Calculus Test Chapter 5 2015 Name You can use a calculator on the whole test. I know! You re welcome! Each question is worth 4 points. 1. A table of values for a continuous function is shown. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of a. 8 b. 12 c. 16 d. 24 e. 32 X 0 0.5 1.0 1.5 2.0 f(x) 3 3 5 8 13 2. The table gives selected values for a continuous function. If is increasing over the closed interval [0, 3], which of the following could be the value of a. 50 b. 62 c. 77 d. 100 e. 154 X 0 0.5 1 1.5 2 2.5 3 f(x) 0 4 10 18 28 40 54 3. The rate at which water is sprayed on a field is given by =2 1+5, where t is in minutes and R(t) is in gallons per minute. During the time interval 0 t 4, what is the average rate of water flow, in gallons per minute a. 8.458 b. 13.395 c. 14.691 d. 18.916 e. 35.833 4. A particle moves along the x-axis so that its velocity at any time t 0 is given by =5 1. What is the total distance traveled by the particle from t = 0 to t = 4 a. 0.366 b. 0.542 c. 1.542 d. 1.821 e. 2.821 5. A spherical tank contains 81.637 gallons of water at time t = 0 minutes. For the next 6 minutes, water flows out of the tank at the rate of 9 +1 gallons per minute. How many gallons of water are in the tank at the end of the 6 minutes a. 36.606 b. 45.031 c. 68.858 d. 77.355 e. 126.668
6. The graph of a function is shown above. What is the value of a. 6 b. 8 c. 10 d. 14 e. 18 7. What is the area of the region between the graph of y = x 2 and y = -x from x = 0 to x = 2 a. 2/3 b. 8/3 c. 4 d. 14/3 e. 16/3 8. Let and be continuous functions such that =21 What is the value of a. 3 b. 7 c. 11 d. 15 e. 19, =8, and =2. 9. The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown. Of the following, which best approximates the total number of barrels of oil that passed through the pipeline that day a. 500 b. 600 c. 2,400 d. 3,000 e. 4,800
10. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of, each using the same number of subintervals. The graph of the function is shown in the figure. Which of the sums give an underestimate of the value of i. Left sum ii. Right sum iii. Trapezoidal sum a. I only b. II only c. III only d. I and III only e. II and III only 11. A differentiable function f has the property that f(5) = 3 and f (5) = 4. What is the estimate for f(4.8) using the local linear approximation for f at x = 5 a. 2.2 b. 2.8 c. 3.4 d. 3.8 e. 4.6 12. = a. 6 b. 6 c. sin 2 (x 3 ) d. -6 e. -2 13. An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of 2π cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters (Note: For a sphere of radius r, the surface area is 4 πr 2 and the volume is. a. b. 40 π c. 80 π 2 d. 100 π 14. If = +1+ 3, then f has a local maximum at x = a. -2.314 b. -1.332 c. 0.350 d. 0.829 e. 1.234
For 0 t 6, a particle is moving along the x-axis. The velocity of the particle is given by =2 +1. The acceleration of the particle is given by =. 15. Is the speed of the particle increasing or decreasing at time t = 5.5 Give a reason for your answer. 16. Find the average velocity of the particle for the time period 0 t 6. 17. Find the total distance traveled by the particle from time t = 0 to time t = 6. 18. For 0 t 6, when, if any time, does the particle change direction Why
The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W, where W(t) is measured in degrees Fahrenheit and t is measured in minutes. At time t = 0, the temperature of the water is 55 F. The water is heated for 30 minutes, beginning at time t = 0. Values of W(t) at selected times t for the first 20 minutes are given in the table below. t (minutes) 0 4 9 15 20 W(t) (degrees Fahrenheit) 55.0 57.1 61.8 67.9 71.0 19. Use the data in the table to estimate W (12). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. 20. Use the data in the table to evaluate. Using correct units, interpret the meaning of in the context of this problem. 21. For 0 t 20, the average temperature of the water in the tub is. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes Explain your reasoning. 22. For 20 t 25, the function W that models the water temperature has first derivative given by =0.4 0.06. Based on the model, what is the temperature of the water at time t = 25
On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by =90+45, where t is measured in hours and 0 t 8. At the beginning of the workday (t = 0), the plant has 500 tons of unprocessed gravel. During the hours of operation, 0 t 8, the plant processes gravel at a constant rate of 100 tons per hour. 23. Find G (5). Using correct units, interpret your answer in the context of the problem. 24. Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. 25. Is the amount of unprocessed gravel at the plant increasing or decreasing at time t = 5 hours Show the work that leads to your answer.