Spring 1999 Math 151 Common Exam 3 Test Form A PRINT Last Name First Name Signature ID Instructor s Name Section INSTRUCTIONS 1 In Part 1 Problems 1 10 mark the correct choice on your ScanTron form using a 2 pencil For your own records also record your choices on your exam The ScanTrons 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 use the back of any page for scratch work but all work to be graded must be shown in the space provided CLEARLY INDICATE YOUR FINAL ANSWERS 1 Multiple Choice 5 points each Part I No Calculators 1 Consider f x 2x3 9x2 12x 3 on the interval 0 x 4 The absolute maximum occurs at A x 0 B x 4 C a point x c where f 0 c 0 D a point x c where f 00 c 0 E There is no absolute maximum 2 sin 1 sin 3 4 A 12 2 B 3 4 C 21 2 D 1 1 sin 2 2 E 4 3 Find The derivative of f x tan 1 x2 1 2x x2 1 B f 0 x 2x csc2 x2 1 A f 0 x C f 0 x csc2 x2 1 tan 1 2x D f 0 x 2 2x 2 x 1 1 1 E f 0 x 2x 2 1 4 sin tan 1 21 A 1 2 B 2 5 C 41 2 q D 2 5 E 2 5 lim tan x3 x x 0 x B 13 A 0 C does not exist D 21 E 1 6 The inflection points of f x 3x5 10x4 7 occur at A x 38 B x 0 C x 2 7 Consider the function defined by f x D x 0 and x 2 x2 x 2 2 x 1 9 x x2 x 1 Find the x values where the local maxima occur A x 0 and x 2 B x 1 2 C x 1 1 3 3 D x 1 E x 1 3 E x 0 and x 8 3 Z 8 Let f x be a continuous function on 0 x 4 If Z 1 f x dx 2 0 Z f x dx 3 1 4 Z f x dx 1 and 4 2 f x dx 2 then 0 A 0 B 1 9 lim tan 1 x 0 1 x2 C 2 B 2 C 2 1 10 Find an anti derivative of f x ln x x A x ln x x ln x 1 x 1 x2 C x ln x x D E 1 x ln x 1 2 2 ln x E 4 A 0 B D 3 1 x2 ln x 4 D E Part II Partial Credit Calculators Permitted Calculators are permitted for checking answers but not for supporting them Show your work to obtain credit In particular 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 the hypotenuse and the base is equal to 1 8 points 1 on the interval 1 x 2 Partition the interval into 5 equal sub intervals and calculate 12 Consider the function f x x the Riemann sum associated with evaluating f x at the mid point of each sub interval 8 points 5 13 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 6 15 Let f x 2 x 2 x Calculation shows that f 0 x 1 x 3 and f 00 x x 1 2 1 x 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 1 and x 2 are roots By using a theorem explain why the polynomial f 0 x has a root that is strictly between 1 and 2 8 points 7
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