Name of Student: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Your Instructor’s Name: . . . . . . . . . . . . . . . . . . . . . . . . . . )Final Examination of Math 1aJanuary 23, 2000 (Tuesday)9:15 a.m., Emerson 105 & Harvard 104Instructors: Yum-Tong Siu (course head), Peter Clark, Kim FroyshovDeepee Khosla, Yang Liu, Russell MannDavid Savitt, Kiril Selverov, Yuhan ZhaQuestion Points Score1 202 123 204 125 206 167 168 259 1210 1611 1512 16Total 200• You have THREE hours to complete this examination.• No calculators are allowed.• No partial credit can be given for unsubstantiated answers.• Use the back of the page if more space is needed for your answer(with an indication that your answer is continued on the back of the page).1. Compute the following limits.(a)limx→2+|x|(x + 1)(x − 2)(x + 3).(b)limx→−∞|x|(x + 1)(x − 2)(x + 3).1(c)limx→01 + x − exx2.(d)limx→0+hxsin xi.22. Consider the functionf(x) =−|x| for x < −π2cos x for −π2≤ x ≤π3x −π3+12for x >π3.(a) Identify all points x where f(x) is not continuous. Justify youranswer.(b) Identify all points x where f(x) is not differentiable. Justify youranswer.33. Calculate the following derivatives.(a)ddxhx2+ cos x + ln xi.(b)ddx[exarctan x] .4(c)ddxx2+ 1x3.(d)ddxZ7x2sint3dt.(Hint: use the Fundamental Theorem of Calculus.)54. Letf(x) = ex+ x + arctan xfor −∞ < x < ∞.(a) Prove that y = f(x) is invertible for −∞ < x < ∞ by verifyingthat it is strictly increasing. What is the domain of the inversefunction f−1(as a function of y)? Justify your answer.(b) Calcuate the derivative of f−1(as a function of y) at y = 1. (Notethat y = 1 when x = 0.)65. For the functionf(x) =x3+ x2− 1x3,answer parts (a) through (h) below.(Hint: f (x) can be written as 1 +1x−1x3.)(a) Find any vertical asymptotes for the graph y = f (x).(b) Find any horizontal asymptotes for the graph y = f(x).(c) For which values of x, if any, is f (x) discontinuous?7(d) Find intervals on which f is increasing and intervals on which fis decreasing.(e) Find any critical points and determine whether they are relativemaxima or minima or neither.(f) Find intervals where f is concave up and intervals which f isconcave down.8(g) Find any points of inflection.(h) Give a rough sketch of the graph of y = f (x) illustrating all of thefeatures you found in parts (a) through (g).96. Water is being drained from an upside-down circular cone at the con-stant rate of 3 cubic feet per minute. When the water level h in theupside-down cone is 9 feet, the water level is decreasing at the rate of1 foot per minute.(a) What is the area A of the circular top of the cone at water levelh when h is 9 feet?Hint: (i) The area A of the circular top of the cone at waterlevel h is proportional to h2(with the constant of proportionalitydepending on the cone). (ii) The volume of a cone with base areaA and height h is13Ah.(b) How fast is the water level decreasing (in terms of the number offeet per minute) when the water level is 5 feet?10(c) How fast is the area of the circular top of the cone at water leveldecreasing (in terms of the number of square feet per minute)when the water level is 5 feet?117. An offshore oil well W is located 4 miles from the closest point A ona straight shoreline. A pipeline is to be contructed from W to a shorepoint B which is 5 miles from A. The pipeline will go along a straightline under water from W to some shore point P between A and B andthen go along a straight line over land along the shore from P to B.The cost of of laying the pipeline is 5 million dollars a mile under waterbut is only 3 million dollars over land.(a) How far away should P be from A (in miles) to minimize theconstruction cost of the pipeline?(b) What is the minimum construction cost?128. Compute the following indefinite and definite integrals.(a)Zx2+ exdx.(b)Zsin3x cos x dx.13(c)Zarctan x1 + x2dx.(d)Z2−11 − e−xdx.14(e)Z232√2x − x2dx[Hint: Complete the square for 2x − x2, write the integral as thearea of the sector ABC minus the area of the triangle ADC. Thearea of a sector (or a fan) of angle α (measured in radians) in adisk of radius r is12r2α.]AαBCD159. (a) FindI =Z101dxx(b) Write down the Riemann sum R ofZ101dxxwith 9 subintervals, using right end-points.16(c) Write down the Riemann sum L ofZ101dxxwith 9 subintervals, using left end-points.(d) Which of the three numbers I, R, L is the largest and which isthe smallest? Justify your answer.1710. A creature moves along the x-axis in such a way that its accelerationis given by a(t) = 2 − 6t. The time variable t is allowed to take onlynonnegative values. When the creature starts out at t = 0, it is at theposition x = 2 and has velocity 1 at that moment.(a) Where will the creature be when t = 2?(b) At what time t is the creature closest to the point (4, 1) in the xy-plane? What is the distance between the creature and the point(4, 1) at that time?1811. A ball is thrown directly upward with an initial velocity of 128 feetper second and is released from a point that is 144 feet above ground.Assume that the free-fall model applies and that the acceleration dueto gravity is 32 feet per second per second (i.e., the ball falls withconstant acceleration due to gravity of 32 feet per second per second).(a) In the terms of the number of feet above ground, what is thehighest point which the ball reaches?(You may leave your answer as the product of two integers.)(b) What is the velocity of the ball when it reaches the highest point?19(c) How much time elapses after the ball is thrown and before itreaches the ground on its fall?2012. (a) State the following three theorems, including all necessary condi-tions.(i) The Intermediate-Value Theorem.(ii) The Extreme-Value Theorem.(iii) The Mean-Value Theorem.(b) Two particles move along a straight line during the time intervalfrom t = a to t = b. Suppose the two particles have the same aver-age velocity for that time interval. Use one of the three theoremsin (a) to show that there is a certain point in that time intervalwhen both particles have the same instantaneous velocity.21(c) A hare is racing a tortoise along a straight line. At the start ofthe race the hare is 10 meters behind the tortoise and at the endof the race the hare is 20 meters ahead of the tortoise. Use one ofthe three theorems in (a) to show that there is a