MATH 1271 Spring 2011, Midterm #3Handout date: Thursday 31 March 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) Computeddx(1 + x2)100(sin x) .(a) 100(1 + x2)99(cos x)(b) 100(1 + x2)99(2x)(cos x)(c) 100(1 + x2)99(sin x) + (1 + x2)100(cos x)(d) 100(1 + x2)99(2x)(sin x) + (1 + x2)100(cos x)(e) NONE OF THE ABOVEB. (5 pts) (no partial credit) Computeddxexx2+ 5.(a)(x2+ 5)(ex) − (ex)(2x)(x2+ 5)2(b)ex2x(c)(ex)(2x) − (x2+ 5)(ex)(x2+ 5)2(d)2xex(e) NONE OF THE ABOVEC. (5 pts) (no partial credit) Computeddx(1 + x2)2x+5 .(a) (2x)2(b) (2x + 5)2+ (2x)2x+5(c)(1 + x2)2x+5 2ln1 + x2+ (2x + 5)2x1 + x2(d)(1 + x2)2x+5 [2x]2ln1 + x2+ (2x + 5)2x1 + x2(e) NONE OF THE ABOVED. (5 pts) (no partial credit) Computeddx(ln 4)cos 7 .(a)(ln 4)cos 7 (− sin 7)(ln(ln(4)) + (cos 7)1/4ln 4(b)(ln 4)cos 7 (− sin 7)(ln(ln(4)) + (cos 7)1/4ln(ln 4)(c) (1/4)sin 7(d) (1/4)− sin 7(e) NONE OF THE ABOVEE. (5 pts) (no partial credit) Computeddxln(x2+ 2x + 5) .(a)2x + 2x2+ 2x + 5(b)2x + 21 + (x2+ 2x + 5)2(c)2x + 2p1 − (x2+ 2x + 5)2(d)1x2+ 2x + 5(e) NONE OF THE ABOVEF. (5 pts) (no partial credit) Computeddxarctan(x2+ 2x + 5) .(a)2x + 2x2+ 2x + 5(b)2x + 21 + (x2+ 2x + 5)2(c)2x + 2p1 − (x2+ 2x + 5)2(d)1x2+ 2x + 5(e) NONE OF THE ABOVEII. True or false (no partial credit):a. (6 pts) If f is a rational function and a is any real number, then limx→a+f(x) exists; itmight be ∞, it might be −∞ or it might be a finite number, but it will exist.b. (6 pts) Every local extremum occurs at a critical number.c. (6 pts) Every global extremum is a local extremum.d. (6 pts) For any b > 0, the function f(x) = bxis one-to-one.e. (6 pts) For any two functions f and g, we have (fg)0= (f0)(g0). That is, the derivativeof the product is the product of the derivatives.THE BOTTOM OF THIS PAGE IS FOR TOTALING SCORESPLEASE DO NOT WRITE BELOW THE LINEI. A,B,CI. D,E,FII. a,b,c,d,eIII. 1III. 2III. 3abcIII. 4abcIII. 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) Compute limx→0[2x + cos x]1/x.2. (10 pts) Find an equation of the tangent line to the graph of y3+ y − 8 = x7+ x at thepoint (1, 2).3. Let f(x) = x3+ x + 1. Then f : R → R turns out to be one-to-one and onto. Also,f(−2) = −9. Let g : R → R be the inverse of f .a. (3 pts) Find f0(−2).b. (3 pts) Find g(−9).c. (4 pts) Find g0(−9).4. (10 pts) Let h(x) = 3x4− 4x3+ 36x2.a. (2 pts) Find all critical numbers for h.b. (2 pts) Find all critical numbers for h|[−1, 1].c. (6 pts) Find the global maximum and minimum values for h(x) on −1 ≤ x ≤
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