# Final Exam - Final Exam Laurence Field Math 152 Section 32...

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Final Exam Laurence Field — Math 152, Section 32 — March 21, 2013 Instructions: This exam has 10 questions, each worth 10 points, for a total of 100 points. The value of each part of each question is stated. Please show your working. You have two hours. 1. Evaluate the following indefinite integrals: (a) 5 Z . sin x C 3/ cos x dx ; (b) 5 Z x.5 C x 2 / 6 dx . 2. Evaluate the following definite integrals: (a) 5 Z 2 0 x 2 e x 3 dx ; (b) 5 Z =6 0 sec .2x/ dx . 3. (a) 5 Evaluate Z log x x dx . (b) 5 Differentiate f .x/ D x x 2 . Hence find Z x 1 x 2 .1 C 2 log x/ dx . 4. (a) 3 Give the definition of the natural logarithm function (log x ) and state its domain and range. (b) 3 Give the definition of the exponential function (exp x or e x ) and state its domain and range. (c) 4 Find all real numbers x such that log x .e 8 / D log .e 2 x/ . 5. 10 Sketch the graph of f .x/ D 1 C x 2 e x , showing any asymptotes, vertical tangents and cusps, local extrema and points of inflection. You need not calculate the y -coordinates of important points; the x -coordinates will suffice. 6. Suppose that P