MATH 1271 Fall 2011, Midterm #2Handout date: Thursday 10 November 2011PRINT YOUR NAME:PRINT YOUR TA’S NAME:WHAT SECTION ARE YOU IN?Closed book, closed notes, no calculators/PDAs; no reference materials of any kind. Turnoff all handheld devices, including cell phones.Show work; a correct answer, by itself, may be insufficient for credit. Arithmetic need notbe simplified, unless the problem requests it.I understand the above, and I understand that cheating has severe consequences, from afailing grade to expulsion.SIGN YOUR NAME:I. Multiple choiceA. (5 pts) (no partial credit) Suppose f0(x) = −x2+ 3x − 2. At most one of the followingstatements is true. If one is, circle it. Otherwise, circle “NONE OF THE ABOVE”.(a) f is increasing on (−∞, 1], decreasing on [1, 2] and increasing on [2, ∞).(b) f is decreasing on (−∞, 1], increasing on [1, 2] and decreasing on [2, ∞).(c) f is increasing on (−∞, −2], decreasing on [−2, −1] and increasing on [−1, ∞).(d) f is decreasing on (−∞, −2], increasing on [−2, −1] and decreasing on [−1, ∞).(e) NONE OF THE ABOVEB. (5 pts) (no partial credit) Find the logarithmic derivative of x2+ 3x − 8 w.r.t. x.(a)2x + 3x2+ 3x − 8(b)x2+ 3x − 82x + 3(c) (ln(x2)) + 3(ln x) − (ln 8)(d) ln(2x + 3)(e) NONE OF THE ABOVEC. (5 pts) (no partial credit) Find the slope of the tangent line to y = (x3+ 4)e2xat thepoint (0, 4).(a) 2(b) 4(c) 6(d) 8(e) NONE OF THE ABOVED. (5 pts) (no partial credit) Find the logarithmic derivative of (2 + sin x)xw.r.t. x.(a) [(2 + sin x)x](ln(2 + sin x)) +x cos x2 + sin x(b) (ln(2 + sin x)) +x cos x2 + sin x(c) ln(cos x)(d) cos x(e) NONE OF THE ABOVEE. (5 pts) (no partial credit) Find the derivative of (2 + sin x)xw.r.t. x.(a) [(2 + sin x)x](ln(2 + sin x)) +x cos x2 + sin x(b) (ln(2 + sin x)) +x cos x2 + sin x(c) ln(cos x)(d) cos x(e) NONE OF THE ABOVEF. (5 pts) (no partial credit) Compute limx→0sin2x4x3+ 2x2.(a) 2(b) 1(c) 1/2(d) 1/4(e) NONE OF THE ABOVEII. True or false (no partial credit):a. (5 pts) If f0(3) = 0 and f00(3) < 0, then f has a local maximum at 3.b. (5 pts) Every local extremum occurs at a critical number.c. (5 pts) Every global extremum occurs at a critical number.d. (5 pts) If f is increasing on an interval I, then f0> 0 on I.e. (5 pts) If f and g are differentiable, thenddx[(f(x))(g(x))] = [f0(x)][g0(x)].THE BOTTOM OF THIS PAGE IS FOR TOTALING SCORESPLEASE DO NOT WRITE BELOW THE LINEVERSION AI. A,B,CI. D,E,FII. a,b,c,d,eIII. 1.III. 2.III. 3,4.III. 5.III. Computations. Show work. Unless otherwise specified, answers must be exactly cor-rect, but can be left in any form easily calculated on a standard calculator.1. (10 pts) Computeddx2x3− 8arctan x+ xesin x2. (10 pts) Using implicit differentiation (and logarithmic differentiation), find y0= dy/dx,assuming that (2 + y2)xy= 9.3. (5 pts) Suppose f is 1-1 and g = f−1is the inverse of f. Suppose f(3) = 4 andf0(3) = 27. Compute g(4) and g0(4).4. (10 pts) Find the maximal intervals of increase and decrease for f(x) = x3− 6x2+ 5.5. (10 pts) Among all pairs of positive numbers x and y such that xy = 100, find theglobal maximum value of x + 4y, provided it exists. Then find the global minimum value,provided it exists. (NOTE: If the global maximum value does not exist, you need to statethat clearly to receive full credit. If it does exist, for full credit, you’ll need to computex + 4y; computing x and/or y alone is insufficient. These same comments apply to theglobal minimum
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