Engineering Mathematics (E35 317) Final Exam December 18, 2007

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1 Engineering Mathematics (E35 317) Final Exam December 18, 2007 This exam contains 18 multile-choice roblems orth to oints each, five short-anser roblems orth one oint each, and nine true-false roblems orth one oint each, for an exam total of 50 oints. art I. Multile-Choice Clearly circle the only correct resonse. Each is orth to oints. 1. Find the solution of the folloing initial value roblem. C *C œ Ð> Ñ CÐÑ œ ' C ÐÑ œ (A) CÐ>Ñ œ cos > > / > > (B) CÐ>Ñ œ cos > > / (C) CÐ>Ñ œ cos > Ð> Ñ?Ð> Ñ (D) CÐ>Ñ œ cos > Ð> Ñ?Ð> Ñ (E) CÐ>Ñ œ ' cos > > / > > (F) CÐ>Ñ œ ' cos > > / (G) CÐ>Ñ œ ' cos > Ð> Ñ?Ð> Ñ (H) CÐ>Ñ œ ' cos > Ð> Ñ?Ð> Ñ 2. Find ÐlnÐ= %ÑÑ. _ (A) > cos > (B) > cos > (C) > cos > > (D) cos > (E) > cos > (F) > cos > (G) > > (H) > > (I) > > (J) > (K) > > (L) > > >

2 3. Find the dimension of the null sace of the folloing matrix. (A) (B) (C) (D) (E) % (F) & Ô ( ' Õ Ø Ô 4. Consider the matrix A œ Ö Ù. Õ Ø Which of the folloing vectors are eigenvectors of A? Ô Ô Ô (I) Ö Ù (II) Ö Ù (III) Ö Ù Õ Ø Õ Ø Õ Ø (A) I only (B) II only (C) III only (D) I and II only (E) I and III only (F) II and III only (G) I, II, and III (H) none of these 5. What curve is reresented by the folloing arametric equations? B œ h > C œ & D œ cosh > (A) line (B) ellise (including the ossibility of a circle) (C) hyerbola (or one branch of a hyerbola) (D) helix

3 6. Let r Ð>Ñ œ >ß > ß >. Find the normal acceleration vector a norm at the oint here > œ. & (A) ß ß % % % (B) ß ß (C) ß ß (D) ß ß % % (E) ß ß (F) ß ß % (G) ß ß (H) ß ß % % (I) ß ß % (J) ß ß (K) ß ß (L) ß ß 7. Let 0 be the scalar function given by 0ÐBßCßDÑœBC D. Find the direction of greatest increase of 0 at the oint Ðß ß Ñ. (A) Òßß Ó (B) ÒßßÓ (C) ÒßßÓ (D) Òßß Ó (E) ÒßßÓ (F) ÒßßÓ (G) Òßß Ó (H) ÒßßÓ (I) ÒßßÓ

4 8. Let F be the vector function (in the lane) given by F ÐBß CÑ œ ÒB Cß BC Ó, and let G be the curve hich follos the straight line segments from Ðß Ñ to Ðß Ñ to Ðß Ñ back to Ðß Ñ. Which of the folloing exressions ould give the correct value of the line integral ) F. r? (A) ' ' (B) ' ' (C) ' ' (D) ' ' (E) ' ' (F) ' ' (G) ' ' (H) ' ' C C ÒC B Ó.B.C B B ÒC B Ó.B.C C C ÒC B Ó.C.B B B ÒC B Ó.C.B C C Ò BC BCÓ.B.C B B Ò BC BCÓ.B.C C C Ò BC BCÓ.C.B B B Ò BC BCÓ.C.B G. Suose a surface is reresented arametrically by r Ð?ß@ÑœÒ@ cos?ß@?ß@ó. Find a normal 1 vector to the surface at the oint here?œ œ. (A) Ò ßß Ó (B) Ò ßßÓ (C) Òß ß Ó (D) Òß ß Ó (E) Òßß Ó (F) ÒßßÓ (G) Òßß Ó (H) ÒßßÓ

5 10. Consider the eighth of a solid ball given by B C D Ÿ ', B, C, D, and let W be the surface of this solid, oriented ith the ositive direction outard. Let F be the vector function given by F ÐBßCßDÑœÒB C ßC D ßD B ÓÞ Which of the folloing integrals ill give the correct value of the surface integral '' Ð F n Ñ.E? W 1 1 ' (A) ' ' ' 1 1 ' (B) ' ' ' 1 1 ' (C) ' ' ' 1 1 ' (D) ' ' ' ) ). ( ' 3. 3.). ( 3. 3.). (E) ' ' ' 3. 3.). 1 1 ' (F) ' ' ' 1 1 ' % 3. 3.). (G) ' ' ' ). 1 1 ' % (H) ' ' ' 3. 3.) œ 7œ 11. Consider the differential equation B C BC Ð B ÑCœ. It has a solution of the form CœB 7 B œ 7 B. (You do not need to verify this.) It turns out that œ, and that for 7, 7 œ 7 7. (You do not need to verify this either.) Find a solution of the differential equation. % '% % ' % % '% % % ' % '% (A) CœB B B â (B) CœB B B â (C) C œ B B â (D) C œ B B â (E) Cœ B (F) Cœ B B â % ' % % % % '% ' (G) Cœ B (H) Cœ B B â B â B â

6 12. Ho many of the folloing are even functions? (I) 0ÐBÑ œ & (II) 0ÐBÑ œ B (III) 0ÐBÑ œ / B (A) (B) (C) (D) (E) % (F) & (IV) % & 0ÐBÑ œ Bcos B (V) 0ÐBÑ œ B B 13. Which ords ould best describe grahs I, II, and III resectively? I II III (A) Bessel, Fourier, gamma (B) Bessel Fourier, Legendre (C) Bessel, gamma, Fourier (D) Bessel, gamma, Legendre (E) Bessel, Legendre, Fourier (F) Bessel, Legendre, gamma (G) Fourier, Bessel, gamma (H) Fourier, Bessel, Legendre (I) Fourier, gamma, Bessel (J) Fourier, gamma, Legendre (K) Fourier, Legendre, Bessel (L) Fourier, Legendre, gamma (M) gamma, Bessel, Fourier (N) gamma, Bessel, Legendre (O) gamma, Fourier, Bessel () gamma, Fourier, Legendre (Q) gamma, Legendre, Bessel (R) gamma, Legendre, Fourier (S) Legendre, Bessel, Fourier (T) Legendre, Bessel, gamma (U) Legendre, Fourier, Bessel (V) Legendre, Fourier, gamma (W) Legendre, gamma, Bessel (X) Legendre, gamma, Fourier

7 14. Find the Fourier series for the odd extension of the folloing function. 0ÐBÑ œ ß B 1 ß 1 B 1 ' B B (A) 0ÐBÑ œ 1 B â ' B B (B) 0ÐBÑ œ 1 B â ' B B &B (C) 0ÐBÑ œ 1 & â ' B B &B (D) 0ÐBÑ œ 1 & â ' (E) 0ÐBÑ œ 1 B % B ' B â ' (F) 0ÐBÑ œ 1 B B B â 15. Consider the folloing artial differential equation. `? `? `> `B > œ Write the solution as?ðbß >Ñ œ J ÐBÑKÐ>Ñ, and continue ith the method of searation of variables. What air of ordinary differential equations results? 5 > (A) J J œ K 5K œ 5 > (B) J J œ K 5K œ (C) J 5J œ K 5>K œ (D) J 5J œ K 5>K œ 5 (E) J 5J œ K > K œ 5 (F) J 5J œ K > K œ (G) J 5>J œ K 5K œ (H) J 5>J œ K 5K œ

8 _ 16. Suose G cos œ 0ÐBÑ. Find a formula for G. 8œ 81 81B 8 8 (A) G8 œ cos 8 1 ' 8 B 0ÐBÑ 1.B (B) G8 œ 8 1 ' 8 B 0ÐBÑ cos 1.B (C) G8 œ cos 8 1 B ' 8 0ÐBÑ 1.B (D) G8 œ 8 1 B ' 8 0ÐBÑ cos 1.B 81 81B ' 81B cos 81B 8 cos ' 81 81B 8 ' 81 cos (E) G œ cos ' 0ÐBÑ.B (F) G œ 0ÐBÑ.B (G) G œ 0ÐBÑ.B (H) G œ 0ÐBÑ.B 17. Consider the folloing boundary value roblem for heat distribution in the vicinity of a shere of radius Vœ. `? `? `? cot `? ` < < `< < ` < ` œ?ð2ßñœ0ðñœ% cos lim?ð<ß Ñ œ?ðß Ñ <Ä_ bounded Find the solution in the interior of the shere. (A)?Ð<ß Ñ œ < (B)?Ð<ß Ñ œ cos (C)?Ð<ß Ñ œ < cos (D)?Ð<ß Ñ œ arccos (E)?Ð<ß Ñ œ < arccos (F)?Ð<ß Ñ œ %< (G)?Ð<ß Ñ œ 4cos (H)?Ð<ß Ñ œ 4< cos (I)?Ð<ß Ñ œ 4 arccos (J)?Ð<ß Ñ œ 4< arccos

9 18. As usual, denote the zeros of the Bessel function N ÐBÑ by ß ß ßá. It turns out that the zeros ß ßá are not integer multiles of. This fact has significant consequences for hat hysical alication? (A) vibration of a violin string (B) vibration of a circular drumhead (C) heat conduction along a long bar or ire (D) steady-state heat distribution on a rectangular surface (E) steady-state heat distribution in the vicinity of a shere art II. Short Anser Briefly anser each of the folloing. You do not need to sho any ork. 1. The inverse Lalace transform of a roduct is the of the individual inverse Lalace tranforms. Ô Ô % % 20. Consider the vectors v œ Ö Ù and v2 œ Ö Ù in. Find a vector v3 in such that the set Õ Ø Õ Ø Öv ßv ß v is linearly deendent. Ô Ô % % 21. Consider the vectors v œ Ö Ù and v2 œ Ö Ù in. Find a vector v3 in such that the set Õ Ø Õ Ø Öv ßv ß v is linearly indeendent. 0 ÐB? BßCÑ 0 ÐBßCÑ?BÄ? B B B 22. Let Dœ0ÐBßCÑ. Then lim œ. `? `? `? `B `C `D 23. Let? be a function of B, C, and D. The exression is called the of?.

10 art III. True-False Write out the ord true or false for each of the folloing. Each is orth one oint. > > 24. ' : ' > : / cosð> :Ñ.: œ / cos :.: 25. The set of all solutions of the differential equation C BC C œ is a vector sace of dimension to. Ô 26. A basis for the ro sace of the matrix A œ ÖÒ ÓßÒ Ó Õ Ø is. 27. If Ais a ( x ( nongular matrix, then Ahas rank (. 28. A stochastic matrix alays has - œ as an eigenvalue. 2. For every vector field v, grad(curl v ) œ. 30. In the evaluation of a line integral over a curve from a oint E to a oint F, the same value ill be obtained regardless of hich curve from E to F is used. 31. The folloing differential equation does not have any oer series solutions about B œ but does have at least one Frobenius series solution about B œ. Ð B ÑC BC C œ 32. As usual, let and reresent the first to zeros of the Bessel function N, and let V be a V ositive constant. Then ' <N Š < N Š <.< œ. V V

11 Table of Lalace Transforms 0 Ð>Ñ _ Ð0 Ð>ÑÑ œ J Ð=Ñ (1) / > = (2) >, 8œßßßá 8 8x = 8 (3) (4) (5) (6) cos = > = > cosh > h > = = = = = = = = = (7) > / 0Ð>Ñ J Ð= Ñ (8) () / > > cos = > / = > = Ð= Ñ = = Ð= Ñ = 8 > (10) > /, 8 œ ß ß ß á 8x Ð= Ñ 8 (11) 0Ð>Ñ =JÐ=Ñ 0ÐÑ (12) (13) 0 Ð>Ñ = J Ð=Ñ =0ÐÑ 0 ÐÑ = 0Ð> Ñ?Ð> Ñ / JÐ=Ñ (14)?Ð> Ñ = / = (15) (16) = 0Ð>Ñ?Ð> Ñ / _ Ð0Ð> ÑÑ = Ð> Ñ / (17) (18) Ð0 1ÑÐ>Ñ >0Ð>Ñ JÐ=Ñ KÐ=Ñ. J Ð=Ñ œ.= JÐ=Ñ (1) 0Ð>Ñ (ith eriod :) / ' : := => / 0Ð>Ñ.>

12 Formula List : : : (1) If 0 is an even function, then ' 0ÐBÑ.B œ ' 0ÐBÑ.B. : ' : (2) If 0 is an odd function, then 0ÐBÑ.B œ. B B (3) ' cos B.B œ B (4) ' B.B œ cos B (5) ' B cos B.B œ cos B B (6) ' B B.B œ B cos B (7) ' B B B cos B.B œ cos B B (8) ' B B B B.B œ B cos B () ' / B / cos,b.b œ, Ð cos,b,,bñ (10) ' / B /,B.B œ, Ð,B, cos,bñ (11) ' cos B.B œ B B B % % (12) ' B.B œ B B (13) B C œ Ò cosðb CÑ cosðb CÑÓ (14) cos B cos C œ ÒcosÐB CÑ cosðb CÑÓ (15) B cos C œ ÒÐB CÑ ÐB CÑÓ TÐBÑœ TÐBÑœB Legendre olynomials TÐBÑœ B TÐBÑœ B B & TÐBÑœ B B % & % & ) % ) T ÐBÑ œ B B B & ' & & & ) % )