MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2005 EXAM-3 FALL 2005 EXAM-III EXAMINATION COVER PAGE Professor Moseley
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1 MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2005 EXAM-3 FALL 2005 EXAM-III EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Friday, October 28, 2005 I swear and/or affirm that all of the work presented on this exam is my own and that I have neither given nor received any help during the exam. SIGNATURE INSTRUCTIONS DATE 1. Besides this cover page, there are 11 pages of questions and problems on this exam numbered 1-10 and 5b. MAKE SURE YOU HAVE ALL THE PAGES. If a page is missing, you will receive a grade of zero for that page. Read through the entire exam. If you cannot read anything, raise your hand and I will come to you. 2. Place your I.D. on your desk during the exam. Your I.D., this exam, and a straight edge are all that you may have on your desk during the exam. NO CALCULATORS! NO SCRATCH PAPER! Use the back of the exam sheets if necessary. You may remove the staple if you wish. Print your name on all sheets. 3. Explain your solutions fully and carefully. Your entire solution will be graded, not just your final answer. SHOW YOUR WORK! Every thought you have should be expressed in your best mathematics on this paper. Partial credit will be given as deemed appropriate. Proofread your solutions and check your computations as time allows. GOOD LUCK!! REQUEST FOR REGRADE Please regrade the following problems for the reasons I have indicated: (e.g., I do not understand what I did wrong on page.) page points score (Regrades should be requested within a week of the date the exam is returned. Attach additional sheets as necessary to explain your reasons.) I swear and/or affirm that upon the return of this exam I have written nothing on this exam except on this REGRADE FORM. (Writing or changing anything is considered to be cheating.) Date Signature Total 101 Scores Total 100
2 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 1 Let P be the point in R 3 (i.e. 3-space) which has rectangular coordinates ( 1, 1, 2 ) R. Give the cylindrical and spherical coordinates of P. Begin by drawing a picture. Be sure to give the coordinates in the correct form. 1. ( 4 pts.) The cylindrical coordinates of P are..a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..BCD. BDE. CDE. 2. ( 4 pts.) The spherical coordinates of P are..a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..BCD. BDE. CDE. A.( 2, π/4,0) C, B.( 2, π/4,π/4,) C, C.( 2, π/4,1) C, D.( 2, π/4, 2 ) C, E.( 2, π/4,π/4) C, AB.( 2, π/3, 2 ) C, AC.( 2, π/3,1) C, AD.( 2, π/4,2) C, AE.(1, π/4,1) C, BC.(1, π/4,0) C, BD.( 2, π/4,0) S, BE.( 2, π/4,π/4,) S, CD.( 2, π/4,1) S, CE.( 2, π/4, 2 ) S, DE. ( 2, π/4,π/4) S, ABC.( 2, π/3, 2 ) S, ABD.( 2, π/3,1) S, ABE.( 2, π/4,2) S, BCD.(1, π/4,1) S, BCE.(1, π/4,0) S,0) CDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE =
3 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 2 ( 12 pts.) The six figures (numbered 1,2,3,4,5,& 6) below are computer generated sketches of six functions z = f(x,y). The six figures (labeled A,B,C,D,E,&AB) on the next page are level curves for the six functions given below (not to the same scale). Match the function (surface) to the level curves by placing an A,B,C,D,E, or AB in the blank by the appropriate number below. Possible points this page = 12. POINTS EARNED THIS PAGE =
4 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 3 The six figures (numbered 1,2,3,4,5,& 6) on the previous page are computer generated sketches of six functions z = f(x,y). The six figures (labeled A,B,C,D,E,&AB) below are level curves for the six functions given on the previous page (not to the same scale). Match the function (surface) to the level curves by placing an A,B,C,D,E, or AB in the blank by the appropriate number on the previous page. Possible points this page = 12. POINTS EARNED THIS PAGE =
5 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 4 Compute the following limits. 9. ( 3pts. ) lim ( 2e xy + 2x 2 y ) =.A. B. C. D. E. AB. AC. AD. AE. BC. ( x, y) ( 1, 1) BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. BCD. BCE. BDE. CDE 10. ( 4pts. ) lim ( e xy sin (xy) / (xy) ) =.A. B. C. D. E. AB. AC. AD. AE. BC. ( x, y) ( 0, 0) BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. BCD. BCE. BDE. CDE A. 0 B.1 C.2e 2 D. 2e+2 E. 2e 1 2 AB. 2e 1 +2 AC. 2e + 1 AD. 2e 1 AE. 2e 1 +1 BC. 2e 1 1 BD. 1, BE. 2 CD. 3 CE. 2 DE.3 ABC.4 ABD. 4 ABE. 5 ACD.10 ACE. 10 BCD BCE., BDE. Does not exist but not or, CDE None of the above Possible points this page = 7. POINTS EARNED THIS PAGE =
6 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 5 Let f(x,y) = 3x 2 y 3-2xy + 7. Compute the following: 11. ( 3 pts. ) f x (x,y) =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. BCD. BCE. BDE. CDE 12. ( 3 pts. ) f y (x,y) =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. BCD. BCE. BDE. CDE 13. ( 3 pts. ) f xy (x,y) =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. BCD. BCE. BDE. CDE 14. ( 3 pts. ) f yx (x,y) =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. BCD. BCE. BDE. CDE A. 6xy 3 - xy + 7, B. 6xy 3-2y, C.3xy 3-2y, D. 6x 2 y 3-2xy + 7. E. 9x 2 y 2-2x + 7, AB.9x 2 y 2-2x, AC.6x 2 y 2-2x, AD.6x 2 y - 2x, AE. 9x 2 y 4-2x, BC. 3x 2 y 4 /4 - xy 2 + 7y, BD.x 3 y 3 - x 2 y + 7x, BE. 6y 3, CD. 18x 2 y, CE.18xy 3-2x, DE.18xy 2, ABC.18x 2 y 2-2y, ABD. 18xy 2-2, ABE.9xy 2-2, ACD.18xy- 2, ACE. 18x 2 y 3-2xy, BCD 2x 2 y 3-3xy, BCE 6x 2 y 3-2xy + 7, BDE 9x 2 y 3-2, CDE.None of the above. Possible points this page = 12. POINTS EARNED THIS PAGE =
7 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 6 Let S be the surface defined by the function z = f(x,y) = 4 x 2 y 2. and let (x,y,z) be a point P on the surface ( 3 pts.) The gradient of f ( f) is.a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD 16. ( 2 pts.) A formula for the normal to the tangent plane to the surface S at the point P is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD 17. ( 4 pts.) The set of points on S where the tangent plane to S is horizontal is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD A. 2x î + 2y ĵ + ˆk, B.2x î + 2y ĵ ˆk, C.2x î + 3y ĵ + ˆk, D.2 î + 2 ĵ + 3 ˆk E.2 î + 2 ĵ 3 ˆk, AB. 2 î + 2 ĵ ˆk, AC. x 2 î + y 2 ĵ + 3 ˆk AD. 2 î + 2 ĵ 3 ˆk AE.2 î + 2 ĵ + 3 ˆk BC.2 î + 2 ĵ BD.2x î + 2y ĵ, BE.x 2 î + y 2 ĵ, CD.2x î + 2y ĵ + 3 ˆk, CE.2x î + 2y ĵ + 3 ˆk, DE..2x î + 2y ĵ + 3 ˆk, ABC.2x î + 2y ĵ + 3 ˆk, ABD. {(0,0,4)}, ABE. {(0,0,4),(0,0, 4)}, ACD. {(0,0,1),(0,0, 1)}, ACE.{(0,0, 4)} ADE.{(0,0,4),(0,0, 4)}, BCD. {(0,0,4),(0,0, 4)}, BCE. {(0,0,4),(0,0, 4)}. BDE {(0,0,4),(0,0, 4)} CDE. {(0,0,4),(0,0, 4)} ABCD. None of the above Possible points this page = 9. POINTS EARNED THIS PAGE =
8 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 7 PRINT NAME ( ) SS No. 18. ( 3 pts.) If w = sin( 3xy ), then dw =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD 19. ( 3 pts.) If w = 3x 3 y 2 z + 2xy, then dw =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD 20. ( 3 pts.) If w = ln( 3xy + z ), then dw =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD A.[3y sin(3xy)]dx+[3x sin(3xy)]dy C.[3x sin(3xy)]dx+[3y sin(3xy)]dy B.[3x cos(3xy)]dx+[3y cos(3xy)]dy D.[3y cos(3xy)]dx+[3x cos(3xy)]dy E. [3 cos(3xy)]dx+[3 cos(3xy)]dy AB. [y cos(3xy)]dx+[x cos(3xy)]dy AC. [3y cos(3xy)]+[3x cos(3xy)] AD.[3y cos(xy)]dx+[3x cos(xy)]dy AE. [9x 2 y 2 z + 2y]dx + [6x 3 yz + 2x]dy BC. [9x 3 y 2 z + 2xy]dx + [6x 3 y 2 z + 2xy]dy +3x 3 y 2 z dz BD.[9x 2 y 2 z + 2y]dx + [6x 3 yz+2x]dy +3x 3 y 2 dz BE.[x 4 y 2 z+xy]dx + {(3/2)x 3 y 3 z + xy 2 ]dy +[(3/2)x 3 y 2 z 2 dz CD. [6x 3 y 2 z+2xy]dx +[9x 3 y 2 z+2xy]dy +3x 3 y 2 dz CE. [9x 3 y 2 z+2xy]dx +{6x 3 y 2 z +2xy]dy +3x 3 y 2 dz DE.[3x 3 y 2 z+2xy]dx +[3x 3 y 2 z+2xy]dy+3x 3 y 2 z dz ABC. [3x 2 y 2 z+2xy]dx +{3x 3 yz+2xy]dy +3x 3 y 2 z dz 3ydx 3xdy 3xdx 3ydy dz 3ydx 3xdy dz 3ydx 3xdy dz ABD., ABE.,.ACD., ACE., 3xy z 3xy z 3xy z 3xyz z ydx xdy dz 3ydx 3xdy dz 3ydx 3xdy dz ydx xdy dz ADE., BCD., BDE., CDE, 3xy z xy z 3xy 3z xy z ABCD None of the above. Possible points this page = 9. POINTS EARNED THIS PAGE =
9 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 8 Let w = f(x,y) = 2x 2 e y where x = g(t) and y = h(t). Hence w = f(g(t),h(t)). Assume g(0) = 1, h(0) = 0, g'(0) = 2, and h'(0) =3. You are to compute w x (x,y) (1,0) 21. (3 pts.) =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. w dw dt t 0 DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD ABCE. ABDE. ACDE. BCDE.. =. 22. (3 pts.) y A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. (x,y) (1,0) DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD ABCE. ABDE. ACDE. BCDE (4 pts.) dw dt =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. t 0 DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BDE. CDE ABCD ABCE. ABDE. ACDE. BCDE.. A.0 B.1 C.2 D.3 E.4 AB.5 AC.6 AD.7 AE.8 BC.9 BD.10 BE.11 CD.12 CE.13 DE.14 ABC.15 ABD. 1 ABE. 2.ACD. 3 ACE. 4 ADE. 5 BCD. 6 BDE. 7 CDE 8 ABCD 9 ABCE. 10 ABDE. 11 ACDE. 12 BCDE. None of the above. Possible points this page = 10. POINTS EARNED THIS PAGE =
10 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 9 (13 pts.) Sketch and describe (i.e. label and provide information as appropriate) the graph of the equation: x 2 - y 2 - z 2 = 1. Remember grading of sketches is subjective. Draw the best sketch you can in the time allotted. Possible points this page = 13. POINTS EARNED THIS PAGE =
11 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 10 PRINT NAME ( ) SS No. ( 8 pts.) Sketch and describe (i.e. label and provide information as appropriate) the graph of the equation in spherical coordinates: φ = π/4. Remember grading of sketches is subjective. Draw the best sketch you can in the time allotted. Possible points this page = 8. POINTS EARNED THIS PAGE =
12 MATH 251 EXAM III Fall 2005 Prof. Moseley Page 11 ( 12 pts.) Sketch and describe (i.e. label and provide information as appropriate) the graph of the function: f(x,y) = 6-3x - 2y. Remember grading of sketches is subjective. Draw the best sketch you can in the time allotted. Possible points this page = 12. POINTS EARNED THIS PAGE =
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