Name: ID#:Midterm IMath 1aIntroduction to Calculus27 October 2004Show all of your work. Full credit may not be given for an answer alone.You may use the backs of the pages or the extra pages for scratch work. Do notunstaple or remove pages.This is a non-calculator exam.Please check your section: 1.0 MWF10 Tatyana Chmutova 4.0 TΘ10 Dawei Chen 1.1 MWF10 Matthew Leingang 4.1 TΘ10 Jerrel Mast 2.0 MWF11 Ethan Cotterill 4.2 TΘ10 Chun-Chun Wu 3.0 MWF12 Matt Bainbridge 5.0 TΘ11:30 Derek Bruff 5.1 TΘ11:30 Sonal JainStudents who, for whatever reason, submit work not their own willordinarily be required to withdraw from the College.—Handbook for StudentsProblem Possible PointsNumber Points Earned1 122 103 104 155 106 87 158 109 10Total 1001 11. (12 Points) Find the following limits.(i) limx→1x19+ x − 1arctan(x)(ii) limx→∞px4+ 19 − x211 1(iii) limx→19x3− 19x2− x + 19x − 19/ 1222 22. (10 Points) We will show thatlimx→∞sin(x19)log x(a) (7 points) Find two functions f and g such thatf(x) ≤sin(x19)log x≤ g(x)for all x > 0 andlimx→∞f(x) = 0 limx→∞g(x) = 0.Justify your answers.(b) (3 points) Finish up the proof with two magic words./ 1033 33. (10 Points) Define the signum functionsgn(x) =0 if x = 0;|x|xif x 6= 0.(i) (2 points) What is sgn(2)? sgn(−5)?(ii) (8 points) Where is sgn continuous? Use limits to justify your answer./ 1044 44. (15 Points) Determine the vertical and horizontal asymptotes for each of thefollowing functions. For each vertical asymptote, also determine the left-handand right-hand limits at that asymptote. (For example, if x = 4 is a verticalasymptote for f, find limx→4−f(x) and limx→4+f(x).)(a) f(x) =5x2− 452x2− 7x + 3(b) f(x) =√x2+ 13x − 154 4(c) f(x) = e2xcos x/ 1565 55. (10 Points) Let f (x) =√2x. Compute f0(x) using the definition of deriva-tive./ 1076 66. (8 Points) Below are the graphs of f and its derivatives f0, f00, and f000. Inthe blanks below the graphs, write the letter of the graph that corresponds toeach of the functions f , f0, f00, and f000.-2-1 12-4-22-2-1 120.511.522.5-2-1 12-2.5-2-1.5-1-0.50.51-2-1 12-1-0.50.511.5(A)-2-1 12-4-22-2-1 120.511.522.5-2-1 12-2.5-2-1.5-1-0.50.51-2-1 12-1-0.50.511.5(C)-2-1 12-4-22-2-1 120.511.522.5-2-1 12-2.5-2-1.5-1-0.50.51-2-1 12-1-0.50.511.5(B)-2-1 12-4-22-2-1 120.511.522.5-2-1 12-2.5-2-1.5-1-0.50.51-2-1 12-1-0.50.511.5(D)ff0f00f000/ 887 77. (15 Points) Take the following derivatives.(i)ddxx17+ 3x4+ 2x − 1(ii)ddxx1 + x297 7(iii)ddxex7x2+ ex/ 15108 88. (10 Points) The profits of a small company for each of the first five years ofits operation are given in the following table.Year Profit in $1000s2000 102001 642002 1012003 1092004 86The company’s accountant plotted p oints representing the profit as a functionof years since 2000 and joined the points by a smooth curve. This graph isshown below.12 3 420406080100(i) Find the slope of the secant line between the points (2, 101) and (4, 86).What does this slope represent in terms of profit?(ii) Use the accountant’s graph to estimate the rate at which profits werechanging in 2002. Clearly describe the method you use to do so./ 10119 99. (10 Points) Ferdbert Freshman decides to go on a hike. On a Saturdaymorning he leaves at 8:00am and climbs Mt. Pennypacker, arriving at 5:00pm.He spends the night at the top of the mountain. The next day, he climbs downthe mountain, arriving back at his dorm room at 5:00pm.Upon Ferdbert’s return, his roommate Egbert (taking a break from studyinghis calculus) says, “Did you know that at some time today you were at theexact same elevation as you were 24 hours before?” “That’s impossible,” saysFerdbert. “I slept late this morning and didn’t start walking until 10:00am!”Who’s right, and why?Hint. Let f be the function which is Ferdbert’s elevation at time t on Satur-day, and g Ferdbert’s elevation at time t on Sunday. Let f (8:00am) = 0 andf(5:00pm) = M. What are g(8:00am) and g (5:00pm)?/ 1012(This page intentionally left
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