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MATH 101 SAMPLE MIDTERM 2 For full credit, please show all work. Time: 50 minutes. Books, notes, calculators, or other aids are not allowed. 1. (18 marks, 6 for each part) Evaluate the integrals: Z (a) xe−2x dx Z (b) x Z (c) p 9 − x2 dx ex dx e2x − 3ex + 2 2. (12 marks, 6 for each part) (a) Evaluate the integral Z 1/2 0 dx , x ln x or show that it diverges. Z (b) Use the comparison test to decide whether the integral ∞ (x3 + x)−1/2 dx is conver- 1 gent. 3. We want to evaluate numerically the integral Z 2 1 sin x dx x using the Midpoint Approximation. (a) (4 marks) Write out (but do not evaluate) the approximation M5 . (b) (6 marks) Find an n such that Mn approximates the actual value of the integral with accuracy 10−5 or better. (The error estimate for the Midpoint Rule is K(b − a)2 /24n2 , where K = max |f 00 (x)| on [a, b].) 1