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Math 121A - Fall 2001 - Borcherds - Final

# Math 121A - Fall 2001 - Borcherds - Final - TUE 09:51 FAX...

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Unformatted text preview: 04/23/2002 TUE 09:51 FAX 6434330 MOFFITT LIBRARY 001 Math 121AFinal, 2001 December 14 8:00-11:00. Borchemls Please make sure that your name is on everything you hand in. You are allowed calculators and 1 page of notes. Answer as many questions as you can. All questions have about the same number of marks. 1. Evaluate sin(6) + sin(29) + - - - + sin(n9). 2. Evaluate the integral fax/ﬂ (1 +37)? 3. Expand the function f in a sine-cosine Fourier series, where f is 1ingac<7r,Oif—ngcc<0,andf(\$+27r)=f(a:). 4. Calculate the Laplace transform fooo e“P’f(t)dt of f(t) = 6“” sin(bt). 5. Use Laplace transforms to solve the differential equation y" — 4y’ + 4y = 4, y(0) = 0, y'(0) = —2. (If y has Laplace transform Y then y’ has Laplace transform pY — 11(0) and y” has Laplace transform p2Y — py(0) — y’ Also 1 has Laplace transform 1/}: and 6“” has Laplace transform 1/(p + (1).) 6. Find the exponential Fourier transform 9(a) = fax—"Maire for the function f(a:) deﬁned by f(:1:) = :r if < 1, f(:1:) = 0 if 2 1. 7. Write and solve the Euler equations (d/dx)(8F/8y’) = 8F/3y to make the following. integral stationary: [mm/2 + Jim? 1:1 04/23/2002 TUE 09:51 FAX 6434330 MOFFITT LIBRARY 002 8. Change the dependent variable from y to a: in the following integral, then write and solve the Euler equation to make it stationary. / (3/2 + 2/2)!” 9. Calculate the inverse Laplace transform 1 c+ioo f0?) — 2m a F(z)e”dz when F is the function F(z) z 1/(24 —- 1). 10. Find the shortestdistance from the origin to the surface 3:52 +y2 — 4232 = 4. ...
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