Math 116 Practice for Exam 1

Transcription

1 Math 116 Practice for Exam 1 Generated October 7, 2016 Name: Instructor: Section Number: 1. This exam has 10 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 Fall garden 14 Winter island 12 Winter Winter Winter Winter sand dune 12 Fall bucket 16 Winter bullet 10 Fall ditch 14 Winter shark 9 Total 124 Recommended time (based on points): 112 minutes

2 Math 116 / Exam 1 (October 12, 2011) page 7 6. [14 points] A botanical garden has the shape of the region in the xy-plane bounded by the curve y = x 2 and the x-axis, with 0 x 8. One of the responsibilities of the gardener, is to keep the garden free of a poisonous weed. The density δ of the weed at any point in the garden depends on the distance x from the y-axis. Values of δ are given in kg of plants per meter square in the table below. x (meters) δ(x) a. [3 points] Write an integral that computes the total amount of weed in the garden. Include units. b. [3 points] Compute RIGHT(4) for the integral in (a). Write out all the terms in the sum. Does this sum give an overestimate or an underestimate for the total amount of weed in the garden? Justify. Fall, 2011 Math 116 Exam 1 Problem 6 (garden)

3 Math 116 / Exam 1 (October 12, 2011) page 8 c. [2 points] Which of the following approximations to (a) are computable with the given data? Circle all that apply. MID(1) MID(2) MID(3) MID(4) d. [1 point] Which Riemann sum gives the best estimate for the integral in (a)? Circle one. RIGHT(4) LEFT(4) TRAP(4) e. [5 points] The gardener built a fence around the garden. How long is the fence? Include units. You may use your calculator. Fall, 2011 Math 116 Exam 1 Problem 6 (garden)

4 Math 116 / Exam 1 (February 6, 2012) page 4 3. [12 points] A boat travels in a straight line toward an island d km away. The velocity v(t) (toward the island is positive velocity) in km/hr, t hours after departing from its starting position. The velocity v(t) during the first three hours is recorded at half hour intervals, and is given in the table below: t v(t) a. [8 points] Find an estimate for how far the boat is from the starting point after 3 hours using the four approximations LEFT, RIGHT, MID and TRAP. Use the maximum number of subintervals possible. Write each sum, and justify whether the sum is an underestimate or an overestimate. Assume the velocity is always decreasing and has no inflection points. Circle your answers. b. [4 points] If it takes the boat 5 hours to reach the island, write an expression involving integrals for the distance S(t) between the island and the boat for 0 t 5. Winter, 2012 Math 116 Exam 1 Problem 3 (island)

5 Math 116 / Exam 1 (February 2011) page 4 3. [15 points] Use the graphs of f(x) and g(x) to find the EXACT values of A,B, and C. Show all your work f(x) 2 1 g(x) Area = 21 4 a. [2 points] A = 6 3 f(x) dx b. [5 points] B = 2 0 xg (x 2 )dx c. [8 points] C = 3 0 2xg (x)dx Winter, 2011 Math 116 Exam 1 Problem 3

6 Math 116 / Exam 1 (February 10, 2014) page 6 5. [10 points] Suppose that f(x) and g(x) are twice differentiable functions defined for all x with the following properties: f(0) = g(0) and f(1) = g(1). f(x) and g(x) are increasing. f(x) is concave down and g(x) is concave up. For each of the following questions, circle the correct answer. No justification is necessary. a. [2 points] Which is larger, 0 f(x)dx or 0 g(x)dx? 0 f(x)dx 0 g(x)dx Equal Impossible to determine b. [2 points] Which is larger, 0 f(x) dx or 0 g(x) dx? 0 f(x) dx 0 g(x) dx Equal Impossible to determine c. [2 points] Which is larger, 0 f (x)dx or 0 g (x)dx? 0 f (x)dx 0 g (x)dx Equal Impossible to determine d. [2 points] Which is larger, 0 xf (x)dx or 0 xg (x)dx? 0 xf (x)dx 0 xg (x)dx Equal Impossible to determine e. [2 points] Which is larger, 0 f(x)f (x)dx or 0 g(x)g (x)dx? 0 f(x)f (x)dx 0 g(x)g (x)dx Equal Impossible to determine Winter, 2014 Math 116 Exam 1 Problem 5

7 Math 116 / Exam 1 (February 6, 2012) page 6 5. [12 points] The graph of f(x) and a table of values for the continuous functions g(x) and h(x) are given below. The function h(x) is an antiderivative of g(x). 1 y 2 1 y f x x Compute the exact value of each of the following expressions: x g(x) h(x) a. [1 point] 7 0 f(x) dx b. [4 points] e 2 1 f(ln x) x dx c. [7 points] Find 2 1 xg (2x)dx Winter, 2012 Math 116 Exam 1 Problem 5

8 Math 116 / Exam 1 (February 2011) page [12 points] Sand dunes come in many shapes. Barchan dunes, which have the shape shown on the left, are studied extensively by geomorphologists. Horizontal cross-sections of these dunes are crescent-shaped (the dashed line encloses one such cross-section), and can be approximated as the shape on the right. The area of this shape is given by the formula A h = K( π 2 Q Q 1). h Q 1 K Q 2 You are studying a barchan dune of 10 meters height, for which the values of Q 1, Q 2, and K vary with respect to the height h (in meters) of the cross-section according to the functions Q 1 (h) = 10 h, Q 2 (h) = 20 2h, K(h) = 100 h 2. The density of sand in the dune is δ = 1600 kilograms per cubic meter. a. [5 points] Write an expression for the volume of one slice of sand dune h meters above the ground and h meters thick. b. [5 points] Write a definite integral that represents the total mass of sand in the dune. You do not need to evaluate this integral. c. [2 points] Write an expression (involving integrals) for the height of the center of mass of the sand dune. You do not need to evaluate this integral. Winter, 2011 Math 116 Exam 1 Problem 8 (sand dune)

9 Math 116 / Exam 2 (November 12, 2008) page 5 4. [16 points] An entrepreneurial University of Michigan Business Squirrel is marketing childrens buckets with curved sides, as shown in the figure to the right, below. The figure gives the radius of the bucket, r, at different heights, z, from the bottom of the bucket. All lengths are given in inches. Suppose that a child fills one of these buckets with muddy water. a. [4 points] If the density of the water in the bucket is δ(z) oz/in 3, write an integral that gives the mass of the water in the bucket. z = 8, r = 6 z = 6, r = 3 z = 4, r = 2.25 z = 2, r = 1.5 z = 0, r = 1 b. [4 points] If δ(z) = (24 z) oz/in 3, estimate the mass using your integral from (a). c. [8 points] Estimate the center of mass of the bucket. Fall, 2008 Math 116 Exam 2 Problem 4 (bucket)

10 Math 116 / Exam 1 (February 6, 2012) page 9 8. [10 points] A bullet, with constant density δ = 3 g/cm 3, has the shape of a solid generated by rotating the region enclosed by piece of the ellipse x 2 +4y 2 = 4 lying in the first quadrant, the x-axis and the y-axis, around the x-axis. The variables x and y are measured in cm. y 1 x 2 4 y 2 = x a. [5 points] Write an expression that approximates the mass of one slice of the bullet perpendicular to the x-axis at x cm from the y-axis and x cm thick. Let ( x,ȳ, z) be the coordinates of the bullet s center of mass. b. [3 points] Find a formula for x. c. [2 points] What can you say about ȳ. Justify. Winter, 2012 Math 116 Exam 1 Problem 8 (bullet)

11 Math 116 / Exam 1 (October 8, 2014) page 5 4. [14 points] Franklin, your robot, has been digging a ditch in the backyard. The ditch is a trapezoidal prism with length 7 feet, width at the bottom 3 feet and width at the top 4 feet. The ditch descends 6 feet underground. You may assume that each cubic foot of dirt weighs 94 pounds. Be sure to include units in your answers. 4ft. h 6ft. 3ft. a. [5 points] Write an expression that approximates the work that Franklin does lifting a slice of dirt h feet thick from h feet below the top of the ditch, to the top of the ditch. 7 ft. b. [3 points] Using your answer to part (a), write an integral that computes the total work required to move all of the dirt out over the top of the ditch. c. [3 points] To lift the dirt, Franklin uses your old rusty bucket. For Franklin s last bucketfull of dirt, the bucket starts with 30 pounds of dirt and loses 10 pounds of dirt at a constant rate over the 6 feet that it travels vertically. When the bucket is x feet above the bottom of the ditch, what is the weight of the dirt in the bucket? d. [3 points] Using your answer from part (c), find the work required to lift the bucket-full of dirt from the bottom of the ditch to the top. Fall, 2014 Math 116 Exam 1 Problem 4 (ditch)

12 Math 116 / Exam 1 (February 9, 2015) page 9 8. [9 points] Sally, the marine scientist, is reeling in a large shark she caught onto her boat. The edge of her boat lies 5 meters above the water as shown in the figure below. The total length of the sharking line is 30 meters. The shark weighs 500 newtons in water, and her sharking line weighs 30 newtons per meter out of water, and 10 newtons per meter in water. The figure below depicts this situation - the sharking line is the thick dark line and the boat is shaded. Write an expression which gives the work Sally does pulling the shark s snout to the surface of the water. 5 m 30 m Winter, 2015 Math 116 Exam 1 Problem 8 (shark)