MATH 151, SPRING SEMESTER 2009COMMON EXAMINATION II - VERSION BName (print):Signature:Instructor’s name:Section No:INSTRUCTIONS1. In Part 1 (Problems 1–1 1), mark your responses on your ScanTron form using a No:2 pencil. For your own record, mark your choices on the exam as well. Collectedscantrons will not be returned after the examinati on.2. Calculators should not be used throughout the examination.3. In Part 2 (Problems 12–16), present your solutions in the space provided. Show allyour work neatly and concisely, and indicate your final answer clearly. You willbe graded, not merely on the final answer, but also on the quality and correctness ofthe work leading up to it.4. Be sure to write your name, section number, and version letter of the examon the ScanTron form.1Part 1 – Multiple Choice ( 44 points)Each question is worth 4 points. Mark your responses on the ScanTro n form and on theexam itself .1. Differentiate the function 2 cos x − sin x with resp ect to x.(a) −2 sin x −cos x(b) 2 cos x + sin x(c) −2 cos x + sin x(d) −2 sin x + cos x(e) 2 cos x −sin x2. Compute the slope of the tangent t o the curve y = x − sec x at the point (0, −1).(a) −1(b) 1(c) −2(d) 2(e) 03. Given that f(x) = (3x − 1)8, find the value of f′(0).(a) −24(b) −16(c) −8(d) 16(e) 2424. Compute the 11th derivative of f(x) = sin x with respect to x.(a) sin x + cos x(b) −sin x(c) sin x(d) cos x(e) −cos x5. Determine the tangent vector to the vector function r(t) = ht1/3, e2ti, correspondingto t = 1.(a) h2e2, 1/3i(b) h1/3, e2i(c) he2, 1/3i(d) h1/3, 2e2i(e) h1, 1i6. Suppose that one begins to solve the equation x3−2x−5 = 0 using Newton’s method.If x1is chosen to be 2, what is x2?(a) 5/2(b) 19/10(c) 21/10(d) 9/4(e) 7/337. Compute limx→0−12 + 3(1/x).(a) 1/2(b) 2(c) −2(d) −1/2(e) 08. Consider the functions f1(x) = x2, f2(x) = cos(x), a nd f3(x) = sin(x). Which ofthese is not one-to-one on the interval [−π/2, π/2]?(a) f1and f2(b) f1and f3(c) f2and f3(d) f1, f2, and f3(e) All three functions are one-to-one9. Compute the linear approximati on to f(x) = esin xat x = π.(a) L(x) = π + 1 − x(b) L(x) = x + 1 − π(c) L(x) = 1 + e−1(x − π)(d) L(x) = 1 + e(x − π)(e) L(x) = x − π410. The function f(x) =21 + x2can be shown to be one-to-one on the interval [0, ∞).Determine f−1(the inverse of f), and state its domain.(a) f−1(x) =r1 − 2xx; domain=(0, 1/2](b) f−1(x) =1 − 2xx; domain=(0, ∞)(c) f−1(x) =r2 − xx; domain=(0, 2](d) f−1(x) =2 − xx; domain=(0, ∞)(e) f−1(x) =1 + x22; domain=(−∞, ∞)11. Computelimh→0tan2π4+ h− 1h.(Hint: Recall the definition of a derivative at a poi nt.)(a) ∞(b) 2(c) 0(d) −∞(e) 45Part 2 ( 61 points)Present your solutions to the following problems (12–16) in the space provided. Show allyour work neatly and concisely, and indicate your final answer clearly. You will begraded, not merely on the final answer, but also on the quality and correctness of the workleading up to it.12. (12 points) Verify that the function y = x2exsatisfies the following equation:y′′′− 3y′′+ 3y′− y = 0.613. (12 points) Given thaty√x − 1 + xpy − 1 = 3x y,employ implicit differentiation to computedydx.714. A heap of rubbish in the shap e of a cube is being compacted into a smaller cube insuch a way that the volume decreases at the rate of 2 cubic meters per minute. (Youmay assume that the shape of the heap remains cubical at every instant.)(i) (6 points) Find the rate of change of the length of an edge of the cube when thevolume is 64 cubic meters.(ii) (5 points) How fast is the surface area of the cube changing at that instant (i.e.,when the volume is 64 cubic meters)?815. Let C denote the parametri c curve determined by the equationsx(t) =1 − t1 + t, y(t) =1 −√t1 + t, t ≥ 0.(i) (7 points) Compute x′(t).(ii) (7 points) Compute y′(t).(iii) (6 points) Calculate the slope of the tangent to C at the point (0, 0).916. (6 points) It can be shown that the functionf(x) = ex3+3x, −∞ < x < ∞,is one-to-one. Let g denote the inverse o f f. Find an equatio n of t he tangent to thegraph of y = g(x), when x = 1.10QN
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