Math 116 Practice for Exam 1

Transcription

1 Math 116 Practice for Exam 1 Generated September 3, 218 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 4 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 2. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam. 3. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter tank 11 Fall spring 11 Fall olive oil 13 Winter sand 9 Total 44 Recommended time (based on points): 5 minutes

2 Math 116 / Exam 1 (February 6, 212) page 7 6. [11 points] The lateral faces of a tank are determined by the curve y = 1 x 4 and the x-axis (where x and y are measured in meters). The length of the tank is 1 meters. Be sure to include units in your answers. 1 y y 1 x x a. [5 points] The tank is filled with water to a height of one half a meter. If the density of water is 1, kg/m 3, write an expression that approximates the mass of one slice of water y meters above the ground and y meters thick. Solution: Mass is density times volume. The density is constant, and the volume of one slice at height y with thickness y is Therefore, the mass of the slice is 2(1 y) 1/4 y m 3. 2,(1 y) 1/4 y kg. b. [2 points] Write a definite integral that represents the total mass of water in the tank. Solution: We add up all of the slices from to.5:.5 2,(1 y) 1/4 dy kg. c. [4 points] Write a definite integral that represents the amount of work required to pump the water to the top of the tank. Solution: Work is force times distance. The force on a slice at height y with thickness y (or dy) is 9.8 times the mass, which we computed above. This slice travels a distance (1 y) under this force. Therefore the work done on one slice is So we add up all of those slices to get ,(1 y) 1/4 (1 y) y N ,(1 y) 1/4 (1 y) dy J or N m. Winter, 212 Math 116 Exam 1 Problem 6 (tank) Solution

3 Math 116 / Final (December 14, 217) do not write your name on this exam page 8 8. [11 points] Imagine that a one pound ball is attached to a spring. This ball is allowed to move forward and backward on a table, but not up and down (or side to side). When the spring is not stretched at all, we say that the ball is at its starting position. Let x be the displacement of the ball from its starting position in the forward/backward direction. (The value of x is positive if the ball has moved forward from its starting position and negative if the ball has moved backward from its starting position.) a. [4 points] Let F(x) be the magnitude of the force, measured in pounds, that the spring exerts on the ball when the ball has been pulled x feet from its starting position. Suppose F(x) = 5x. i. Which of the following best estimates the work, in foot-pounds, needed to move the ball a very small distance x feet forward from a position x? Circle one choice. I. 5 II. 5x III. 2.5x 2 IV. 5 x V. 5x x VI. 2.5x 2 x ii. Use your answer to part i. to write an expression involving one or more integrals that gives the total work needed to move the ball from its starting position forward a distance of one half of one foot (i.e. 6 inches). Then compute the value of your integral (either by hand or using your calculator). Include units on your answer. Answer: Integral Expression: Numerical Answer (with units): b. [4 points] After stretching the spring as described above, you release it from a starting position of x = 1/2. The ball oscillates backwards and forwards (in the x-direction), and its position x = x(t) satisfies the differential equation x +5x =. Note that x = d2 x. dt 2 For what values of A, B, and k will the function x(t) = Asin(kt)+Bcos(kt) be a solution to the differential equation x +5x = with the initial conditions x() = 1/2 and x () =? Answer: A = and B = and k = c. [3 points] Using the particular solution that you found in part b, find the first time t > when the ball reaches the position x =. Answer: t = Fall, 217 Math 116 Exam 3 Problem 8 (spring) Solution

4 Math 116 / Final (December 17, 213) page [13 points] Olive oil have been poured into the Math Department s starfish aquarium! The shape of the aquarium is a solid of revolution, obtained by rotating the graph of y = x 4 for x 1 around the y-axis. Here x and y are measured in meters. The aquarium contains water up to a level of y =.6 meters. There is a layer of oil of thickness.2 meters floating on top of the water. The water and olive oil have densities 1 and 8 kg per m 3, respectively. Use the value of g = 9.8 m per s 2 for the acceleration due to gravity. a. [6 points] Give an expression involving definite integrals that computes the total mass of the water in the aquarium. Solution: Mass water =.6 π( 4 y) 2 (1)dy =.6 π y(1)dy b. [7 points] Give an expression involving definite integrals that computes the work necessary to pump all the olive oil to the top of the aquarium. Solution: Work oil =.8.6 π( 4 y) 2 (8)(9.8)(1 y)dy =.8.6 π y(8)(9.8)(1 y)dy Fall, 213 Math 116 Exam 3 Problem 9 Solution

5 Math 116 / Final (April 23, 215) page 7 7. [9 points] Gwen lifts a bucket of sand straight up from the ground to a height of 1 meters at a constant speed of.5 meters per second. The sand is leaking out of the bucket at a rate of r(t) = 1 t+1 kilograms per second, t seconds after she begins lifting. The bucket and the sand in the bucket together weigh 1 kg when she starts lifting. Recall the gravitational constant is g = 9.8 m/s 2. a. [4 points] Suppose M(x) is the mass of the bucket of sand (in kilograms) when she has lifted it x meters from the ground. Find an expression involving integrals for the work Gwen does lifting the bucket. Your answer can include the function M. Solution: The work Gwen does lifting the bucket is 1 M(x)gdx. b. [5 points] Find an expression, possibly involving integrals, for M(x), the mass of the bucket of sand after Gwen has lifted it x meters. Solution: The mass of the bucket after Gwen has lifted it x meters is M(x) = 1 x 2 2s+1 ds. Winter, 215 Math 116 Exam 3 Problem 7 (sand) Solution