Spring 2003Math 151COMMON EXAM 1Test Form APRINT: Last Name: First Name:Signature: ID:Instructor’s Name: Section #INSTRUCTIONS1. In Part 1 (Problems 1–11), mark the correct choice on your ScanTron form usinga #2 pencil. For your own records, also record your choices on your exam! TheScanTrons will be collected after 1 hour; they will NOT be returned.2. In Part 2 (Problems 12–16), write all solutions in the space provided. You may usethe back of any page for scratch work, but all work to be graded must be shown inthe space provided. CLEARLY INDICATE YOUR FINAL ANSWERS.1Part I: Multiple-Choice Problems. Each problem is worth 4 points. No partial credit will be given.Calculators may not be used on this part. Scantron forms will be collected after one hour.1. Calculate limx→∞sin x.(a) 0(b) 1(c) −1(d) The limit does not exist.(e)122. Find the derivative of f (x)=3x3−5x2+x−7.(a) x2−52x +1(b) 9x2− 10x +1(c)34x4−53x3+12x2− 7x(d) 3x2− 5x +1(e) 6x2− 5x3. Evaluate limx→∞5x2+2x+33x2+5x+2(a)32(b)25(c) 0(d)53(e) ∞4. A line is given by the parametric equations x =2t+3,y=7t−2. Find the slope of this line.(a)72(b) −27(c)32(d) −23(e)2725. The polynomial x3+3x−5has only one real root. In which interval does it lie?(a) 0 <x<1(b) −1 <x<0(c) 1 <x<3(d) −5 <x<−1(e) 3 <x<76. Evaluate limx→−1x3− xx2+5x+4.(a) 0(b) 1(c)23(d)25(e) ∞7. Consider the function f (x)=x2−c, x ≤ 12c − x, x > 1.Forwhatvalueofcis f (x) continuous at x =1?(a) c =1(b) c =23(c) c =0(d) c =12(e) c =138. Find the vertical asymptotes of y =2x2+5x+23x2+7x+2.(a) x = −13(b) x = −2 and x = −13(c) x = −12(d) x = −12and x = −2(e) There are no vertical asymptotes.39. Determine which vector is perpendicular to the line passing through the points (2,1) and (3,5).(a) h2, 1i(b) h−1, 2i(c) h1, 4i(d) h−4, 1i(e) h5, 3i10. With h(x)=f(x)g(x), suppose f (2) = 1, f0(2) = 3, g(2) = 5,andg0(2) = 1.Then(a) h0(2) = 3(b) h0(2) = 16(c) h0(2) = 8(d) h0(2) = 4(e) h(x) is nondifferentiable at x =2.11. With h(x)=f(x)g(x), suppose limx→cf (x)=5,f(c)=3,limx→cg(x)=7,andg(c)=2.Then(a) limx→ch(x)=31(b) limx→ch(x)=35(c) limx→ch(x)=6(d) limx→ch(x)=7(e) limx→ch(x) does not exist.4Part II: Work-Out Problems.Partial credit is possible. Calculators are permitted during the second hour only. Show your work.Ananswer with no work is not acceptable.12. Consider the function f (x)=12x+3.(a) Calculate f0(1) using only the definition of derivative. (6 points)(b) Find the equation of the line tangent to the curve y =12x +3at the point1,15. (4 points)13. Find parametric equations for the line passing through the points (−1, 2) and (5,1). (8 points)514. Find the derivative of f (x) in each case. (4 points each)(a) f (x)=5x−2x2+3x+1.(b) f (x)=(3x3−2x2+x−7)(x2+4x−3)(c) f (x)=x3√x.15. Find limx→0(sin x)sin1x. Justify your answer. (7 points)616. Consider the triangle whose vertices are (−1, 3), (3,1), and (−2, −2). Choosing the segment (−1, 3)(3, 1) asthe base of the triangle, find the altitude. (9 points)yx17. Calculate the following limits. (5 points each)(a) limx→4√x − 2x2− 16(b) limx→∞√x2+ x
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