Name: ID#:Final ExamMath 1aIntroduction to Calculus21 January 2005Show 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 182 83 204 85 156 187 188 89 810 1511 14Total 1501 11. (18 Points) Compute the following limits, with justification.(i) limx→∞√x2+ 1x2+ 7(ii) limx→π/ 2+tan(x)11 1(iii) limx→1x − 1ex− 1(iv) limx→02 sec(x) − 2 − x2x2/ 1822 22. (8 Points) Let f be the function f (x) = sin(x) −x3.(a) Use the Intermediate Value Theorem to show that there exists a point c inπ2, πsuch that f(c) = 0.(b) Suppose that there was another point d in the sam e interval such thatf(d) = 0. What fact would the Mean Value Theorem allow you to conclude?Why is this “fact” impossible?Once you have shown this, you will have prove n that c is the unique solutionin the interval to the equation f (x) = 0./ 833 33. (20 Points) Find the following derivatives.(i)ddx(3x2+ 4x + 6)(ii)ddx2x1 + ln x43 3(iii)ddxcosx1/3(iv)ddxx√x/ 2054 44. (8 Points)(a) Write down the function which is the linear approximation to the squareroot function at a =94.(b) Use this function to approximate√2./ 865 55. (15 Points) The relation y2= x(x − 1)2defines a curve in the plane.0.51 1.52-1.5-1-0.50.511.5(i) Solve for y in terms of x, and use it to finddydxat the point14, −38.75 5(ii) Finddydximplicitly in terms of y and x. What is its value at14, −38?(iii) A parametrization of the curve is given byx(t) = t2y(t) = t(t2− 1).Finddydxin terms of t. What is its value at t =12? (This corresponds tothe point14, −38)./ 1586 66. (18 Points) The dreaded graphing problem. Letf(x) =1x + 1−1(x + 1)2.(a) Find all horizontal and vertical asymptotes of f .96 6(b) The derivative of f isf0(x) = −1(x + 1)2+2(x + 1)3.Find the intervals of increase or decrease.Increasing on :Decreasing on :(c) Find any local maxima or minima.106 6(d) The second derivative of f isf00(x) =2(x + 1)3−6(x + 1)4Find the intervals of concavity.Concave up on :Concave down on :(e) Find any inflection point(s).116 6(f) Sketch the graph of f . Label all the significant points you have foundpreviously.(g) Find the global minimum and maximum, if they exist./ 18127 77. (18 Points) Apex Corporation is planning to sell sponges on television. Thesponges cost $2 per package. Stacy believes that if they set the selling price at$20 per package, they will sell 1000 packages, and for every dollar they increasethe price, the quantity they will sell will decrease by 50 packages.We will find the price at which profit will be maximized.(a) Assuming Stacy’s assumptions about the market are true, show that thedemand curve (price in terms of quantity sold) is given byp(x) = 40 −150x.How many packages will be sold if the price is set at $27?137 7(b) Now show that the profit (this is revenue minus costs, remember) is givenbyK(x) = −150x(x − 1900)(c) What price maximizes profit? Make sure you show it’s maximal and notminimal!/ 18148 88. (8 Points) Ferdbert Freshman is studying for his Math 1a Final. He startsstudying at midnight and does problems at the rate ofr(t) =60π(t2+ 1)problems per hour, where t is measured in hours after midnight. How manyproblems has he done by 1:00am?/ 8159 99. (8 Points) Evaluate the following definite integrals.(i)Z41(2x + x2) dx(ii)Ze1711xdx/ 81610 1010. (15 Points) Compute the following integrals. For definite integrals, youranswer should be a number. For indefinite integrals, your answer should be themost general antiderivative as a function of x.(i)Z3xpx2+ 1dx(ii)Zln xxdx1710 10(iii)Z10e4x1 + e4xdx/ 151811 1111. (14 Points) Suppose that f is the differentiable function shown in the graphbelow2468-2-11234H1,1LH2,2LH3,3LH5,2L(The function is a straight line from (0, 0) to (3, 3), and is differentiable at x = 4,even though the graph looks a little pointy.) Suppose the the position at timet seconds of a particle moving along a coordinate axis iss(t) =Zt0f(x) dxmeters. Use the graph to answer the following questions. Give reasons for youranswers.(i) What is the particle’s velocity at time t = 5?(ii) Is the acceleration of the particle at time t = 5 positive or negative?(iii) What is the particle’s position at time t = 3?1911 11(iv) At what time during the first 9 seconds does s have its largest value?(v) Approximately when is the acceleration zero?(vi) When is the particle moving toward the origin? Away from the origin?(vii) On which side (p ositive or negative) of the origin does the particle lie attime t = 9?/
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