Math 116 Practice for Exam 2

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1 Math 6 Practice for Exam 2 Generated November 9, 206 Name: Instructor: Section Number:. This exam has 0 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 heart 2 Winter ant 0 Fall Winter can opener 6 Winter Leia Han Luke 3 Winter pond rock 0 Fall Fall Winter Fall spirals 0 Total 04 Recommended time (based on points): 96 minutes

2 Math 6 / Exam 2 (March 20, 203) page 7 5. [2 points] A particle moves according to the following parametric equations x = x(t) and y = y(t) for 2 t 2, where the graphs of x(t) and y(t) are shown below. a. [2 points] Is there a value of t at which the particle is at the point (0,2)? If so, find the value of t where this happens. b. [3 points] At which value(s) of t does the particle on the x-axis? c. [4 points] At what points (x,y) does the curve traveled by the particle have a horizontal tangent line? Include the time of each point. d. [3 points] For which of values of t is the slope of the tangent line to the curve positive? University of Michigan Department of Mathematics Winter, 203 Math 6 Exam 2 Problem 5 (heart)

3 Math 6 / Exam 2 (March 20) page 7 5. [3 points] When an ant finds food, it leaves a trail of chemicals from its nest to the food source. Other ants follow this chemical trail using their two antennae. When an ant possesses both antennae, it will walk in a straight line to the food. If you remove (amputate) the left antenna of an ant, it will walk in a pattern like the one shown in the second figure. If you cross the ant s left antenna with its right antenna and glue them in that position, the ant walks erratically but eventually reaches the food, as in the third figure. y y y food (a,b) food (a,b) food (a,b) nest Healthy x nest Amputated x nest Crossed Antennae x a. [4 points] Write a parametric equation for the path of a healthy ant that starts at its nest at (0, 0) and arrives at the food at (a, b). b. [3 points] Suppose the parametric equation for the amputated ant is given by x = x(t) y = y(t). Assume the ant starts walking at t = 0, arrives at the food at t =, and never pauses or backtracks. For each blank below, determine whether the number on the left is greater than, less than, or equal to the number on the right, and fill the blank with the symbol >, <, or = respectively. Justify your answers. (x (t)) 2 +(y (t)) 2 dt a 2 +b 2 0 x (c) (for any 0 <c<) 0

4 Math 6 / Exam 2 (March 20) page 8 (problem 5 continued) c. [6points] Supposethepathoftheantwithcrossedantennaeisdescribedbytheparametric equation x = f(t) y = g(t) and that the ant reaches the food at t = 0. Sketch a graph of f(t). The graph of its path is below, with its position after 5 seconds marked. Assume that the ant walks at a constant speed during its journey (it never backtracks or pauses along the way). f(t) a 5 0 t y food (a,b) t=5 nest x Crossed Antennae

5 Math 6 / Exam 2 (November 4, 202) page 8 6. [3 points] A particle moves along the path given by the parametric equations x(t) = acost y(t) = sin2t for 0 t 2π. where a is a positive constant. The graph of the particle s path in the x-y plane is shown below. In the questions below, show all your work to receive full credit. a. [2 points] At which values of 0 t 2π, does the particle pass through the origin? b. [5 points] For what values of a are the two tangent lines to the curve at the origin perpendicular? Hint: Two lines are perpendicular if the product of their slopes is equal to. c. [4 points] At what values of 0 t 2π, does the curve have horizontal tangents? d. [2 points] Find an expression that computes the length of the curve. University of Michigan Department of Mathematics Fall, 202 Math 6 Exam 2 Problem 6

6 Math 6 / Final (April 2, 206) DO NOT WRITE YOUR NAME ON THIS PAGE page 9 9. [6 points] O-guk is creating a can opener to open his many cans of juice. The opener is in the shape of the shaded region enclosed by the two loops of the polar curve r = 2sin(θ)+ and the x- and y-axes. y r = 2sin(θ)+ x Write an expression involving integrals that gives the total area of the shaded region.

7 Math 6 / Exam 2 (March 2, 206) DO NOT WRITE YOUR NAME ON THIS PAGE page 3 2. [3 points] Leia and Han are imprisoned in a cell whose door is made out of steel and has a thickness of 3 feet. Luke uses his lightsaber to cut through the door in the shape of the curve given by the polar coordinates equation r = 5 3+2cos ( θ + π 4 ) where r is measured in feet. a. [6 points] Write an expression involving integrals for the volume of the piece that Luke cuts out of the door. b. [7 points] Still considering the polar curve r = 5 3+2cos ( θ + π 4 ) graphed in the xy-plane, write an explicit expression involving integrals for the length of the part of the curve that lies to the right of the y-axis. University of Michigan Department of Mathematics Winter, 206 Math 6 Exam 2 Problem 2 (Leia Han Luke)

8 Math 6 / Exam 2 (March 24th, 204) page 8 9. [0 points] Linda is designing a pond with a flat rock at one end. The rock plus the pond are in the shape of a cardioid. Plans for her pond design are depicted below. The cardioid has equation r = 20+40sinθ where r is in feet and θ is in radians. The inner loop of the cardioid forms the shape of the rock and the outer loop forms the boundary of the pond a. [2 points] Find all values of θ between 0 and 2π for which r = 0. b. [4 points] Write an integral or sum of integrals which give(s) the perimeter of the boundary of the pond. Note this is the perimeter of the part of the cardioid drawn with a solid line. c. [4 points] Write an integral or sum of integrals which give(s) the area of the top of the rock. Note this is the area enclosed by the dashed part of the cardioid. University of Michigan Department of Mathematics Winter, 204 Math 6 Exam 2 Problem 9 (pond rock)

9 Math 6 / Exam 2 (November 2, 204) page 0 9. [0 points] Determine whether the following integrals converge or diverge. Justify your answer. x 2 a. [5 points] e x +3x 3dx b. [5 points] ( Hint : 0 d dx xcos(x) sin(x) x 2 dx ( ) sin(x) x = xcos(x) sin(x) ) x 2. University of Michigan Department of Mathematics Fall, 204 Math 6 Exam 2 Problem 9

10 Math 6 / Exam 2 (November 2, 204) page 8 7. [8 points] Suppose that f(x) is a differentiable function, defined for x > 0, which satisfies the inequalities 0 f(x) x for x > 0. Determine whether the following statements are always, sometimes or never true by circling the appropriate answer. No justification is necessary. a. [2 points] f(x)dx converges. Always Sometimes Never b. [2 points] (f(x)) 2 dx converges. Always Sometimes Never c. [2 points] 0 f(x)dx converges. Always Sometimes Never d. [2 points] e f(x) dx converges. Always Sometimes Never University of Michigan Department of Mathematics Fall, 204 Math 6 Exam 2 Problem 7

11 Math 6 / Exam 2 (March 24th, 204) page 9 0. [2 points] Suppose that g(x) and h(x) are positive continuous functions on the interval (0, ) with the following properties: g(x) dx converges. 0 g(x)dx diverges. e x h(x) x for all x in (0, ). For each of the following questions, circle the correct answer. a. [2 points] Does the integral h(x) 2 dx converge? Converge Diverge Cannot determine b. [2 points] Does the integral 0 h(x) dx converge? Converge Diverge Cannot determine c. [2 points] Does the integral h(/x) dx converge? Converge Diverge Cannot determine d. [2 points] Does the integral 0 g(x)h(x) dx converge? Converge Diverge Cannot determine e. [2 points] Does the integral g(x)h(x) dx converge? Converge Diverge Cannot determine f. [2 points] Does the integral e x g(e x )dx converge? Converge Diverge Cannot determine University of Michigan Department of Mathematics Winter, 204 Math 6 Exam 2 Problem 0

12 Math 6 / Exam 2 (November 6, 20) page 9 8. [0 points] For α > 0, consider the family of spirals given by r = θ α in polar coordinates. a. [2 points] Write down an integral that gives the length L of a spiral in this family for θ b. No credit will be given if you just write down the formula given in part (b). b. [8 points] It can be shown that the length L of the spiral in part a) may also be written as b L = θ α + α2 θ 2dθ. Use this formula for L to find all values of α > 0 for which the length of the spiral is infinite for θ. For which values of α is the length finite? Justify all your answers using the comparison test. University of Michigan Department of Mathematics Fall, 20 Math 6 Exam 2 Problem 8 (spirals)

Math 116 Practice for Exam 2

Math 116 Practice for Exam 2 Math 116 Practice for Exam Generated October 1, 015 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 7 questions. Note that the problems are not of equal difficulty, so you may want to skip