Spring 1999Math 151Common Exam 3Test Form APRINT: Last Name: First Name:Signature: ID:Instructor’s Name: Section #INSTRUCTIONS1. In Part 1 (Problems 1–10), 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 11–16), 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 IMultiple Choice(5 points each) No Calculators1. Consider f(x)=2x3−9x2+12x+3on the interval 0 ≤ x ≤ 4. The absolute maximum occurs atA. x =0B. x =4C. a point x = c where f0(c)=0D. a point x = c where f00(c)=0E. There is no absolute maximum.2. sin−1(sin(3π/4)) =A.12√2 B. 3π/4 C. −12√2 D.1sin12√2E.π43. Find The derivative of f (x)=tan−1(x2+1).A. f0(x)=2xx2+1B. f0(x)=−2xcsc2(x2+1)C. f0(x)=−csc2(x2+1)+tan−1(2x)D. f0(x)=2x(x2+1)2+1E. f0(x)=1(2x)2+14. sintan−112=A.1√2B.2√5C.14D.q25E.π225. limx→0tan x − xx3=A. 0 B.13C. does not exist D.12E. 16. The inflection points of f(x)=3x5−10x4+7occur at:A. x =83B. x =0 C. x =2 D. x =0and x =2 E. x =0and x =837. Consider the function defined byf(x)=x2(x+2)2,x≤1,9+x−x2,x>1.Find the x-values where the local maxima occur.A. x =0and x = −2B. x =12C. x = −1 ±13√3D. x = ±1E. x = −138. Let f(x) be a continuous function on 0 ≤ x ≤ 4.IfZ20f(x)dx =3,Z41f(x)dx =1,andZ42f(x)dx =2,thenZ10f(x)dx =A. 0 B. 1 C. 2 D. 3 E. 49. limx→0tan−11x2=A. 0 B.π2C. −π2D. ∞ E. −∞10. Find an anti-derivative of f(x)=lnx−1x.A. x ln x − x − ln xB.1x+1x2C. x ln x − x +1x2D.1x+lnxE.12(ln x)2− ln x4Part II Partial Credit Calculators PermittedCalculators are permitted for checking answers but not for supporting them. Show your work to obtain credit. Inparticular, no credit will be given for derivatives found solely by formal differentiation on your calculators.11. Find the dimensions of a right triangle which maximize the area with respect to the constraint that the sum of thehypotenuse and the base is equal to 1. (8 points)12. Consider the function f(x)=1xon the interval 1 ≤ x ≤ 2. Partition the interval into 5 equal sub-intervals, and calculatethe Riemann sum associated with evaluating f(x) at the mid-point of each sub-interval. (8 points)513. A bacterial culture starts with 500 bacteria and after 3 hours there are 8000 bacteria.(a) Find an expression for the number of bacteria after t hours. (4 points)(b) When will the population reach 30,000? (4 points)14. Find the derivative of f(x)=(x2+x+1)sin x. (8 points)615. Let f (x)=x(x+1)2. Calculation shows that f0(x)=1−x(1 + x)3and f00(x)=2(x − 2)(x +1)4.(a) List the regions where f(x) is increasing or decreasing. (5 points)(b) List the regions where the curve y = f(x) is concave up or concave down. (3 points)(c) Find the horizontal and vertical asymptotes. (2 points)16. Let f(x) be a polynomial and suppose x =1and x =2are roots. By using a theorem, explain why the polynomialf0(x) has a root that is strictly between 1 and 2. (8
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