MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2005 FINAL EXAM FALL 2005 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley

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1 MATH 5 MATH 5: Multivariate Calculus MATH 5 FALL 5 FINAL EXAM FALL 5 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Tuesday, December 3, 5 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. Besides this cover page, there are 6 pages of questions and problems on this exam. 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.. 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!! page points score Total

2 MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page Matrix algebra. Circle the correct answer from the choices below to fill in the blank.. Using the abbreviated (tensor) notation for a matrix discussed in class, let A = [a ij ], B=[b ij ], C=[c ij ], D=[d ij ], and E=[e ij ] be nxn square matrices..( pt.) If α is a scalar and C = αa, then c ij =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE.. ( pt.) If D = A + B, then d ij =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. 3. ( pts.) If E = AB, then e ij =.A. B. C. D. E. AB. AC. AD. AE. BC. Possible Answers for questions 7, 8, and 9. BD. BE. CD. CE. DE. A. αa ij, B. βa ij, C. b ij a ij, D. b ij + a ij, E.a ij /b ij, AB.= a b, AC. a b, AD. a ij b ij, AE. a ij, BC. a ij +c ij BD. b ij, BE. b ij d ij, CD. b ij + e ij, CE. b ij a ij, DE. None of the above ( pts.) True or False. Matrix Algebra. Circle True or False, but not both. If I cannot read your answer, it is WRONG. n i ij ij n k ik kj True or False True or False True or False True or False True or False 4. Matrix addition is associative. 5. Matrix addition is not commutative. 6. α,βr and AR m n, α(βa) = (αβ)a. 7. Multiplication of square matrices is associative. 8. Multiplication of square matrices is commutative. True or False 9. If A and B are invertible square matrices, then (AB) - exists and (AB) - = A - B -. True or False. If A is an invertible square matrix, then (A - ) - exists and (A - ) - = A. True or False. If A and B are square matrices, then (AB) T exists and (AB) T = A T B T. True or False. If A is a square matrix, then A T and (A T ) T exist and (A T ) T = A.

3 True or False 3. If A is an invertible square matrix, then (A T ) - exists and (A T ) - = (A - ) T. Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page On the back of the previous sheet, solve the x + x + x 3 - x 4 = system of linear algebraic equations Be sure to write you answer in the correct form. Circle the correct answer x + x + x 3 = from the possibilities below x 3 + x 4 = x - x 3 + x 4 = 4. (3 pts.) x =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE. 5. (3 pts.) x =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE. 6. (3 pts.) x 3 =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE. 7. (3 pts.) x 4 =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE. Possible answers for questions 4, 5, 6, and 7. A., B., C. 3, D. 4, E. 5, AB. 6, AC. 7, AD. 8, AE. 9, BC., BD., BE., CD.3,

4 CE.4, DE.5, ABC.6, ABD.7, ABE 8, BCD.9, BCE., CDE. None of the above. Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 3 ( 5 pts.) True or false. Solution of Linear Algebraic Equations having possibly complex coefficients. Assume A is an m n matrix of possibly complex numbers, that is an n column vector of (possibly complex) unknowns, and that b A x x is an m (possibly complex valued) column vector. Now consider. (*) mxn nx b mx Under these hypotheses, determine which of the following is true and which is false. It true, circle True. It false, circle False. If I can not read your answer, it is wrong. b 8. True or False, If, then (*) always has an infinite number of solutions. 9. True or False, The vector equation (*) always has exactly one solution.. True or False, If A is square (n=m) and nonsingular, then (*) always has a unique solution.. True or False, The equation (*) can be considered as a mapping problem from one vector space to another. A i i. True or False, If then (*) has a unique solution.

5 Total points this page = 5. TOTAL POINTS EARNED THIS PAGE MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 4 i x You are to solve A x b where A,, and. Be sure you write your x x x i x y b i answer according to the directions given in class (attendance is mandatory) for these kinds of problems. A b i U c U c 3. (4 pts.) If is reduced to using Gauss elimination, then =. i i i i A., B., C., D., E., AB. None of the above are possible. 4. ( 4 pts.) The general solution of can be written as. A x b x x x i i i A. No Solution, B. x, C., D., E., x y x y x i i i x y x y x y AB., AC., AD., BC. None of the above correctly describes the solution or set of solutions.

6 Total points this page = 8. TOTAL POINTS EARNED THIS PAGE MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page a b Let A = and A =. Compute the inverse of A. Be sure to explain clearly 9 c d and completely your method. Circle the correct values of a, b, c, and d from the possiblities below: 5. ( pts.) a =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE.. 6. ( pts.) b =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE.. 7. ( pts.) c =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE.. 8. ( pts.) d =.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE. AC. ABD. ABE BCD. BCE. CDE.. Possible answers for questions 5, 6, 7, and 8. A., B., C. 3, D. 4, E. 5, AB. 8, AC. 9, AD.!, AE., BC., BD., BE., CD. 5,

7 CE. 6, DE. 3, ABC. 4, ABD. 4, ABE. 4, BCD. 45, BCE. 55. CDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 6 {v,v,...,v } 9. ( pts.) Let S = n V where V is a vector space. Choose the completion of the following definition of what it means for S to be linearly independent. {v,v,...,v } Definition. The set S = n V where V is a vector space is linearly independent if A. The vector equation c has an infinite number of solutions. v +cv +...+cnv n = B. The vector equation c has a solution other than the trivial solution. v +cv +...+cnv n = C. The vector equation c has only the trivial solution c = c = = c n =. v +cv +...+cnv n = D. The vector equation c has at least two solutions. v +cv +...+cnv n = E. The vector equation c has no solution. v +cv +...+cnv n = AB. The associated matrix is nonsingular. AC. The associated matrix is singular On the back of the previous sheet, determine Directly Using the Definition (DUD) if the following sets of vectors are linearly independent. As explained in class, circle the appropriate answer that gives an appropriate method to prove that your results are correct (Attendance is mandatory). Be careful. If you get them backwards, your grade is zero. 3. (4 pts.) Let S =.{[, 4, 8] T, [3, 6, ] T }. Circle the correct answer A. S is linearly independent as c [, 4, 8] T + c [3, 6, ] T = [,,] implies c = and c =. B. S is linearly independent as 3[, 4, 8] T + () [3, 6, ] T = [,,]. C. S is linearly dependent as c [, 4, 8] T + c [3, 6, ] T = [,,] implies c = and c =. D. S is linearly dependent as 3[, 4, 8] T + () [3, 6, ] T = [,,]. E. S is neither linearly independent or linearly dependent as the definition does not apply. 3. (4 pts.) Let S = {[,, 6] T, [3, 3, 9] T }. Circle the correct answer A. S is linearly independent as c [,, 6] T + c [3, 3, 9] T = [,,] implies c = and c =. B. S is linearly independent as 3[,, 6] T + () [3, 3, 9] T = [,,]. C. S is linearly dependent as c [,, 6] T + c [3, 3, 9] T = [,,] implies c = and c =. D. S is linearly dependent as 3[,, 6] T + () [3, 3, 9] T = [,,]. E. S is neither linearly independent or linearly dependent as the definition does not apply.

8 Total points this page =. TOTAL POINTS EARNED THIS PAGE MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 7 3 Let A = On the back of the previous sheet, compute the determinant of A ( 3 pts.) The first step of the Laplace Expansion in terms of the first column yields det(a) =. 33. (3 pts.) The first step in using Gauss Elimination to find det(a) yields det(a) =.: 34. (3 pts.) The numerical value of det(a) is det(a) = A. () (3), B. () (3) 4, C. () 4 (), D. (3) 4 (), E. () 4 (3), AB. () ( 3) 4, AC. () () 4, AD. (3) ( 3) 4, AE. () 4 ( 3) 4, BC., BD., BE., CD., CE., DE., ABC., ABD., ABE., ACD., ACE., 4 ADE. 3, BCD. 4, BCE. 5, BDE. 8, CDE. 9, ABCD., ABCE., ABDE., ACDE.3 BCDE. None of the above.

9 Possible points this page = 9. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 8 PRINT NAME ( ) SS No. Let a and b be the vectors, a = <,-,> = (,,) = [,,] T = î ĵ + ˆk and b = <,,3> = (,,3) = [,,3] T = ĵ + 3 ˆk. 35. (3 pts.) Then the dot product is a b = ( a, b ) = a, b. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. BCD. BCE, CDE, ABCD. 36. (5 pts.) The cross product is a b. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. BCD. BCE, CDE, ABCD. Possible answers for questions 35 and 36. A., B., C.3, D.4, E.5 AB. AC. AD.3 AE.4 BC. 3i ˆ j ˆ k ˆ, BD. 3i ˆ j ˆ kˆ, BE. 3i ˆ j ˆ k ˆ,CD. 3i ˆ j ˆ kˆ, CE. 3i ˆ j ˆ kˆ, DE. 3i ˆ j ˆ kˆ, ABC. 3i ˆ j ˆ kˆ, ABD. 3i ˆ j ˆ kˆ,

10 ABE. 3i ˆ j ˆ k ˆ, BCD. 3i ˆ j ˆ kˆ, BCE. 3i ˆ j ˆ kˆ, CDE. 3i ˆ j ˆ kˆ, ABCD. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 9 You are to find an equation for a plane and an equation for a sphere. Recall that these equations are not unique. To get the equations given in the answers below, you should use the procedures illustrated in class (attendance is mandatory). Choose the answer that best fills in the blank from the possibilities below. Then circle the appropriate letter after the question. 37. (4 pts.) Let P be the plane through the origin and parallel to the plane with equation 4x + y = 3 z +. An equation for P is.a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. 38. (4 pts.) Let S be the sphere of radius 3 with center at (,3,). An equation for S is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. Possible answers for this page. A. 4x + y + 3 z = B. 4x + y = 3 z +, C. 4x + y 3 z =, D. x + 3 y = E. 4x + y = AB. 4x + y 3 z =, AC. 4x + y + 3z =, AD. x +3 y = 3 AE. (x +) + (y + 3) = 9, BC. (x ) + (y + 3) + z = 9, BD. (x +) + (y 3) + z = 9, BE. (x ) + (y 3) = 3, CD.(x ) + (y 3) + z = 9, CE. (x +) + (y 3) + z = 3,

11 DE. (x +) + (y + 3) = 3, ABC. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page Suppose that the position vector for a point mass M as a function of the time t is given by: r = (e t ) i + (t 3 + 3t ) j + ( 3 sin(t) ) k You are to compute the velocity and acceleration for M. Choose the answer that best fills in the blank from the possibilities below. Then circle the appropriate letter after the question.(be careful. Remember once you make a mistake, everything beyond that point is wrong.) 39. ( 5 pts.) Let the velocity vector for the point mass M be v(t). Then v(t) =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..BCD. BDE. CDE. v(t) = 4. ( 5 pts.) Let the acceleration vector for the point mass M be a(t). Then a(t) =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..BCD. BDE. CDE. a(t) = Possible answers for this page. A. (e t ) î + (t 3 + 3t ) ĵ + (3 sin(t)) ˆk B. (e 3 t ) î + (t 3 + 3t ) ĵ + ( 3 sin(t)) ˆk C. (4e t ) î + (6t + 6t ) ĵ + (3 cos(t) ) ˆk D. (4e 3 t ) î + (6t 3 + 6t ) ĵ + ( 3 cos(t) ) ˆk D. (e 3 t ) î + (6t 3 + 6t ) j + (3 sin(t) ) ˆk E. (e 3 t ) î + (t 3 + 3t ) ĵ + (3 sin(t) ) ˆk AB (e 3 t ) î + (6t + 6t ) ĵ + (3 cos(t) ) ˆk AC (8e t ) î + ( t + 6 ) ĵ (3 sin(t) ) ˆk AD. (8e t ) î + (6 t + 3 ) ĵ (3 sin(t) ) ˆk AE (e 3 t ) î + (t + 6t ) ĵ + (3 sin(t) ) ˆk

12 BC. (4e t ) î + (6t + 3t ) ĵ + (3 sin(t) ) ˆk BD. None of the above. Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page Let L and L be the two intersecting lines (check t = ) whose parametric equations are given by: L : x = 3t +, y =, z = t L : x =, y = t, z = t where t R. You are to find an equation of the plane P that contains L and L. 4. ( pts.) The point where the lines intersect is.a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. 4. ( pts.) A vector in the direction of L is.a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. 43. ( pts.) A vector in the direction of L is.a. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. 44. (4 pts.) A normal to the plane P is. (Recall that a normal vector to a plane is not unique.) A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. 45. (4 pts.) An equation for the plane P is.(recall that an equation for the plane is not unique.) A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. Possible answers for this page. A. (,,) B.(,,) C.(,,) D.((,,) E. (3,,) AB..(,,) AC. î + ĵ + 3 ˆk AD. 3 î + 3 ˆk AE. ĵ + 3 ˆk BC. 3 î + ˆk BD. î + ĵ + 3 ˆk BE. ĵ ˆk CD. ĵ + 3 ˆk CE.. î + ĵ + 3 ˆk DE.. î + ĵ + 3 ˆk ABC. î + ĵ + 3 ˆk ABD. î + 3 ĵ + 3 ˆk ABE. x + 3y + 3z = 3 ACD. x + 3y + 3z = ACE. x + 3y + z = 3 ADE. x 3y + 3z = BCD. x + 3y + 3z = BCE. x +3y + z = 3 BDE x + 3y + 3z = CDE. None of the

13 above. Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page Suppose that the position vector for a point mass as a function of the time t is given by: r = 3 t i + cos(t) j + sin(t) k. (Be careful! If you make a mistake, the rest is wrong.) 46. (3 pts) The velocity v(t) at time t = is v() =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. v(t) v() 47. (3 pts) The acceleration a(t) at time t = is a() =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. a(t) a() 48. (4 pts) a() v() is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. a() v() 49. (3 pts) v() is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. v() 5. (4 pts) The curvature at t = of the curve traced out by the particle is κ =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE..BCD. BCE. BDE. CDE. κ = Possible answers for this page. A. 3 î B. 3 î + 4 ˆk C. 3 î + 4 ĵ + 4 ˆk D. 3 î + 3 ˆk E. 3 ĵ + 4 ˆk AB 3 î + 4 ĵ + 4 ˆk AC. 8 ĵ AD. 8 ˆk AE. 8 î BC. 8 ĵ BD. 8(4 î + 3 ˆk ) BE. 8(3 î + 4 ĵ ) CD. 8( 4 ĵ + 3 ˆk ) CE. 8(4 î 3 ˆk ) DE. ABC. 3 ABD. 4 ABE. 5 ACD. ACE. ADE. BCD

14 4 5 BCD. BDE. CDE None of the above. Possible points this page = 7. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 3 Let S be the surface defined by the function z = f(x,y) = 4 x y. and let (x,y,z) be a point P on the surface.. 5. ( 3 pts.) Using geometric notation ( î and ĵ, or î, ĵ, and ˆk ), 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 5. ( 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 53. ( 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. x î + y ĵ + ˆk, B.x î + y ĵ ˆk, C.x î + 3y ĵ + ˆk, D. î + ĵ + 3 ˆk E. î + ĵ 3 ˆk, AB. î + ĵ ˆk, AC. x î + y ĵ + 3 ˆk AD. î + ĵ 3 ˆk AE. î + ĵ + 3 ˆk BC. î + ĵ BD.x î + y ĵ, BE.x î + y ĵ, CD.x î + y ĵ + 3 ˆk, CE.x î + y ĵ + 3 ˆk, DE..x î + y ĵ + 3 ˆk, ABC.x î + y ĵ + 3 ˆk, ABD. {(,,4)}, ABE. {(,,4),(,,4)}, ACD. {(,,),(,,)}, ACE.{(,,4)}

15 ADE.{(,,4),(,,4)}, BCD. {(,,4),(,,4)}, BCE. {(,,4),(,,4)}. BDE {(,,4),(,,4)} CDE. {(,,4),(,,4)} ABCD. None of the above Possible points this page = 9. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 4 Let w = f(x,y) = x e y where x = g(t) and y = h(t). Hence w = f(g(t),h(t)). Assume g() =, h() =, g'() =, and h'() =3. You are to compute w x (x,y) (,) 54. (3 pts.) =. A. B. C. D. E. AB. AC. AD. AE. BC. 55. (3 pts.) w y dw dt BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. t CDE ABCD ABCE. ABDE. ACDE. BCDE.. (x,y) (,) =. 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 ABCE. ABDE. ACDE. BCDE (4 pts.) dw dt =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. t BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD ABCE. ABDE. ACDE. BCDE..

16 A. B. C. D.3 E.4 AB.5 AC.6 AD.7 AE.8 BC.9 BD. BE. CD. CE.3 DE.4 ABC.5 ABD. ABE..ACD.3 ACE.4 ADE.5 BCD.6 BDE.7 CDE 8 ABCD 9 ABCE. ABDE. ACDE. BCDE. None of the above. Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 5 Let P be the point in R 3 (i.e. 3-space) which has rectangular coordinates (,, ) R. Give the cylindrical and spherical coordinates of P. Begin by drawing a picture. Be sure to give the coordinates in the correct form. 57. ( 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. 58. ( 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.(, π/4,) C, B.(, π/4,π/4,) C, C.(, π/4,) C, D.(, π/4, ) C, E.(, π/4,π/4) C, AB.(, π/3, ) C, AC.(, π/3,) C, AD.(, π/4,) C, AE.(, π/4,) C, BC.(, π/4,) C, BD.(, π/4,) S, BE.(, π/4,π/4,) S, CD.(, π/4,) S, CE.(, π/4, ) S, DE. (, π/4,π/4) S,

17 ABC.(, π/3, ) S, ABD.(, π/3,) S, ABE.(, π/4,) S, BCD.(, π/4,) S, BCE.(, π/4,) S,) CDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 6 Let w = f(x,y) = 5 e 3x cos(y), P be the point (,), and be a unit vector in the direction of. û v 3iˆ 4j ˆ 59. (4 pts.) Using geometric notation (i.e. î and ĵ or î, ĵ, and ˆk ), f. (x, y) (,) A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD, ABCE, ABDE, ACDE, BCDE. D f(p) = D f. uˆ uˆ 6. (4 pts.) A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. (x,y) (,) CD. CE. DE. ABC. ABD. ABE.. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD, ABCE, ABDE, ACDE, BCDE. Possible answers. A. 5 î + 5 ĵ + ˆk, B. 5 î + 5 ĵ, C. î 5 ĵ ˆk, D.5 î 5 ĵ ˆk, E. 5 î 5 ĵ, AB. 5 î, AC. 5 î 5 ĵ ˆk, AD. 5 î 5 ĵ ˆk, AE. 5 î 5 ĵ ˆk, BC., BD., BE. 5, CD. 9, CE., DE., ABC.3, ABD.45, ABE. 6, ACD.

18 ACE. 5, ADE. 9, BCD., BCE.45, BDE. 8/5, CDE. 4/(5), ABCD. 8/5, ABCE.4/(5), ABDE. 8/ 5, ACDE 8/ 5, BCDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 7 Let S be the surface which is defined by the graph of the function z = f(x,y). Suppose using geometric notation (i.e. î and ĵ or î, ĵ, and ˆk ), that 6iˆ 4ˆj. and that f(,) = f (x, y) (,) 6. (4 pts.) Using geometric notation, a normal to the surface S when x = and y = is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD. ABCE. ABDE. 6. (4 pts.)the equation of the tangent plane to the surface S at the point on the surface where x = and y = is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE..ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD. ABCE. ABDE. Possible answers. A. 6 î + 4 ĵ + ˆk, B. 6 î + 4 ĵ, C. 6 î 6 ĵ ˆk, D.6 î 4 ĵ ˆk, E. 6 î 4 ĵ, AB. 6 î, AC. 6 î 4 ĵ ˆk, AD. 6 î + 4 ĵ ˆk, AE. 6 î 4 ĵ ˆk, BC. 6x 4y z =, BD. 6x + 4y z =, BE. 6x + 4y z =, CD. 4x + 4y +z =, CE. 6x + 4y z =, DE. 6x + 4y z = 4, ABC.5x + 4y z = 4, ABD. 6x + 4y z =, ABE. 6x + 4y z =, ACD. 6x + 4y z = 5, ACE. 6x + 4y z = 5, ADE.6x + 4y z =,

19 BCD. 3x + 4y z =, BCE. 3x + 4y z =, BDE. 3x 4y z =, CDE. 3x + 7y z =, ABCD. 6x + 7y z =, ABCE. 3x + 7y z =, ABDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 8 Consider the function f:r R defined by z = f(x,y) = x 3 + (3/)x + y (4 pts.) Using geometric notation, a formula for f is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE. ABDE. 64. ( 4 pts.) The set of critical points of this function is. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD.. BCE BDE. CDE ABCD ABCE. ABDE Possible answers. A. (x 6x) î + y ĵ, B. (3x6) î + y ĵ, C. (3x 6) î + y ĵ, D. (3x 3x) î + y ĵ, E. (3x 3x) î + y ĵ + z ˆk, AB. (3x 3) î + (y+) ĵ, AC. (x6) î + (y+) ĵ + z ˆk, AD. (x6) î + (y+) ĵ + z ˆk, AE. (3x +3) î + y ĵ, BC. (3x 6) î + (y+) ĵ, BD. (3x3) î + (y+) ĵ + z ˆk, BE. (3x3) î + (y+) ĵ + ˆk, CD. (3x6) î + (y+) ĵ, CE., DE. {(3,,),(3,,)}, ABC.R, ABD. {(,),(,)}, ABE. {(,),(,)}, ACD. {(,),(,)}, ACE. {(3,,),(3,,)}, ADE.{(3,,),(3,,)}, BCD. {(3,,),(3,,)}, BCE. {(3,,),(3,,)},

20 BDE. {(3,,),(3,,)}, CDE. {(3,,),(3,,)}, ABCD. {(3,,)}, ABCE. {(3,,)}, ABDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 9 3 x You are to evaluate the iterated integral I (6x y ye )dydx. 65. (4pts.) Doing the first step in the evaluation results in the single integral I =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE ABDE. ACDE. BCDE. ABCDE. 66. (4pts.) The final numerical value of I is I =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD. ABCE. ABDE. ACDE. BCDE. ABCDE. Possible answers. 3 x 3 x 3 x 3 x 3 x A. (x e )dy, B. (3x e )dx, C. (6x 4e )dy, D. (6x e )dy E. (3x e )dx, 3 x 3 x 3 x 3 x AB. (x e )dx, AC. (8x 4e )dx, AD. (6x e )dx AE. (6x 4e )dx, BC. (6x xe)dy, 3 x 3 x BD. (x e )dy, BE. (x e )dy CD., CE., DE. 5, ABC. 8, ABD., ABE.,

21 ACD. 7e, ACE. 7e, ADE. 73e, BCD. BCE. 5, BDE. 8, CDE.4/(5) ABCD., ABCE.7e, ABDE. 7e, ACDE. 73e, BCDE. 78e/5, ABCDE. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 4 x x 3 3 Assume 4x y dydx g(x, y)dxdy. On the back of the previous sheet you are find g(x,y), α, β, γ, and δ, x that is, you are to reverse the order of integration in the integral. DO NOT EVALUATE EITHER INTEGRAL. Begin by drawing an appropriate picture. 67. ( pt.) g(x,y) =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE BDE. CDE ABCD ABDE. ABDE. ACDE 68. (4 pts.) α =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE BDE. CDE ABCD ABDE. ABDE. ACDE 69. (4 pts.) β =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE BDE. CDE ABCD ABDE. ABDE. ACDE 7. ( pts.) γ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE BDE. CDE ABCD ABDE. ABDE. ACDE. 7. ( pts.) δ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE BDE. CDE ABCD ABDE. ABDE. ACDE, Possible answers. A. 4 x 3 y 3, B. x 3 y 3, C. 4x-x, D. x, 4 6 4y 4 6 4y 4 6 4y E., AB., AC., 4 6 4y 4 6 4y 6 4y 4 6 4y 4 6 4y AD., AE., BC. BD., BE., CD y 4 4, CE. x, DE. y, ABC. x/, ABD. y/, ABE. x/3, ACD. y/3, ACE.3x, 4

22 ADE. 3y, BCD., BCE., BDE., CDE. 3, ABCD., ABCE., ABDE. 3, ACDE. None of the above. Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page g(x, y)dydx Let A be the area of the region in the first quadrant bounded by the curves y = x, x + y =, and y =. On the back of the previous sheet, determine g(x,y), α, β, γ, and δ. Begin by drawing an appropriate sketch. DO NOT EVALUATE THE INTEGRAL. 7. ( pts.) g(x,y) =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE. ABDE 73. (4 pts.) α =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD..BCE BDE. CDE ABCD ABCE. ABDE 74. (4 pts.) β =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD..BCE BDE. CDE ABCD ABCE. ABDE 75. ( pts.) γ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD..BCE BDE. CDE ABCD ABCE. ABDE 76. ( pts.) δ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD..BCE BDE. CDE ABCD ABCE. ABDE Possible answers. A. 4 x 3 y 3, B. x 3 y 3, C. 4x-x, D. x, E., AB., AC., AD., AE., 3 4 x 3 x 4 x x x

23 BC. x, BD. x/, BE. x/3, CD. x/4, CE. x/5, DE.3x, ABC.x, ABD., ABE., ACD., ACE. 3, ADE., BCD., BCE.3, ABDE. None of the above. Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page x xy On the back of the previous sheet, you are to evaluate the iterated integral I 5xyz dzdydx. 77. (3pts.) Doing the first step in in the computation results in the double integral I =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE ABDE. ACDE. BCDE. 78. (3pts.) Doing the second step in the computation results in the single integral I =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE ABDE ACDE. BCDE 79. (3pts.) After the computation is complete, the numerical value of I is I =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE. ABDE ACDE. BCDE Possible answers. x x A. 5xydydx B. 5x y dydx C. 5x y dydx D. 5x y dydx E. x x x x 4 4 5x y dydx AB. 5x y dydx AC. 5x y dydx, AD. 5x y dydx, AE. 5x y dydx, BC. 8x dx, x BD. 6x dx, BE. 6x dx, CD. 6x dx, CE. 3x dx, DE. 3x dx, ABC. 4x dx, x x

24 9 ABD. 6y dy, ABE. (6x dx, ACD., ACE., ADE., BCD., BCE., BDE. 8/5, CDE.6/5, ABCD.8/5, ABCE.5/3, ABDE.4/3, ACDE. 5/4, BCDE. None of the above. Possible points this page = 9. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Fall 5 Prof. Moseley Page 3 ν δ β Let V = g(x,y,z) dz dy dx be the volume of the solid in the first octant bounded by µ γ α the planes x + y +3 z = 6, x =, y =, and z =. On the back of the previous sheet determine g(x,y,z), α, β, γ, δ, µ, and ν (i.e. set up an iterated integral in rectangular coordinates which gives the value of V). Begin by drawing an appropriate sketch. DO NOT EVALUATE. 8. ( pts.) g(x,y) =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 8. ( pts.) α =. A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 8. (4 pts.) β =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 83. ( pts.) γ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 84. (4 pts.) δ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 85. ( pts.) µ =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE 86. ( pts.) ν =.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE ABCD ABCE Possible answers. A. 4 x 3 y 3, B. x 3 y 3, C. 4x-x, D. 3 (x/3) (y/6), E. (x/3), AB. (y/6), AC x (y/6) AD. (x/3) y, AE. (x/3) (y/6), BC. (x/3) (y/6), BD. (x/) (y/6),

25 BE. (x/3) (y/6), CD. x, CE. y, DE. x/, ABC. y/, ABD. 6 x, ABE. y/3, ACD. 3x, ACE. 3y, ADE., BCD., BCE., BDE. 3, CDE., ABCD., ABCE. 3, ABDE. None of the above. Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 4 Let C be the curve that is the path of a point mass whose position vector is given by: r = t i + t j, t [, ], (i.e. < x<). Also let F (x,y) = xy i + j. You are to compute I F(x, y) dr. C 87. ( pts.) A parameterization of the curve is x(t) = and y(t) =. A. x(t) = t, y(t) = t, B. x(t) =, y(t) = t, C. x(t) = t, y(t) = t, D. x(t) = t /, y(t) = t 3 /3, E. x(t) = t, y(t) = t, AB. x(t) =, y(t) = 3t, AC. x(t) = t, y(t) = 3t, AD. None of the above. 88. (3 pts.) With the above parameterization, F(x(t), y(t)) along the curve C is F(x(t), y(t)) =. A. t 3 î + j, B. t î + t ĵ, C. t î + t 3 ĵ, D. (t /) î +( t 3 /3) ĵ, E. t î +3t ĵ, AB. t 3 î + t 3 ĵ, AC. t î + 3t 3 ĵ, AD. t î + t 3 ĵ, AE. None of the above. 89. (3 pts.) The numerical value for I is I =. A., B., C. 3, D. 4, E. 5, AB. 6, AC. 7, AD. 8, AE., ABC., ABD., ABE., ACD., ACE., ADE., BCD., BCE., BDE., CDE., ABCD. None of the above

26 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 FINAL EXAM Fall 5 Prof. Moseley Page 5 ( pts.) Let F (x,y) = M(x,y) i + N(x,y) j where M(x,y) = 6xy 3 and N(x,y) = 9x y M 9. ( pts.). A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. x ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD. M 9. ( pts.). A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. y ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD N 9. ( pts.). A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. x ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD N y 93. ( pts.). A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD 94. ( pts.) A potential function for F (x,y) = M(x,y) i + N(x,y) j is f(x,y) =. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD Possible answers this page A. 8y 3, B. xy 3, C. 8xy D. 8xy 3, E. xy 3, AB. 3x y 3 AC. 9x y, AD. 8x y, AE. 8x y ABC. 3x 3 y, ABD. 8x y 3, ABE. 3x y, ACD. 8x y, ACE.8x 3 y, ADE. 8x 3 y 3, BCD. 3xy, BDE. None of the above could possibly be a potential function as F (x,y) is not conservative.

27 CDE. F (x,y) is conservative but none of the above functions is a potential function for F (x,y). ABCD. None of the above. Possible points this page =. POINTS EARNED THIS PAGE =