MATH 151, FALL SEMESTER 2004COMMON EXAMINATION II - VERSION BName (print):Signature:Instructor’s name:Section No:Seat No:INSTRUCTIONS1. In Part 1 (Problems 1–13), mark the correct choice on your ScanTron form using a No:2 pencil.For your own record, mark your choices on the exam itself. ScanTrons will be collected fromall examinees after one hour, and will notbe returned. No student will be allowed to turn inhis/her scantron before the end of the first hour.2. Calculators may notbe used in Part 1. The use of calculators is permitted only after the first hourhas elapsed and all ScanTrons have been collected.3. In Part 2 (Problems 14–18), present your solutions in the space provided. Show all yourwork neatly and concisely, and indicate your final answer clearly. You will be graded,not merely on the final answer, but also on the quality and correctness of the work leading upto it.4. Be sure to write your name, section number, and version letter of the exam on theScanTron form.QN PTS1–131415161718TOTAL1Part 1 – Multiple Choice (52 points)Read each question carefully; each problem is worth 4points. Calcluators are not allowed for thispart of the exam.1. Evaluate limx→∞3−x.(a) ∞(b) 1(c) −1(d) 0(e) −∞2. Determine the domain of the function f (x)=ln(|x|).(a) x>0(b) x>1(c) x 6=0(d) x<0(e) all real numbers3. If log3(3x + 2) = 2, what is x ?(a) 0(b) 4/3(c) 3/2(d) 7/3(e) 5/24. Given that f (x)=√1−x, find the range of f−1(the inverse of f ).(a) (−∞, ∞)(b) [1, ∞)(c) (−∞, 1](d) [−1, 1](e) [0, ∞)25. Suppose that f(x)=x5+x3and let g denote its inverse. What is g0(2) ? (Note: f (1) = 2 andf(2) = 40 .)(a) 1/92(b) 1/8(c) −92/1600(d) 1/(5(40)4+ 3(40)2)(e) cannot be determined with the information given6. Differentiate the function f (x)=x2tan x with respect to x .(a) 2x sec2x(b) 2x tan x + x2sec x tan x(c) 2x tan x + x2sec2x(d) 2x tan x − x2sec2x(e) 2x tan x − x2sec x tan x7. Differentiate the function f (x)=2√exwith respect to x .(a) 2√ex(b)√ex(c) e√x(d)√ex/2(e) 2e√x8. Let r(t)=hcos2t, ti be a vector function representing the position of a particle at time t.Findthe acceleration of the particle at t = π/3.(a) h−1, 1i(b) h1, 1i(c) h−1, 0i(d) h1, 0i(e) h−√3, 0i9. Find D28(cos x)(i.e., the twenty-eighth derivative of cos x).(a) sin x(b) cos x(c) −sin x(d) −cos x(e) sin x cos x310. Find the linear (i.e., tangent-line) approximation L(x) to the function f (x)=sin(x2)ata=√π.(a) L(x)=√π−x(b) L(x)=−2√πx(c) L(x)=−2π+2√πx(d) L(x)=2π−2√πx(e) L(x)=2√π−2x11. Suppose that x>0andletsdenote the distance between the points (x, 0) and (0, 1). If x ischanging with time anddxdt=2,thendsdt=?(a) 2x(b) 2(c)x(x2+1)(d)2x√x2+1(e)4x2x2+112. 2−x+2−x=?(a) 2−2x(b) 4−2x(c) 2x2(d) 21−x(e) 4−x13. Let f be a differentiable function, and let C denote the graph of y = f(x) . It is known thatC passes through the point P (2, 2), and that the slope of the tangent to C at P is a numberbetween 3 and 5. If one begins Newton’s method (to solve the equation f(x)=0)withx1=2,which of the following intervals will contain the next iterate x2?(a) (−1, 0)(b) (0, 1)(c) (1, 2)(d) (2, 3)(e) (3, 4)4Part 2 (54 points)The use of a calculator is permitted for this part of the exam, but all work must be shownin orderto receive credit. Refer to the front page for further instructions.14. (12 points)LetCbe the curve given by the equation(x2+ y2)2=2(x2−y2)+6x+7.Use implicit differentiation to find the slope of the tangent to C at the point (2, 1).Present your work methodically and clearly; merely relaying an answer reported by yourcalculator is unacceptable.515. (14 points) Consider the curve C given by the parametric equationsx(t)=te−t,y(t)=(t+1)1/3, −∞ <t<∞.(i) (8 points)Finddxdtanddydt.(ii) (3 points)Finddydxin terms of the parameter t.(iii) (3 points) Obtain an equation of the tangent line to C at the point (0, 1).616. (12 points) Suppose that a and b are constants, neither of which is zero. Compute the firstand second derivatives of the function f(x)=ebxsin(ax) . Show all your steps clearly.717. (10 points) A television camera is positioned 4000 feet from the base of a rocket launchingpad. A rocket rises vertically and its speed is 600 feet per second when it has risen 3000 feet.Assuming that the camera is always kept focused on the rocket, how fast is the camera’s angleof elevation changing at that same moment? Include units with your answer.818. (6 points) The linear (i.e., tangent-line) approximation to a function f at a =1isgivenby L(x)=3x+ 2 . Find the linear approximation to the function H =√f at a =1.Explain your reasoning clearly and
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