Math121SampleFinalSols

Math121SampleFinalSols - FINAL (SAMPLE SOLUTIONS) USE A...

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FINAL (SAMPLE SOLUTIONS) USE A SCIENTIFIC CALCULATOR! 1) Approximate the area under the graph of fx x () = ln from a = 4 to b = 10 by finding a Left Riemann Sum using 3 rectangles of the same width. Round off to four decimal places whenever you need to round off. (10 points) Step 1: Find D x , the width of each rectangle. D x ba n = - = - == 10 4 3 6 3 2 Step 2: Find the left breakpoints. ax x x æ Æ æ= æ Æ ++ 1 2 2 2 3 468 Step 3: Find the Left Riemann Sum. 42 62 82 2 7726 3 5835 4 1589 ◊+ () ◊ = () + ª++ = DDD ln ln ln ... 10.5150 square units Note: The exact value is closer to 11.4807.
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2) Find the integrals. Simplify wherever possible. (32 points total) a) 33 12 1 5 xx d x - - () [] Ú cont. on 1,5 4 43 44 (6 points) =- = -- + 3 35 5 31 1 35 125 0 1 125 1 3 1 5 ln ln . ln ln ln ln x x ln on 1,5 3ln5 124 or ln5 124 or ln125 124 3 b) d x 2 6 4 - Ú (6 points) ux du x dx = 2 4 2 = = Ê Ë Á ˆ ¯ ˜ + =+ + Ú Ú 1 2 24 1 2 1 27 1 14 2 6 6 7 7 d x ud u u C uC 1 14 4 2 7 xC c) ex d x 3 2 61 2 +- + Ú (6 points) ux x du x xd x =+- 3 2 2 36 32 =◊ + = Ú Ú 1 3 1 3 1 3 3 2 d x ed u eC u u 1 3 3
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d) ln x x dx Ú (6 points) ux du x dx = = ln 1 = () = =+ = + Ú Ú ln x x dx udu u C 1 2 2 ln 2 2 x C e) x x dx
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Math121SampleFinalSols - FINAL (SAMPLE SOLUTIONS) USE A...

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