Calculus Test 2013- Chapter 5 Name You can use a calculator on all of the test. Each multiple choice & each part of the free response is worth 5 points. 1. A bug begins to crawl up a vertical wire at time t = 0. The velocity of the bug at time t, 0 t 8, is given by the function whose graph is shown. What is the total distance the bug traveled from t = 0 to t = 8? A. 14 B. 13 C. 11 D. 8 E. 6 2. The velocity, in ft/sec, of a particle moving along the x-axis is given b the function v(t) = e t + te t. What is the average velocity of the particle from time t = 0 to time t = 3? A. 20.086 ft/sec B. 26.447 ft/sec C. 32.809 ft/sec D. 40.671 ft/sec E. 79.342 ft/sec 3. If a trapezoidal sum overapproximates, and a right Riemann sum underapproximates, which of the following could be the graph of? 4. The function f is continuous on the closed interval [0, 6] and has the values given in the table. The trapezoidal approximation for with 3 subintervals of equal length is 52. What is the value of k? A. 2 B. 6 x 0 2 4 6 C. 7 f(x) 4 k 8 12 D. 10 E. 14 5. Oil is leaking from a tanker at the rate of = 2000. gallons per hour, where t is measured in hours. How much oil leaks out of the tanker from time t = 0 to t = 10? A. 54 gallons B. 271 gallons C. 865 gallons D. 8647 gallons E. 14778 gallons
6. The area of the region in the first quadrant that is enclosed by the graphs of y = x 3 + 8 and y = x + 8 is A. 0.25 B. 0.5 C. 0.75 D. 1 E. 16.25 7. Which of the following represents the area of the shaded region in the figure below? A. B. C. f (b) f (a) D. E. c 8. What is the value of f ( x ) dx if the area of A = 15 and the area of B = 3? a A. -12 B. 6 C. 12 D. 18 E. 24 9. To the right is the graph of the velocity, in feet per second, of a hat that is thrown up in the air from ground level. Positive velocity means upward motion. About how high is the hat at the top of its flight? A. 30 B. 35 C. 40 D. 45 E. 50 10. What does the following figure represent? A. The left-hand Riemann sum for the function f on the interval 0 t 12 with t = 3. B. The left-hand Riemann sum for the function f on the interval 0 t 12 with t = 6. C. The right-hand Riemann sum for the function f on the interval 0 t 12 with t = 3. D. The right-hand Riemann sum for the function f on the interval 0 t 12 with t = 6.
11. t 0 2 4 6 8 10 12 P(t) 0 46 53 57 60 62 63 The figure to the right shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and height of 4 feet. The pool contains 1000 cubic feet of water at time t = 0. During the time interval 0 t 12 hours, water is pumped into the pool at the rate P(t) cubic feet per hour. The table above gives values of P(t) for selected values of t. During the same time interval, water is leaking from the pool at the rate R(t) cubic feet per hour, where R(t) = 25e -0.05t. (a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval 0 t 12 hours. Show the computation that lead to your answer. (b) Calculate the total amount of water that leaked out of the pool during the time interval 0 t 12. (c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time t = 12 hours. Round your answers to the nearest cubic foot. EXTRA CREDIT!!! (d) Find the rate at which the volume of water in the pool is increasing at time t = 8 hours. How fast is the water level in the pool rising at t = 8 hours? Indicate units of measure in both answers. The volume V of a cylinder with radius r and height h is given by V = πr 2 h.
12. The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The t(minutes) 0 2 5 7 11 12 radius of the balloon, in feet, is modeled by a twicedifferentiable function r of time t, where t is measured in minutes. For 0 < t <12, the graph of r is concave r (t)(feet per minute) 5.7 4.0 2.0 1.2 0.6 0.5 down. The table above gives selected values of the rate of change, r (t), of the radius of the balloon over the time interval 0 t 12. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by.) (a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer. (b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. (c) Use a right Riemann sum with five subintervals indicated by the data in the table to approximate. Using correct units, explain the meaning of in terms of the radius of the balloon. (d) Is your approximation in part (c) greater than or less than answer.? Give a reason for your
13. A particle moves along the x-axis with velocity at time t 0 given by v(t) = -1 + e 1-t. (a) Find the acceleration of the particle at time t = 3 (b) Find all values of t at which the particle changes direction. Justify your answer. (c) Find the total distance traveled by the particle over the time interval 0 t 3.