consumption for b this plane. Indicate units of measure in both answers.
y y=8-3 y=f() R S 39. Let f be the function given by f() = 4 3, and let L be the line y = 8 3, where L is tangent to the graph of f. Let R be the region bounded by the graph of f and the -ais, and let S be the region bounded by the graph of f, the line l, and the -ais, as shown above. (a) Show that L is tangent to the graph of y = f() at the point = 3. (b) Find the area of S.
4. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by F (t) = 8 + 4 sin ( ) t for t 3, where F (t) is measured in cars per minute and t is measured in minutes. (a) To the nearest whole number, how many cars pass through the intersection over the 3-minute period? (b) Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer. (c) What is the average value of the traffic flow over the time interval t 5? Indicate units of measure. (d) What is the average rate of change of the traffic flow over the time interval t 5? Indicate units of measure.
4 y 3 y=f() - 3 4 5 6 7 4. Let f be the function defined on the closed interval [, 7]. The graph of f, consisting of four line segments, is shown above. Let g be the function given by g() = (a) Find g(3), g (3), and g (3). f(t) dt. (b) Find the average rate of change of g on the interval 3. (c) For how many values c, where < c < 3, is g (c) equal to the average rate found in part (b)? Eplain your reasoning. (d) Find the -coordinate of each point of inflection of the graph of g on the interval < < 7. Justify your answer.
y y=f() R - y=g() 4. Let f and g be the functions given by f() = ( ) and g() = 3( ) for. The graphs of f and g are shown in the figure above. Find the area of the region R enclosed by the graphs of f and g.