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m 408L hw 8

# m 408L hw 8 - wwm364 – Homework 8 – Cepparo –(58400 1...

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Unformatted text preview: wwm364 – Homework 8 – Cepparo – (58400) 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 , 4) , ( 1 2 , 2) , (1 , 5) , ( 3 2 , 8) , (2 , 9) 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 f ( x ) dx . 1. I ≈ 11 2. I ≈ 41 4 3. I ≈ 21 2 4. I ≈ 45 4 5. I ≈ 43 4 correct 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 ≈ 43 4 . 002 10.0 points The graph of a function f is shown in 2 4 6 8 10 2 4 6 8 Use Simpson’s Rule with n = 6 to estimate the integral I = integraldisplay 9 3 f ( x ) dx . 1. I ≈ 74 3 2. I ≈ 73 3 correct 3. I ≈ 76 3 4. I ≈ 24 5. I ≈ 25 Explanation: Simpson’s Rule estimates the integral I = integraldisplay 9 3 f ( x ) dx by I ≈ 1 3 braceleftBig f (3) + 4 f (4) + 2 f (5) + 4 f (6) + 2 f (7) + 4 f (8) + f (9) bracerightBig , taking n = 6. Reading off the values of f from its graph we thus see that I ≈ 73 3 . 003 10.0 points wwm364 – Homework 8 – Cepparo – (58400) 2 If f is the function whose graph on [0 , 10] is given by 2 4 6 8 2 4 6 8 use the Trapezoidal Rule with n = 5 to esti- mate the definite integral I = integraldisplay 8 3 f ( x ) dx . 1. I ≈ 49 2 2. I ≈ 47 2 3. I ≈ 24 correct 4. I ≈ 45 2 5. I ≈ 23 Explanation: The Trapezoidal Rule estimates the definite integral I = integraldisplay 8 3 f ( x ) dx by I ≈ 1 2 bracketleftBig f (3) + 2 { f (4)+ ··· + f (7) } + f (8) bracketrightBig when n = 5. For the given f , therefore, I ≈ 1 2 bracketleftBig 7 + 2 { 5 + 6 + 5 + 3 } + 3 bracketrightBig = 24 , reading off the values of f from the graph. 004 10.0 points Below is the graph of a function f .- 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 ≈ 11 2. I ≈ 12 3. I ≈ 9 4. I ≈ 8 5. I ≈ 10 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 wwm364 – Homework 8 – Cepparo – (58400) 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- 3 + 2 + 4 + 5 = 10 . 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 ) = ln(1 + 6 x- x 2 ) and the x-axis....
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m 408L hw 8 - wwm364 – Homework 8 – Cepparo –(58400 1...

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