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First Midterm

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Math 115 — First Midterm — October 8, 2018EXAM SOLUTIONS1. Do not open this exam until you are told to do so.2. Do not write your name anywhere on this exam.3. This exam has 11 pages including this cover. There are 10 problems.Note that the problems are not of equal difficulty, so you may want to skip over and return to aproblem on which you are stuck.4. Do not separate the pages of this exam. If they do become separated, write your UMID (not name)on every page and point this out to your instructor when you hand in the exam.5. Note that the back of every page of the exam is blank, and, if needed, you may use this space forscratchwork. Clearly identify any of this work that you would like to have graded.6. Please read the instructions for each individual problem carefully. One of the skills being testedon this exam is your ability to interpret mathematical questions, so instructors will not answerquestions about exam problems during the exam.7. Show an appropriate amount of work (including appropriate explanation) for each problem, so thatgraders can see not only your answer but how you obtained it.8. The use of any networked device while working on this exam is not permitted.9. You may use any one calculator that does not have an internet or data connection except a TI-92(or other calculator with a “qwerty” keypad). However, you must show work for any calculationwhich we have learned how to do in this course.You are also allowed two sides of a single 300× 500notecard.10. For any graph or table that you use to find an answer, be sure to sketch the graph or write out theentries of the table. In either case, include an explanation of how you used the graph or table tofind the answer.11. Include units in your answer where that is appropriate.12. Problems may ask for answers in exact form. Recall that x =√2 is a solution in exact form to theequation x2= 2, but x = 1.41421356237 is not.13. Turn off all cell phones, smartphones, and other electronic devices, and remove all head-phones, earbuds, and smartwatches. Put all of these items away.14. You must use the methods learned in this course to solve all problems.Problem Points Score1 112 123 64 145 16Problem Points Score6 57 98 109 1110 6Total 100Math 115 / Exam 1 (October 8, 2018) page 21. [11 points] Brianna rides her unicycle north from her home to the grocery store and back again. Thedifferentiable function r(t) represents Brianna’s distance in meters from her home t minutes aftershe leaves the house. Some values of r(t) are shown in the table below.t 0 1 5 7 10 12 14 16 17r(t) 0 180 1050 1420 1425 980 570 220 0a. [2 points] What was Brianna’s average velocity between times t = 7 and t = 12? Include units.Solution: Average velocity=980 − 142012 − 7=5= −88 Answer: −88 meters per minute.b. [2 points] Approximate the value of r0(14). Include units.Solution: r0(14) ≈220 − 5702= −175 Answer: −175 meters per minute.c. [3 points] For which of the following time interval(s) is it possible for r(t) to be concave up onthe entire interval? Circle all correct choices.Solution: Computing average rate of changes in consecutive subintervals we see thatIntervals [1,5] [5,7] [10,12] [12,14]Average rate of change8704= 217.53702= 185 −4452= −222.5 −4102= −205Since the average rate of change only increases on [10, 14], then it is possible that r(t) is concaveup on [10, 14].Use the following additional information about Brianna’s ride to answer the questions below:• The grocery store is 1430 meters away from Brianna’s home.• It takes Brianna 8 minutes to get to the store.• On her way to the store, Brianna does not stop at all. On her way back, she only stops once ata traffic light, which is 250 meters from her home.d. [2 points] For which of the following time interval(s) is r0(t) equal to 0 for some value of t inthat interval? Circle all correct choices.Solution: Based on the information given r0(t) 6= 0 on [1, 5] and [10, 12]. r0(8) = 0 sinceit takes 8 minutes to get to the store. Since she stops on her way back, then r0(t) = 0 for14 ≤ t ≤ 16.[1,5][5,10][10,12][12,16]none of thesee. [2 points] For which of the following time interval(s) is r0(t) negative for some value of t in thatinterval? Circle all correct choices.Solution: The derivative of r(t) is negative on her way back.[1,5][5,10] [10,12] [12,16]none of theseMath 115 / Exam 1 (October 8, 2018) page 32. [12 points] On the axes provided below, sketch the graph of a single function y = Q(x) satisfyingall of the following conditions:• The function Q(x) is defined on −8 ≤ x ≤ 8.• On the interval (3, 8), the function Q(x) is equal to thederivative of the function h(x), which is shown in thegraph at the right.• Q0(−6) = 0 and Q(x) is increasing in −8 < x < −5.• Q(x) is not continuous at x = −5 but limx→−5Q(x) exists.• Q(−2) = 3.• Q(x) has an x-intercept at x = 1.• Q(x) = −Q(−x) for −3 < x < 3.3 4 5 6 7 8−2−1123y = h(x)xyMake sure that your graph is large and unambiguous.Solution:−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8−6−5−4−3−2−1123456xyMath 115 / Exam 1 (October 8, 2018) page 43. [6 points]a. [4 points] For which value(s) of the constant A is the functionR(t) =5(13)Atfor t < 2.20 − 3t2for t ≥ 2.continuous? Find your answer algebraically and give your answer in exact form. If no suchvalue exists, write “DNE”. Show all your work step by step.Solution: The function R(t) is continuous on (−∞, 2) and (2, ∞). In order for R(t) to becontinuous at t = 2, R(t) has to satisfy R(2) = limt→2R(t). Since R(2) = 8 = limt→2+R(t),then it is only necessary that limt→2−R(t) = limt→2−5(13)At= 5(13)2A= 8. This yields5(13)2A= 8132A=85= 1.62A ln(13) = ln (1.6)A =ln (1.6)2 ln(13)Answer: A =ln (1.6)2 ln(13)b. [2 points] A different function, f (d), has the property that limd→∞f(d) = 10. What is the value oflimd→∞4f(2d − 14) + 9?Write “DNE” if the limit does not exist or “NI” if there is not enough information to answerthe question. You do not need to show your work.Solution: Since 2d − 14 tends to infinity as d tends to infinity, then limd→∞f(2d − 14) = 10.Hence limd→∞4f(2d − 14) + 9 = 4(10) + 9 = 49.Answer: 49Math 115 / Exam 1 (October 8, 2018) page 54. [14 points] A portion of the function b(x) is depicted in the graph below. This function is definedfor all real numbers x.−5 −4 −3 −2 −1 1 2 3 4

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