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Homework 8

# Homework 8 - tgo72 Homework 8 Gompf(58370 10 1 This...

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tgo72 – Homework 8 – Gompf – (58370) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points If the points (0 , 6) , ( 1 2 , 6) , (1 , 8) , ( 3 2 , 7) , (2 , 4) lie on the graph of a continuous function y = f ( x ), use the trapezoidal rule and all these points to estimate the definite integral I = integraldisplay 2 0 f ( x ) dx . 1. I 25 2 2. I 51 4 3. I 13 correct 4. I 27 2 5. I 53 4 Explanation: The trapezoidal rule estimates the definite integral I as h 2 parenleftBig f (0) + 2 f ( 1 2 ) + 2 f (1) + 2 f ( 3 2 ) + f (2) parenrightBig . With h = 1 2 and the given values of f , there- fore, the area is estimated by I 13 . 002 10.0 points The graph of a function f is shown in -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 6 8 Use Simpson’s Rule with n = 6 to estimate the integral I = integraldisplay 10 4 f ( x ) dx . 1. I 83 3 correct 2. I 79 3 3. I 82 3 4. I 80 3 5. I 27 Explanation: Simpson’s Rule estimates the integral I = integraldisplay 10 4 f ( x ) dx by I 1 3 braceleftBig f (4) + 4 f (5) + 2 f (6) + 4 f (7) + 2 f (8) + 4 f (9) + f (10) bracerightBig , taking n = 6. Reading off the values of f from its graph we thus see that I 83 3 . 003 10.0 points

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tgo72 – Homework 8 – Gompf – (58370) 2 If f is the function whose graph on [0 , 10] is given by -1 0 1 2 3 4 5 6 7 8 9 2 4 6 8 2 4 6 8 use the Trapezoidal Rule with n = 5 to esti- mate the definite integral I = integraldisplay 6 1 f ( x ) dx . 1. I 33 2. I 63 2 3. I 32 correct 4. I 65 2 5. I 31 Explanation: The Trapezoidal Rule estimates the definite integral I = integraldisplay 6 1 f ( x ) dx by I 1 2 bracketleftBig f (1) + 2 { f (2)+ · · · + f (5) } + f (6) bracketrightBig when n = 5. For the given f , therefore, I 1 2 bracketleftBig 7 + 2 { 8 + 6 + 6 + 6 } + 5 bracketrightBig = 32 , reading off the values of f from the graph. 004 10.0 points Below is the graph of a function f . -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 - 1 - 2 - 3 1 2 3 2 4 6 Estimate the definite integral I = integraldisplay 3 - 3 f ( x ) dx using the Midpoint Rule with six equal subin- tervals. 1. I 8 2. I 5 3. I 7 4. I 6 5. I 9 correct Explanation: Since [ - 3 , 3] is subdivided into six equal subintervals, each of these will have length 1 and the six corresponding rectangles are shown as the gray-shaded areas in
tgo72 – Homework 8 – Gompf – (58370) 3 - 1 - 2 - 3 1 2 3 2 4 6 The heights of the rectangles are midpoint sample values of f that can be read off from the graph. Thus, with midpoints, I 3 - 1 - 4 + 1 + 4 + 6 = 9 . 005 10.0 points Use Simpson’s Rule with 2 subintervals to estimate the area of the region in the first quadrant enclosed by the graph of f ( x ) = 3 ln(1 + 2 x - x 2 ) and the x -axis. 1. Area 4 ln 2 correct 2. Area 1 ln 2 3. Area 4 ln 4 4. Area 6 ln 2 5. Area 1 ln 4 6. Area 6 ln 4 Explanation: The graph of f intersects the x -axis when f ( x ) = 3 ln(1 + 2 x - x 2 ) = 0 , which after exponentiation becomes 1 + 2 x - x 2 = 1 .

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