Math 116 Practice for Exam 1

Transcription

1 Math 116 Practice for Exam 1 Generated September 4, 17 Name: Instructor: Section Number: 1. This exam has 5 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.. 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-9 (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 Rodin Coil 1 Fall Winter jewelry 1 Fall thumbtack doorknob 8 Fall Total 49 Recommended time (based on points): 44 minutes

2 Math 116 / Exam 1 (February 6, 17) do not write your name on this exam page 7 6. [1 points] The Rodin Coil is a fantastic device that (supposedly) creates unlimited free energy. The rate at which it creates this energy is a function of the volume of the coil. a. [5 points] Suppose a prototype of the Rodin Coil is the solid whose base is the circle x +y = (where x and y are measured in meters), and whose cross sections perpendicular to the x axis are squares. Write, but do not compute, an expression involving one or more integrals which gives the volume, in cubic meters, of this prototype. b. [5 points] One of Rodin s students was able to come up with an even more efficient free energy machine. Suppose the student s prototype was made by taking the same circle x +y = and rotating it around the vertical line x = 3. Write, but do not compute, an expression involving one or more integrals which gives the volume, in cubic meters, of this prototype. University of Michigan Department of Mathematics Winter, 17 Math 116 Exam 1 Problem 6 (Rodin Coil)

3 Math 116 / Midterm (October 1, 16) DO NOT WRITE YOUR NAME ON THIS PAGE page 8 6. [1 points] a. [3 points] Let f be a positive, continuous function. Which of the following are antiderivatives of f whose graphs go through the point (1,)? Circle all that apply. 1 f(t)dt x f(t)dt+ 1 f(t)dt x f(t)dt x f(t/)dt 1 x f(t/)dt b. [3 points] Let R be the region between the x-axis and the graph of some positive, continuous function from x = a to x = b. If V is the volume of the solid whose base is R and whose cross-sections parallel to the y-axis are semicircles, what is the volume of the solid whose base is R and whose cross-sections parallel to the y-axis are equilateral triangles? 3 4 V 3 π V 4 3 π V πv π V 3 c. [3points] Whichofthefollowingexpressionsgivesthearclengthofthegraphofy = sin(x ) from x = to x = π? π 1+x sin (x )dx π 1+sin (x )dx π 1+4x cos(x )dx π 1+cos (x )dx π 1+4x cos (x )dx d. [3 points] If the average value of a continuous function is A on [,3] and B on [3,5], what is its average value on [,5]? A+3B A+3B 5 A+B 3A+B 5 A+B 5 University of Michigan Department of Mathematics Fall, 16 Math 116 Exam 1 Problem 6

4 Math 116 / Exam 1 (February 8, 16) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 7. [1 points] Maize and Blue Jewelry Company is trying to decide on a design for their signature amaize-ing bracelet. There are two possible designs: type W and type J. The company has done research and the two bracelet designs are equally pleasing to customers. The design for both rings starts with the function C(x) = cos ( π x) where all units are in millimeters. Let R be the region enclosed by the graph of C(x) and the graph of C(x) for 1 x 1. a. [5 points] The type W bracelet is in the shape of the solid formed by rotating R around the line x = 5. Write an integral that gives the volume of the type W bracelet. Include units. b. [5 points] The type J bracelet is in the shape of the solid formed by rotating R around the line y = 5. Write an integral that gives the volume of the type J bracelet. Include units. University of Michigan Department of Mathematics Winter, 16 Math 116 Exam 1 Problem 7 (jewelry)

5 Math 116 / Exam 1 (October 14, 15) DO NOT WRITE YOUR NAME ON THIS PAGE page [8 points] The graph of part of a function g(x) is pictured below. 3 g(x) (1, 1 ) a. [4 points] A thumbtack has the shape of the solid obtained by rotating the region bounded by y = g(x), the x-axis and y-axis, about the y-axis. Find an expression involving integrals that gives the volume of the thumbtack. Do not evaluate any integrals. 7 x b. [4 points] A door knob has the shape of the solid obtained by rotating the region bounded by y = g(x), the x-axis and y-axis, about the x-axis. Find an expression involving integrals that gives the volume of the door knob. Do not evaluate any integrals. University of Michigan Department of Mathematics Fall, 15 Math 116 Exam 1 Problem 1 (thumbtack doorknob)

6 Math 116 / Exam 1 (October 14, 15) DO NOT WRITE YOUR NAME ON THIS PAGE page 4 3. [9 points] For each of the following questions, fill in the blank with the letter corresponding to the correct answer from the bottom of the page. No credit will be given for unclear answers. a. [3 points] The length of the curve y = πe x between x = and x = 4 is: b. [3 points] Consider the region bounded by the x-axis, y = e x, x =, and x = 4. The volume of the solid whose base is that region and whose cross sections perpendicular to the x-axis are semicircles is: c. [3 points] The volume of the solid obtained by rotating the region bounded by y = e x, y = e x, x = and x = 4, around the y-axis is: π (A) e 4x dx (F) π 1+e 4x dx (B) (C) (D) (E) π(e x 4e 4x )dx πx(e x +e x )dx π 1+e x dx πe 4x dx π (G) e x dx (H) (I) (J) 1+π e x dx π(e x e 4x )dx π(e x +e x )dx University of Michigan Department of Mathematics Fall, 15 Math 116 Exam 1 Problem 3