Engineering Mathematics (E35 317) Final Exam December 15, 2006

Transcription

1 Engineering Mathematics (E35 317) Final Exam December 15, 2006 This exam contains six free-resonse roblems orth 36 oints altogether, eight short-anser roblems orth one oint each, seven multile-choice roblems orth one oint each, and nine true-false roblems orth one oint each, for an exam total of 60 oints. (1) ' B B cos +B.B œ + cos +B + sin +B (2) ' B B sin +B.B œ + sin +B + cos +B List of Formulas (3) ' B B + B cos +B.B œ + cos +B + $ sin +B (4) ' B B + B sin +B.B œ + sin +B + $ cos +B (5) ' / +B / cos,b.b œ +, Ð+ cos,b, sin,bñ (6) ' / +B / sin,b.b œ +, Ð+ sin,b, cos,bñ (7) ' cos +B.B œ B sin +B +B +B %+ %+ (8) ' sin +B.B œ B sin +B (9) sin Bsin C œ Ò cosðb CÑ cosðb CÑÓ (10) cos B cos C œ ÒcosÐB CÑ cosðb CÑÓ (11) sin B cos C œ ÒsinÐB CÑ sinðb CÑÓ (12) ' / / BN ÐBÑ.BœBNÐBÑ / / List of Lalace Transforms (next age)

2 0 Ð>Ñ _ Ð0 Ð>ÑÑ œ J ÐÑ (1) (2) (3) / +> > + (4) >, 8œ!ßßßá 8 8x 8 (5) (6) (7) (8) cos > sin > cosh +> sinh +> (9) +> / 0Ð> Ñ JÐ +Ñ (10) (11) +> / cos > +> / sin > + Ð +Ñ Ð +Ñ 8 +> (12) > /, 8 œ!ß ß ß á 8x Ð +Ñ 8 (13) 0Ð>Ñ JÐÑ 0Ð!Ñ (14) (15) (16) 0 Ð>Ñ J ÐÑ 0Ð!Ñ 0 Ð!Ñ 0Ð> +Ñ?Ð> +Ñ / JÐÑ + _ + 0Ð>Ñ?Ð> +Ñ / Ð0Ð> +ÑÑ (17) (18) (19)?Ð> +Ñ $ Ð> +Ñ Ð0 1ÑÐ>Ñ + / / + JÐÑ KÐÑ (20). >0Ð>Ñ J ÐÑ œ. _ Ð0Ð>Ñ Ñ (21) > 0Ð>Ñ, 8 œ!ß ß ß á Ð Ñ. 8 _ Ð0Ð>ÑÑ

3 Part I. Free Resonse In each roblem in this section, follo directions carefully, and sho all the stes arrive at the correct anser. The oint value for each roblem is shon to its left. needed to (6) 1. Use convolution to find _ Š Ð Ñ. * * * Alternative roblem 1 * * * If you do not kno ho to find this inverse Lalace transform using convolution, you may use another method. Hoever, the maximum number of oints that can be earned ith an alternative method is four, not six. (If you attemt both aroaches, I ill grade the convolution one unless you clearly cross it out.)

4 (5) 2. Find the length of the curve (or arc) traced out by the vector function r Ð>ÑœÒ > ß> ß>Ó beteen the origin and the oint Ð*ß*ß'Ñ. $ $ (2) 3. Prove that divðgrad 0Ñ œ f 0 for any scalar function 0.

5 (10) 4. Consider the vector function F ÐBßCßDÑœÒBDß&BCßBÓ, and consider the curve G hich traverses the four straight line segments from Ð!ß!ß!Ñ to Ð%ß!ß Ñ to Ð%ß $ß Ñ to Ð!ß $ß $Ñ and back to Ð!ß!ß!Ñ. Note that the given four oints all lie in the lane B C D œ!. (a) Find a arametric reresentation r for the lane B C D œ! (hence for the surface bounded by G). (b) Use Stokes's Theorem to find the value of the line integral issue of orientation.) ) G F. r. (You may ignore the

6 (4) 5. Evaluate ' BNÐBÑ.B, (here N is the Bessel function of the first kind of order to). & (Hint: See the list of formulas on the first age of the exam.)

7 (9) 6. Find the solution?ðbß >Ñ of the folloing heat initial boundary value roblem on! B 1. `? œ%`? `> `B?Ð!ß>Ñœ! B and?ð B 21ß>Ñœ! for >!?ÐBß!Ñ œ 0ÐBÑ œ!ß! B 1 œ!ß 1 B 1 (In your anser, include all (nonzero) terms through 8œ&.)

8 Part II. Short Anser Anser briefly. Each is orth one oint. For roblems 7 9, consider the folloing system of equations. A B $C D œ! A B 'C %D œ! A B $C Dœ! Ô $ The coefficient matrix for this system is Eœ ' %. Õ $ Ø 7. What is the rank of the matrix E? 8. What is the nullity of the matrix E? 9. What is the dimension of the solution sace of the system of equations? 10. Suose D is a function of B and C, say D œ 0ÐBßCÑ. Write out the limit definition of. `D `B 11. Suose that F is a conservative vector field. What is meant by a otential of F?

9 12. Name one alication of Fourier integrals. 13. Suose that during the solution of a artial differential equation, the folloing oint has been reached. _ 0ÐBÑ œ! F sin 8œ B P P What can be concluded about the constants F 8? You may clearly anser in ords, or you may rite out a formula for. F Please anser one or the other, but NOT BOTH. Make sure your anser is clear, but it does not need to be lengthy. (a) Why does a drum generally not have a definable itch as a violin string has? Your anser should address hat is different about the frequencies of the normal modes for the to instruments. (b) Name one secific asect of the design or laying of a kettle drum (timani) hich gives it a definable itch.

10 Part III. Multile-Choice Clearly circle the only correct resonse. Each is orth one oint. 15. Let 0Ð>Ñ be as ictured. Find _ Ð0Ð>ÑÑ. (A) / Š (B) (C) / Š / Š (D) / Š (E) (F) / Š / Š

11 Ô $! )! % 16. The vector ÒßßßÓ is an eigenvector for the matrix EœÖ Ù. %! Õ ' % $ Ø To hat eigenvalue does it corresond? (A) ' (B) $ (C) (D) (E) (F) (G) $ (H) ' 17. Let u, v, and be vectors in three-sace, let 0ÐBßCßDÑ be a scalar function, and let F be a vector function. Consider the folloing six comutations. (I) Ð u v Ñ (II) Ð u v Ñ (III) Ð u v Ñ (IV) divðcurl F Ñ (V) curlðgrad 0Ñ (VI) f 0 Ho many of these comutations do NOT make sense; in other ords, ho many cannot be done? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 (F) 6

12 18. Consider the folloing ordinary differential equation. *B C *BC ÐB ÑC œ! Since this differential equation has non-constant coefficients, series solutions are aroriate. What ill be the nature of these series solutions about B œ!? (A) They ill be oer series. (B) There ill be no oer series solutions, but there ill be Frobenius series solutions ith no ossibility of terms involving natural logarithms. (C) There ill be no oer series solutions, but there ill be Frobenius series solutions ith the ossibility of terms involving natural logarithms.! 19. Consider the folloing Sturm-Liouville roblem. C -C œ! C Ð!Ñ œ! C Ð1Ñ œ! The function Cœ cos $B is an eigenfunction of this roblem for some eigenvalue -. What is -? (A) 0 (B) 2 (C) 3 (D) 4 (E) 6 (F) 9

13 20. Consider the folloing artial differential equation. C? BB? C œ! Which one of the folloing airs of ordinary differential equations can be obtained from this artial differential equation using the method of searation of variables? (A) J 5J œ! K 5CK œ! (B) J 5J œ! CK 5K œ! (C) J 5J œ! K 5CK œ! (D) J 5J œ! CK 5K œ! (E) J 5J œ! K 5CK œ! (F) J 5J œ! CK 5K œ! (G) J 5J œ! K 5CK œ! (H) J 5J œ! CK 5K œ! 21. Consider the vibrations of a circular membrane under the assumtion that solutions are radially symmetric. The solutions involve the usual sine and/or cosine functions together ith hat other functions? (A) exonential functions (B) hyerbolic sine functions (C) Bessel functions (D) Legendre olynomials

14 Part IV. True-False Write out the ord true or false for each of the folloing. Each is orth one oint. 22. The set of vectors v œ Ò!ß ß!Ó, v œ Òß!ß Ó, v œ Òß ß Ó is linearly indeendent. $ 23. The set of all olynomials of degree five is a vector sace. 24. The set of all solutions of the ordinary differential equation C C $C œ! is a vector sace. Ô!Þ&!!Þ& 25. The matrix!!þ&! is a stochastic matrix. Õ!Þ&!!Þ& Ø 26. Let F œòjßjßjóœòcsin DßBCsin DßBCcos DÓ. The line integral ) F. r is ath indeendent. $ G

15 ' 27. T ÐBÑ T ÐBÑ.B œ! (here T ÐBÑ reresents the Legendre olynomial of degree 8). * 3 8 ' 28. Let 0ÐBÑ be as ictured. Then 0ÐBÑcos B.B œ! 29. Let? and? be solutions of a second-order linear homogeneous artial differential equation. Then by the Princile of Suerosition, the comlete set of solutions of the PDE is given by?œ-? -?. 30. If all other factors (including length and tension) are equal, the fundamental frequency (and hence the itch) of a string ith a greater density is higher than that of a string ith a lesser density.