2. Let 0 be as in problem 1. Find the Fourier coefficient,&. (In other words, find the constant,& the Fourier series for 0.)

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1 Engineering Mathematics (ESE 317) Exam 4 April 23, 200 This exam contains seven multiple-choice problems worth two points each, 11 true-false problems worth one point each, and some free-response problems worth 15 points altogether, for an exam total of 40 points. There is a formula list on the final page. art I. Multiple-Choice Clearly circle the only correct response. Each is worth two points. 1. Let 0 be the function given by 0ÐBÑœ if B œ on and then made if B B periodic with period :œ. Find the Fourier coefficient. (In other words, find the constant in the Fourier series for 0.) (A) (B) (C) (D) (E) 1 (F) 1 (G) 1 2. Let 0 be as in problem 1. Find the Fourier coefficient,&. (In other words, find the constant,& in the Fourier series for 0.) (A) (B) (C) (D) (E) (F) (G) (H) (I) &1 &1 &1 &1 & & & &

2 1 if B 3. Let 0ÐBÑ œ 1 if B otherwise _ Ðcos A Ñsin AB Then the Fourier integral of 0 exists and is given by 0ÐBÑ œ A.A. (You do not need to verify this.) _ Ðcos A Ñsin A A Find the value of the integral.a. (A) 1 (B) 1 (C) (D) (E) (F) (G) (H) (I) (J) 1 1 _ 4. Classify the partial differential equation `? `? `? `? `> œ `B `C `D (A) three-dimensional, linear, homogeneous (B) three-dimensional, linear, nonhomogeneous (C) three-dimensional, nonlinear, homogeneous (D) three-dimensional, nonlinear, nonhomogeneous (E) four-dimensional, linear, homogeneous (F) four-dimensional, linear, nonhomogeneous (G) four-dimensional, nonlinear, homogeneous (H) four-dimensional, nonlinear, nonhomogeneous (I) five-dimensional, linear, homogeneous (J) five-dimensional, linear, nonhomogeneous (K) five-dimensional, nonlinear, homogeneous (L) five-dimensional, nonlinear, nonhomogeneous

3 5. Suppose the fundamental frequency of a vibrating string on the interval B is ) Hz. If the initial displacement is doubled, (for example, changed from?ðbß Ñ œ 0ÐBÑ œ Þ sin 1B to?ðbß Ñ œ 0ÐBÑ œ Þ sin 1 B, what will then be the fundamental frequency of the string? (A) (B) (C) (D) (E) Hz Hz ) Hz Hz $ Hz 6. Suppose that, during the process of solving a partial differential equation, the following point has been reached. _?ÐBß Ñ œ F sin œ 0ÐBÑ œ 1 1B Use this expression to find a formula for the constants F. (A) F œ B 0ÐBÑ sin 1.B (B) F œ B 0ÐBÑ sin 1.B (C) F œ 0ÐBÑ sin.b 1B 1 (D) F œ 0ÐBÑ sin.b 1B 1 (E) F œ 0ÐBÑ sin.b B (F) F œ 0ÐBÑ sin.b B (G) F œ 0ÐBÑ sin.b 1B 1 (H) F œ 0ÐBÑ sin.b 1B 1 7. The solutions of the steady-state two-dimensional heat equation involve which types of functions? (A) cosines and sines (B) cosines and hyperbolic cosines (C) cosines and hyperbolic sines (D) sines and hyperbolic cosines (E) sines and hyperbolic sines (F) hyperbolic cosines and hyperbolic sines

4 art II. True-False Write out the word true or false for each of the following. Each is worth one point.. The following boundary value problem on B is a Sturm-Liouville problem. ww BC -Ð BÑCœ CÐ Ñœ CÐÑœ 9. The functions C œb and C œb are orthogonal on the interval B. $ 10. Consider the following Sturm-Liouville problem on the interval B. ww w C -C œ C ÐÑ œ CÐ Ñ œ The function Cœcos 1B is an eigenfunction of this Sturm-Liouville problem corresponding to the eigenvalue - œ The set of Legendre polynomials is an orthogonal family on the interval B. 1 1 & & sin cos sin cos 1 B B.B œ B B.B 12.

5 13. Any function 0 such that 0 and 0 w are piecewise continuous can be written as a Fourier series. 14. If 0 is an even function which has a Fourier series, then its Fourier series contains no sine terms. 15. The function pictured to the right is absolutely integrable. `? `? 16. Consider Laplaces equation `B `C œ. Under the usual assumption (with the separation of 5B 5B variables method) that?ðbß CÑ œ J ÐBÑKÐCÑ, computations show that J ÐBÑ œ -/ -/ and KÐCÑ œ -$ cos 5C - sin 5C. (The preceding statement is true. You do not need to verify it or judge it. The following statement is the one you must judge.) Thus the complete set of solutions to 5B 5B Laplaces equation is given by?ðbß CÑ œ Ð- / - / ÑÐ- cos 5C - sin 5CÑ. $ 17. If? and? are solutions of a partial differential equation, then any linear combination?œ-? -? is also a solution. 1. If the fundamental frequency of a vibrating string is ) Hz, then 200 Hz is one of the overtones.

6 art III. Free Response Follow directions carefully. The point value for each problem is shown to its left. if () 19. Let 0ÐBÑ œ œ B B if B (a) Draw the odd extension of 0. (b) This function (the odd extension of 0) is absolutely integrable. (You do not need to verify this.) Find its Fourier integral (not its Fourier series). Show all the steps needed to arrive at the correct solution. You do not need to do any algebraic simplifications to your final answer. If EÐAÑ œ or FÐAÑ œ, you may state that without any reason or explanation.

7 ? ÐB? Bß>Ñ? ÐBß>Ñ B B (1) 20. What is the limit of the expression? B as?b approaches zero? (6) 21. Find the solution of the following initial boundary value problem. `? `? `> œ `B?Ðß >Ñ œ?ð 1ß >Ñ œ for > 1?ÐBß Ñ œ 0ÐBÑ œ sin B sin $B for B (Do not take the time to work out the solution from the beginning using separation of variables. Rather, you may work with the solution form which we have already derived, once you recognize which type of equation this is.) (1) EXTRA CREDIT You already earned this point if you had your Formula List in class on Monday and turned it in.

8 Formula List (1) cos B.B œ sin B (2) sin B.B œ cos B (3) B cos B.B œ cos B B sin B (4) B sin B.B œ sin B B cos B (5) B B.B œ B cos cos B B $ sin B (6) B B.B œ B sin sin B B $ cos B (7) / B,B.B œ / cos, Ð cos,b, sin,bñ () / B,B.B œ / sin, Ð sin,b, cos,bñ (9) cos B.B œ B sin B B B (10) sin B.B œ B sin B (11) sin Bsin C œ Ò cosðb CÑ cosðb CÑÓ (12) cos B cos C œ ÒcosÐB CÑ cosðb CÑÓ (13) sin B cos C œ ÒsinÐB CÑ sinðb CÑÓ