Mathematics 1A, Fall 2010 — M. Christ — Practice Final ExaminationFormula Listddxsin(x) = cos(x)ddxcos(x) = −sin(x)ddxtan(x) = sec2(x)ddxsec(x) = tan(x) sec(x)ddxcot(x) = −csc2(x)ddxcsc(x) = −cot(x) csc(x)ddxarcsin(x) =1√1 − x2ddxarctan(x) = (1 + x2)−1ddxarccos(x) = −1√1 − x2Newton’s Method: xn+1= xn−f(xn)f0(xn)Volume of a sphere:43πr3Volume of a right circular cone:13πr2h(1) (3 points) for each part. Calculate the following. Show your steps in an organizedfashion, and place final answers in boxes .(1a) The equation of the line tangent to f (x) = x + exat x = 2.(1b)ddxp3 + ln(ln(x)).(1c) limx→π/2cos(x)x−π/2(1d)ddxxcos(x). (Here x > 0.)(1e)Rddxp|sin(x) + cos(x)| dx.(1f)ddxRsin(2x)0arcsin(t) dt.(1g)Rsin(x) cos(x) dx(2a) (3 points)R10(1 − x2)−1/2dx(2b) (3 points)P4i=1i2cos(πi)(2c) (3 points)R2−2x ln(1 + x4) dx(2d) (4 points)R(1 − x2)−3/2dx (You need not simplify your answer.)(2e) (4 points) limx→∞(x + x1/3)2/3− x2/3(2f) (4 points) Express an approximation toR31ex2dx as a right endpoint Riemann sumwith n = 3. Your answer need not be simplified; it could be expressed as a sum of severalnumbers.(2g) (4 points) limn→∞Pni=1nn2+i2. (Either use a method taught in this course, or justifyyour steps in full detail.)(2h) (4 points) Use Newton’s method with initial approximation x1= 10 and one step toapproximate the cube root of 996.(3) (6 points) A right circular cone has height h and has a circular base of radius r. Itsvolume is13πr2h. Suppose that r2+ h2= 1. For what value of h is the volume of the conemaximized? What is the maximum volume?12(4) (9 points) Sketch a graph of the function f (x) = 4 + xe−1/2x. Indicate any horizontalor vertical asymptotes, but you need not indicate any slant asymptotes. Indicate intervalson which f is increasing and decreasing, local maxima and minima, inflection points, andintervals on which the graph is concave up or down. It is not possible to calculate interceptsexactly. Instead, determine exactly how many intercepts there are, and indicate roughlywhere they are located. You may use the formulas f0(x) = (1 +12x−1)e−1/2xand f00(x) =14x−3e−1/2x.(5) Newton’s law of cooling says: The rate of cooling of a body is proportional to thedifference between that body’s temperature, and the temperature of its environment. In acafe where the ambient room temperature is a steady 70 degrees, a cup of coffee is served at190 degrees. (All temperatures are measured in degrees Fahrenheit.) Assume that Newton’slaw of cooling applies.(5a) (3 points) Let f (t) be the temperature of the coffee at time t. Write a differentialequation satisfied by f (t). Your equation may include one or more unknown constants.(5b) (2 points) Write the general solution of your differential equation.(5c) (3 points) 3 minutes after the coffee is served, its temperature is 180 degrees. At whattime will the coffee cool to 160 degrees? Express your answer in minutes after the coffee isserved.(6) Show your steps in an organized, legible manner to receive credit. Let C be the circlewith radius 1 centered at (2, 0) in the xy plane. The region enclosed by C is revolved aroundthe y axis to generate a three dimensional solid known to mathematicians as a torus, andto law enforcement officers as a staple food.(6a) (3 points) Using the method of cylindrical shells, express the volume of this solid asan integral.(6b) (6 points) Evaluate the integral in (6a) using methods and results taught in this course.(You might be unable to find an antiderivative, but it is possible to evaluate the definiteintegral using material taught in the course.)(7) Short answer questions. 2 points for each part. These questions require only briefanswers, and require little or no calculation.(7a) An emu and a wombat race along a straight line, beginning at time t = 0. The wombatis given a head start. At time t, their positions are E(t) and W (t), respectively. Supposethat W (t) = 1 + t for all t ≥ 0, that E(0) = 0, and that E00(t) < 0 for all t ≥ 0. What isthe maximum possible number of times t > 0 at which E(t) can be equal to W (t)? Explainyour answer briefly.(7b) Define: f has an inflection point at x.(7c) Let t = time, s(t) = the position of a projectile at time t, and v(t) = its velocity. Leta < b be two times. We have defined two kinds of averages of v over the interval [a, b]: (i)The net change in position divided by the elapsed time, and (ii) (b − a)−1Rbav(t) dt. Howare these two averages related to one another? Explain very briefly.3(7d) If f (0) = 0, f0(0) = −1, and f00(x) ≤ 2 for all x, what is the largest possible value off(3)?(7e) If f0(−1) = f0(1), and if f00(x) exists and is a continuous function on [−1, 1], then twoconclusions can be drawn about f00. What are they?(7f) If f and its derivative f0are continuous functions defined for all real numbers, and iff0(x + 1) = f0(x) for all x, what conclusion can be drawn about f ?(7g) Let f be a continuous function defined for all x ≥ 0. For s > 0 let g(s) =s−1Rs0f(x) dx. Suppose that t is a positive number such that g(t) ≥ g(s) for every s > 0.Find an equation relating f(t) to g(t).(7h) Explain briefly how the formula xn+1= xn−f(xn)f0(xn)in Newton’s method is derived.(7i) If you were asked to derive the formula arcsec0(x) =1x√x2−1, assuming that arcsec isdifferentiable, how would you begin? You need not give a proof or do any calculations, butshow that you know what to