M1014FE2009R

# 1 a 5 pts evaluate the sum 99 3 1 1 1 i i i b 5 pts

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1. a) [5 pts] Evaluate the sum: = + - 99 3 ) 1 1 1 ( i i i b) [5 pts] Find the limit: = + n i n n i n 1 2 1 1 lim 2. a) [5 pts] Find the derivative of the function: = π x du u y ) cos( 2 b) [5 pts] Evaluate the definite integral: 2 / 0 2 ) cos( π dx x x 3. Sketch the region in the xy-plane bounded by x – 2y 2 = 0 and 1-x-|y| = 0 , and find its area. 1

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4. Find the volume of the solid obtained by rotating the region bounded by : y = x, y = 1, x = 2, x = 4 about y = 0. Sketch the region, the solid, and a typical washer or shell (i.e. whichever you used). 5. Using the definition ln x = dt t x 1 1 ; a) [2 points] Write an precise expression for ln 2 as the limit of a Riemann sum, R n . b) [4 points] Obtain an approximate value for ln 2 using a Riemann sum with a uniform partition and n = 4 (R 4 ). Take the sample points, x i *, at the right endpoint of each subinterval. c) [3 points] Draw a carefully labelled sketch to illustrate the integral and the approximation, R 4 .
• Fall '09
• ganong
• Calculus, Professor Taylor, Professor Guiasu, Professor Poliakov

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