final_su03 - ( r 2-x 2 ). What is the average value of the...

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Name: Math 20B Final Examination September 5, 2003 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (4 points) The sine integral function Si( x ) = Z x 0 sin t t dt is important in electrical engineering. Note that since lim t 0 sin t t = 1, the integrand sin t t has a removable discontinuity at t = 0. (a) Evaluate Si(0). (b) At what values of x does Si( x ) have local maximum values? # Score 1 2 3 4 5 6 7 8 9 10 Σ
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2. (4 points) Evaluate the following indefinite integrals. (a) Z x 2 1 - x 2 dx (b) Z x x 2 + x dx
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3. (4 points) Evaluate each of the following definite integrals, or determine that it does not exist. (a) Z π 2 0 1 - sin x cos x dx (b) Z 3 0 1 x - 1 dx
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4. (4 points) A hole of radius r is bored through the center of a sphere of radius R > r . Find the volume of the remaining solid. 5. (4 points) The area of a cross section of a sphere of radius r a distance x < r from the center is
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Unformatted text preview: ( r 2-x 2 ). What is the average value of the areas of these cross sections with respect to the distance x from the center of the sphere? 6. (4 points) A radar gun was used to record the speed of a runner during the rst 5 seconds of a race; the data is recorded in the table below. Estimate the distance the runner covered during those 5 seconds using Simpsons Rule. t (sec) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 v (m/sec) 4.7 7.3 8.9 9.7 10.2 10.5 10.7 10.8 10.8 10.8 7. (4 points) Find the solution to the following initial value problem. xy + y = y 2 ; y (1) =-1 . 8. (4 points) Express (1 + i ) 8 in standard ( a + bi ) form. 9. (4 points) Find the area of the region that lies inside both of the polar curves r = sin and r = 3 cos . 10. (4 points) Find the coordinates of the vertices and foci of the ellipse with equation 4 x 2 + 25 y 2 = 100....
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final_su03 - ( r 2-x 2 ). What is the average value of the...

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