Math 251 OLD FINAL EXAM Fall-2005moseley/Math 251 OLD FINAL EXAM-2005-Fall.pdf · FALL 2005 FINAL EXAM FALL 2005 FINAL EXAM EXAMINATION COVER PAGE Professor Moseley ... Matrix algebra

MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2005 FINAL EXAM FALL 2005 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 13, 2005

I swear and/or affirm that all of the work presented on this exam is my ownand that I have neither given nor received any help during the exam.

SIGNATURE DATE

INSTRUCTIONS

1. Besides this cover page, there are 26 pages of questions and problemson this exam. MAKE SURE YOU HAVE ALL THE PAGES. If apage is missing, you will receive a grade of zero for that page. Readthrough the entire exam. If you cannot read anything, raise your handand 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 theexam. NO CALCULATORS! NO SCRATCH PAPER! Use theback of the exam sheets if necessary. You may remove the staple ifyou wish. Print your name on all sheets.

3. Explain your solutions fully and carefully. Your entire solution will begraded, not just your final answer. SHOW YOUR WORK! Everythought you have should be expressed in your best mathematics onthis paper. Partial credit will be given as deemed appropriate. Proof-read your solutions and check your computations as time allows. GOOD LUCK!!

page points score

1 14

2 12

3 5

4 8

5 8

6 10

7 9

8 8

9 8

10 10

11 14

12 17

13 9

14 10

15 8

16 8

17 8

18 8

19 8

20 12

21 14

22 9

23 14

24 8

25 10

26

27

Total 250

MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 1

PRINT NAME _________________________(_________________) ID No. __________________ Last Name, First Name MI, What you wish to be called

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 = [aij], B=[bij], C=[cij], D=[dij],and E=[eij] be nxn square matrices.

1.( 1 pt.) If α is a scalar and C = αA, then cij = __________. A. B. C. D. E. AB. AC. AD. AE.

BC. BD. BE. CD. CE. DE.

2. ( 1 pt.) If D = A + B, then dij = ________________. A. B. C. D. E. AB. AC. AD. AE. BC.

BD. BE. CD. CE. DE.

3. ( 2 pts.) If E = AB, then eij = ________________.A. B. C. D. E. AB. AC. AD. AE. BC.

BD. BE. CD. CE. DE.

Possible Answers for questions 7, 8, and 9.

A. αaij, B. βaij, C. bij aij, D. bij+ aij, E .aij/bij, AB.= , AC. , AD. aijbij, n

ij iji 1

a b

n

ik kjk 1

a b

AE. aij, BC. aij+cij BD. bij, BE. bij dij, CD. bij + eij, CE. bij aij, DE. None of the above

(10 pts.) True or False. Matrix Algebra.Circle True or False, but not both. If I cannot read your answer, it is WRONG.

True or False 4. Matrix addition is associative.

True or False 5. Matrix addition is not commutative.

True or False 6. α,βR and ARm×n, α(βA) = (αβ)A.

True or False 7. Multiplication of square matrices is associative.

True or False 8. Multiplication of square matrices is commutative.

True or False 9. If A and B are invertible square matrices, then (AB)-1 exists and (AB)-1 = A-1 B-1.

True or False 10. If A is an invertible square matrix, then (A-1)-1 exists and (A-1)-1 = A.

True or False 11. If A and B are square matrices, then (AB)T exists and (AB)T = AT BT.

True or False 12. If A is a square matrix, then AT and (AT)T exist and (AT)T = A.

True or False 13. If A is an invertible square matrix, then (AT)-1 exists and (AT)-1 = (A-1)T.Possible points this page = 14. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 2


On the back of the previous sheet, solve the x1 + x2 + x3 - x4 = 1system of linear algebraic equations Be sure to write youanswer in the correct form. Circle the correct answer x1 + 2x2 + x3 = 0from the possibilities below

x3 + x4 = 0

x2 - 2x3 + x4 = 1

14. (3 pts.) x1 = ______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.

AC. ABD. ABE BCD. BCE. CDE.

15. (3 pts.) x2 = ______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


16. (3 pts.) x3 = _______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


17. (3 pts.) x4 = _______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


Possible answers for questions 14, 15, 16, and 17.

A.1, B.2, C. 3, D. 4, E. 5, AB. 6, AC. 7, AD. 8, AE. 9, BC.10, BD.1, BE.2, CD.3,

CE.4, DE.5, ABC.6, ABD.7, ABE 8, BCD.9, BCE.10, CDE. None of the above.Possible points this page = 12. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 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×1 column vector of (possibly

xcomplex) unknowns, and that is an m×1 (possibly complex valued) column vector. Now consider

b

. (*)mxn nx1 mx1A x b

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.

18. True or False, If , then (*) always has an infinite number of solutions. b 0

19. True or False, The vector equation (*) always has exactly one solution.

20. True or False, If A is square (n=m) and nonsingular, then (*) always has a unique solution.

21. True or False, The equation (*) can be considered as a mapping problem from one vector space to another.

22. True or False, If then (*) has a unique solution.A1 ii 1

Total points this page = 5. TOTAL POINTS EARNED THIS PAGE _______



You are to solve where , , and . Be sure you write your2x2 2x1 2x1A x b

1 iA

i 1

xx

y

1b

i

answer according to the directions given in class (attendance is mandatory) for these kinds of problems.

23. (4 pts.) If is reduced to using Gauss elimination, then =___________.A b

U c U c

A. , B. , C. , D. , E. ,1 i 10 0 0

1 i 00 0 0

1 i 10 0 1

0 0 00 0 0

1 i 10 0 i

AB. None of the above are possible.

24. ( 4 pts.) The general solution of can be written as ____________________________.2x2 2x1 2x1A x b

A. No Solution, B. , C. , D. , E. , 1

x0

ix y

1

1 ix y

0 1

ix

1

AB. , AC. , AD. , 1 i

x y0 1

i 1x y

1 0

1 ix y

0 1

BC. None of the above correctly describes the solution or set of solutions.

Total points this page = 8. TOTAL POINTS EARNED THIS PAGE _______



Let A = and A1 = . Compute the inverse of A. Be sure to explain clearly41 99 2

a bc d

and completely your method. Circle the correct values of a, b, c, and d from the possiblities below:

25. (2 pts.) a = _______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.

AC. ABD. ABE BCD. BCE. CDE..

26. (2 pts.) b = _______.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


27. (2 pts.) c = ________.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


28. (2 pts.) d = ________.A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD.CE. DE.


Possible answers for questions 25, 26, 27, and 28.

A.1, B.2, C. 3, D. 4, E. 5, AB. 8, AC. 9, AD. !0, AE. 11, BC. 20, BD. 21, BE. 22, CD. 25,

CE. 26, DE. 30, ABC. 40, ABD. 41, ABE. 42, BCD. 45, BCE. 55. CDE. None of the above.Possible points this page = 8. POINTS EARNED THIS PAGE = ________



29. ( 2 pts.) Let S = V where V is a vector space. Choose the completion of the {v ,v ,...,v }1 2 n

following definition of what it means for S to be linearly independent.

Definition. The set S = V where V is a vector space is linearly independent if {v ,v ,...,v }1 2 n

A. The vector equation has an infinite number of solutions.1 1 2 2 n nc v +c v +...+c v = 0

B. The vector equation has a solution other than the trivial solution.1 1 2 2 n nc v +c v +...+c v = 0

C. The vector equation has only the trivial solution c1 = c2 = = cn = 0.1 1 2 2 n nc v +c v +...+c v = 0

D. The vector equation has at least two solutions.1 1 2 2 n nc v +c v +...+c v = 0

E. The vector equation has no solution.1 1 2 2 n nc v +c v +...+c v = 0

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 ofvectors are linearly independent. As explained in class, circle the appropriate answer that gives anappropriate method to prove that your results are correct (Attendance is mandatory). Be careful. If youget them backwards, your grade is zero.

30. (4 pts.) Let S =.{[2, 4, 8]T, [3, 6, 11]T}. Circle the correct answerA. S is linearly independent as c1[2, 4, 8]T + c2 [3, 6, 11]T = [0,0,0] implies c1 = 0 and c2 = 0.B. S is linearly independent as 3[2, 4, 8]T + (2) [3, 6, 11]T = [0,0,0].C. S is linearly dependent as c1[2, 4, 8]T + c2 [3, 6, 11]T = [0,0,0] implies c1 = 0 and c2 = 0.D. S is linearly dependent as 3[2, 4, 8]T + (2) [3, 6, 11]T = [0,0,0].E. S is neither linearly independent or linearly dependent as the definition does not apply.

31. (4 pts.) Let S = {[2, 2, 6]T, [3, 3, 9]T}. Circle the correct answerA. S is linearly independent as c1[2, 2, 6]T + c2 [3, 3, 9]T = [0,0,0] implies c1 = 0 and c2 = 0.B. S is linearly independent as 3[2, 2, 6]T + (2) [3, 3, 9]T = [0,0,0].C. S is linearly dependent as c1[2, 2, 6]T + c2 [3, 3, 9]T = [0,0,0] implies c1 = 0 and c2 = 0.D. S is linearly dependent as 3[2, 2, 6]T + (2) [3, 3, 9]T = [0,0,0].E. S is neither linearly independent or linearly dependent as the definition does not apply.

Total points this page = 10. TOTAL POINTS EARNED THIS PAGE ________



Let A = On the back of the previous sheet, compute the determinant of A.

1 0 3 00 1 0 20 2 1 43 0 9 1

32. ( 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. , B. , C. , 0 3 0 0 3 0

(1) 1 0 2 (3) 1 0 22 1 4 2 1 4

0 3 0 1 0 2(1) 1 0 2 (3) 2 1 4

2 1 4 0 9 1

1 0 2 0 3 0(1) 2 1 4 (1) 1 0 2

0 9 1 2 1 4

D. , E. , AB. , 1 0 2 0 3 0

(3) 2 1 4 (1) 1 0 20 9 1 2 1 4

1 0 2 0 3 0(1) 2 1 4 (3) 1 0 2

0 9 1 2 1 4

0 3 0 0 1 2(1) 1 0 2 ( 3) 0 2 4

2 1 4 3 0 1

AC. , AD. , AE. , 0 3 0 0 1 2

(1) 1 0 2 (1) 0 2 42 1 4 3 0 1

0 3 0 0 1 2(3) 1 0 2 ( 3) 0 2 4

2 1 4 3 0 1

1 0 2 0 1 2(1) 2 1 4 ( 3) 0 2 4

0 9 1 3 0 1

BC. , BD. , BE. , CD. , CE. , 1 0 3 00 1 0 20 2 1 40 0 9 1

1 0 3 00 1 0 20 2 1 40 1 0 1

1 0 3 00 1 0 20 0 2 43 0 9 1

1 0 3 00 1 0 20 2 1 40 0 0 0

1 0 3 00 1 0 20 2 1 40 0 0 1

DE. , ABC. , ABD. , ABE. , ACD.1, ACE.2, 1 0 3 00 1 0 20 2 1 40 0 0 2

1 0 3 00 1 0 20 0 0 00 0 0 1

1 0 3 00 1 0 20 0 1 00 0 0 2

1 0 3 00 1 0 20 0 1 00 0 0 0

ADE. 3, BCD. 4, BCE. 5, BDE. 8, CDE. 9, ABCD. 0, ABCE. 1, ABDE. 2, ACDE.3

BCDE. None of the above.

Possible points this page = 9. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 8

PRINT NAME _________________________(_________________) SS No. __________________ Last Name, First Name MI, What you wish to be called

Let and be the vectors, = <2,-1,1> = (2,1,1) = [2,1,1]T = 2 + and ab

a i j k

= <0,1,3> = (0,1,3) = [0,1,3]T = + 3 . b j k

35. (3 pts.) Then the dot product is = ( , ) = , ____________. A. B. C. D. E. AB. ab a

b a b

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. C. D. E. AB. AC.a

b

AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. BCD. BCE, CDE, ABCD.

Possible answers for questions 35 and 36.A. 1, B.2, C.3, D.4, E.5 AB.1 AC.2 AD.3 AE.4 BC. , BD. , ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k

BE. ,CD. , CE. , DE. , ABC. , ABD. , ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k

ABE. , BCD. , BCE. , CDE. , ABCD. None of the above.ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2 j k ˆ ˆ ˆ3i 2 j k ˆ ˆ ˆ3i 2j k Possible points this page = 8. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 9


You are to find an equation for a plane and an equation for a sphere. Recall that these equations are notunique. 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 +2 y = 3 z +10. 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 (2,3,0). 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 + 2 y + 3 z =10 B. 4x +2 y = 3 z +10, C. 4x +2 y 3 z = 0, D. 2x + 3 y = 10

E. 4x +2 y = 10 AB. 4x +2 y 3 z = 10, AC. 4x + 2 y + 3z = 0, AD. 2x +3 y = 3

AE. (x +2)2 + (y + 3)2 = 9, BC. (x 2)2 + (y + 3)2 + z2 = 9, BD. (x +2)2 + (y 3)2 + z2 = 9,

BE. (x 2)2 + (y 3)2 = 3, CD.(x 2)2 + (y 3)2 + z2 = 9, CE. (x +2)2 + (y 3)2 + z2 = 3,

DE. (x +2)2 + (y + 3)2 = 3, ABC. None of the above.Possible points this page = 8. POINTS EARNED THIS PAGE = ________



Suppose that the position vector for a point mass M as a function of the time t is given by:

= (2e2 t ) + (2t3 + 3t2 ) + ( 3 sin(t) ) r i j 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 . Then = ________________. v(t) 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)

40. ( 5 pts.) Let the acceleration vector for the point mass M be . Then = _______________. a(t) 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. (2e2 t ) + (2t3 + 3t2 ) + (3 sin(t)) B. (2e3 t ) + (2t3 + 3t2 ) + ( 3 sin(t)) i j k i j k

C. (4e2 t ) + (6t2 + 6t ) + (3 cos(t) ) D. (4e3 t ) + (6t3 + 6t2 ) + ( 3 cos(t) ) i j k i j k

D. (2e3 t ) + (6t3 + 6t2 ) + (3 sin(t) ) E. (2e3 t ) + (2t3 + 3t2 ) + (3 sin(t) )i j k i j k

AB (2e3 t ) + (6t2 + 6t ) + (3 cos(t) ) AC (8e2 t ) + (12 t + 6 ) (3 sin(t) ) i j k i j k

AD. (8e2 t ) + (6 t + 3 ) (3 sin(t) ) AE (12e3 t ) + (12t2 + 6t ) + (3 sin(t) ) i j k i j k

BC. (4e2 t ) + (6t2 + 3t ) + (3 sin(t) ) BD. None of the above.i j k



Let L1 and L2 be the two intersecting lines (check t = 0) whose parametric equations are given by:

L1: x = 3t + 1, y = 0, z = 2t L2: x = 1, y = t, z = t

where t R. You are to find an equation of the plane P that contains L1 and L2.

41. (2 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. 42. (2 pts.) A vector in the direction of L1 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. (2 pts.) A vector in the direction of L2 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. (1,0,0) B.(0,1,0) C.(0,0,1) D.((1,1,1) E. (3,2,2) AB..(1,1,1) AC. 2 + 2 + 3i j k

AD. 3 + 3 AE. 2 + 3 BC. 3 + 2 BD. 2 + 2 + 3 BE. CD. + 3i k j k i k i j k j k j k

CE..2 + 2 + 3 DE..2 + 2 + 3 ABC. 2 + 2 + 3 ABD. 2 + 3 + 3 i j k i j k i j k i j k

ABE. 2x + 3y + 3z = 3 ACD. 2x + 3y + 3z = 2 ACE. 2x + 3y + 2z = 3 ADE. 2x 3y + 3z = 2

BCD. 2x + 3y + 3z = 2 BCE. 2x +3y + 2z = 3 BDE 2x + 3y + 3z = 2 CDE. None of the

above.Possible points this page = 14. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 12


Suppose that the position vector for a point mass as a function of the time t is given by:

= 3 t + 2 cos(2t) + 2 sin(2t) . (Be careful! If you make a mistake, the rest is wrong.)r i j k

46. (3 pts) The velocity at time t = 0 is = ____________________. A. B. C. D. E. AB. v(t) v(0)

AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. .BCD.

BCE. BDE. CDE.

v(t) v(0)

47. (3 pts) The acceleration at time t = 0 is = ___________________. A. B. C. D. E. AB. a(t)a(0)

AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. .BCD. BCE. BDE. CDE. a(t) a(0)

48. (4 pts) is _________________________. A. B. C. D. E. AB. AC. AD. AE. BC. a(0) v(0)

BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. .BCD. BCE. BDE. CDE. a(0) v(0)

49. (3 pts) is ___________________________. A. B. C. D. E. AB. AC. AD. AE. BC. v(0)

BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. .BCD. BCE. BDE. CDE. v(0)

50. (4 pts) The curvature at t = 0 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 B. 3 + 4 C. 3 + 4 + 4 D. 3 + 3 E. 3 + 4 AB 3 + 4 + 4 i i k i j k i k j k i j k

AC. 8 AD. 8 AE. 8 BC. 8 BD. 8(4 + 3 ) BE. 8(3 + 4 ) CD. 8( 4 + 3 ) j k i j i k i j j k

CE. 8(4 3 ) DE. 2 ABC. 3 ABD. 4 ABE. 5 ACD. ACE. ADE. BCD. i k1225

625

13

825

BCD. BDE. CDE None of the above.14

15

Possible points this page = 17. POINTS EARNED THIS PAGE = ________



Let S be the surface defined by the function z = f(x,y) = 4 x2 y2. and let (x,y,z) be a point P on the surface..

51. ( 3 pts.) Using geometric notation ( and , or , , and ), the gradient of f (f) is _________.i j i j k


ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD

52. ( 2 pts.) A formula for the normal to the tangent plane to the surface S at the point P is___________.


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. 2x + 2y + , B.2x + 2y , C.2x + 3y + , D.2 + 2 + 3 i j k i j k i j k i j k

E.2 + 2 3 , AB.2 + 2 , AC. x2 + y2 + 3 AD. 2 + 2 3i j k i j k i j k i j k

AE.2 + 2 + 3 BC.2 + 2 BD.2x + 2y , BE.x2 + y2 , i j k i j i j i j

CD.2x + 2y + 3 , CE.2x + 2y + 3 , DE..2x + 2y + 3 , ABC.2x + 2y + 3 ,i j k i j k i j k i j k

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 abovePossible points this page = 9. POINTS EARNED THIS PAGE = _________



Let w = f(x,y) = 2x2e 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 dwdt

t 0

54. (3 pts.) = ___________________. A. B. C. D. E. AB. AC. AD. AE. BC.wx

(x,y) (1,0)

BD. BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE.

CDE ABCD ABCE. ABDE. ACDE. BCDE. .

55. (3 pts.) = _______________________________. A. B. C. D. E. AB. AC. AD. AE. BC. BD.wy

(x,y) (1,0)

BE. CD. CE. DE. ABC. ABD. ABE. ACD. ACE. ADE. BCD. BDE. CDE ABCD ABCE. ABDE. ACDE. BCDE. .

56. (4 pts.) = _________________________. A. B. C. D. E. AB. AC. AD. AE. BC. BD. dwdt

t 0

BE. CD. CE. 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 = _________



Let P be the point in R3 (i.e. 3-space) which has rectangular coordinates ( 1, 1, )R. Give the2 cylindrical and spherical coordinates of P. Begin by drawing a picture. Be sure to give thecoordinates 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/4,0)C, B.( , π/4,π/4,)C, C.( , π/4,1)C, D.( , π/4, )C, E.( , π/4,π/4)C,2 2 2 2 2 2

AB.( , π/3, )C, AC.( , π/3,1)C, AD.( , π/4,2)C, AE.(1, π/4,1)C, BC.(1, π/4,0)C, 2 2 2 2

BD.( , π/4,0)S, BE.( , π/4,π/4,)S, CD.( , π/4,1)S, CE.( , π/4, )S, DE. ( , π/4,π/4)S,2 2 2 2 2 2

ABC.( , π/3, )S, ABD.( , π/3,1)S, ABE.( , π/4,2)S, BCD.(1, π/4,1)S, BCE.(1, π/4,0)S,0) 2 2 2 2

CDE. None of the above.Possible points this page = 8. POINTS EARNED THIS PAGE = ________



Let w = f(x,y) = 5 e3x cos(y), P be the point (0,0), and be a unit vector in the direction of . u ˆ ˆv 3i 4j

59. (4 pts.) Using geometric notation (i.e. and or , , and ), i j i j k f____________.

(x, y) (0,0)

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.

60. (4 pts.) A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. ˆ ˆu uD f(P) = D f ________________. (x,y) (0,0)

CD. CE. DE. ABC. ABD. ABE.. ACD. ACE. ADE. BCD.

BCE. BDE. CDE ABCD, ABCE, ABDE, ACDE, BCDE.

Possible answers.A. 5 + 5 + , B. 5 + 5 , C. 10 5 , D.15 5 , E. 15 5 , i j k i j i j k i j k i j

AB. 15 , AC. 15 5 , AD. 15 5 , AE. 15 5 , i i j k i j k i j k

BC. 0, BD. 1, BE. 5, CD. 9, CE. 10, DE.20, ABC.30, ABD.45, ABE. 60, ACD.1

ACE. 5, ADE. 9, BCD. 10, BCE.45, BDE. 8/5, CDE. 24/(15), ABCD. 8/5,

ABCE.24/(15), ABDE. , ACDE , BCDE. None of the above.8/ 5 8/ 5

Possible points this page = 8. POINTS EARNED THIS PAGE = _________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 17


Let S be the surface which is defined by the graph of the function z = f(x,y). Suppose using geometric

notation (i.e. and or , , and ), that and that f(1,1) = 10 i j i j k f ˆ ˆ6i 14 j.(x, y) (1,1)

61. (4 pts.) Using geometric notation, a normal to the surface S when x = 1 and y = 1 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.

62. (4 pts.)The equation of the tangent plane to the surface S at the point on the surface where x = 1 and

y = 1 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 + 14 + , B. 6 + 14 , C. 6 6 , D.6 14 , E. 6 14 , i j k i j i j k i j k i j

AB. 6 , AC. 6 14 , AD. 6 + 14 , AE. 6 14 , i i j k i j k i j k

BC. 6x 14y z = 10, BD. 6x + 14y z = 10, BE. 6x + 14y z = 10, CD. 4x + 14y +z = 10,

CE. 6x + 14y z = 0, DE. 6x + 14y z = 4, ABC.5x + 14y z = 4, ABD. 6x + 4y z = 10,

ABE. 6x + 14y z = 20, ACD. 6x + 14y z = 5, ACE. 6x + 14y z = 5, ADE.6x + 14y z = 10,

BCD. 3x + 14y z = 10, BCE. 3x + 4y z = 10, BDE. 3x 14y z = 10, CDE. 3x + 7y z = 10,

ABCD. 6x + 7y z = 10, ABCE. 3x + 7y z = 10, ABDE. None of the above.



Consider the function f:R2R defined by z = f(x,y) = x3 + (3/2)x2 + y2 6.

63. (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. (2x26x) + 2y , B. (3x6) + 2y , C. (3x26) + 2y , D. (3x23x) + 2y , i j i j i j i j

E. (3x23x) + 2y + z , AB. (3x23) + (2y+2) , AC. (2x6) + (2y+2) + 2z , i j k i j i j k

AD. (2x6) + (2y+2) + 2z , AE. (3x2+3) + 2y , BC. (3x26) + (2y+2) , i j k i j i j

BD. (3x3) + (2y+2) + 2z , BE. (3x3) + (2y+2) + , CD. (3x6) + (2y+2) , i j k i j k i j

CE. , DE. {(3,1,2),(3,1,2)}, ABC.R2, ABD. {(0,0),(0,1)},

ABE. {(0,1),(1,0)}, ACD. {(0,0),(1,0)}, ACE. {(3,1,2),(3,1,2)},

ADE.{(3,1,2),(3,1,2)}, BCD. {(3,1,2),(3,1,2)}, BCE. {(3,1,2),(3,1,2)},

BDE. {(3,1,2),(3,1,2)}, CDE. {(3,1,2),(3,1,2)}, ABCD. {(3,1,2)},

ABCE. {(3,1,2)}, ABDE. None of the above.



You are to evaluate the iterated integral .2 1

3 2 x

0 0

I (6x y 2ye )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.

A. , B. , C. , D. E. , 2

3 x

0

(2x 2e )dy2

3 x

0

(3x 2e )dx21

3 x

0

(6x 4e )dy2

3 x

0

(6x e )dy2

3 x

0

(3x e )dx

AB. , AC. , AD. AE. , BC. , 2

3 x

0

(2x e )dx2

3 x

0

(8x 4e )dx1

3 x

0

(6x 2e )dx2

3 x

0

(6x 4e )dx2

2

0

(6x 2xe)dy

BD. , BE. CD. 0, CE. 1, DE. 5, ABC. 8, ABD. 10, ABE.20, 1

3 x

0

(2x e )dy1

3 x

0

(2x 2e )dy

ACD. 7e, ACE. 72e, ADE. 73e2, BCD.1 BCE. 5, BDE. 8, CDE.24/(15) ABCD. 10, ABCE.7e2, ABDE. 72e, ACDE. 73e, BCDE. 78e/5, ABCDE. None of the above.

Possible points this page = 8. POINTS EARNED THIS PAGE = _________



Assume . On the back of the previous sheet you are find g(x,y), α, β, γ, and δ,22 4 x x

3 3

0 2x

4x y dydx g(x, y)dxdy

that is, you are to reverse the order of integration in the integral. DO NOT EVALUATE EITHERINTEGRAL. Begin by drawing an appropriate picture.

67. (2 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.


70. (2 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.

71. (2 pts.) δ = _______________.A. B. C. D. E. AB AC. AD. AE. BC. BD. BE. CD. CE. DE.


,

Possible answers.

A. 4 x3y3, B. x3y3, C. 4x-x2, D. x2, E. , AB. , AC. , 4 16 4y2

4 16 4y2

4 16 4y2

AD. , AE. , BC. BD. , BE. , 4 16 4y2

4 16 4y4

2 16 4y4

4 16 4y2

4 16 4y4

CD. , CE. 2x, DE. 2y, ABC. x/2, ABD. y/2, ABE. x/3, ACD. y/3, ACE.3x, 4 16 4y2

ADE. 3y, BCD. 0, BCE. 1, BDE. 2, CDE. 3, ABCD. 1, ABCE. 2, ABDE. 3,

ACDE. None of the above.Possible points this page = 12. POINTS EARNED THIS PAGE = _________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 21


Let be the area of the region in the first quadrant bounded by the curves y = x2,A g(x, y)dydx

x + y = 2, and y = 0. On the back of the previous sheet, determine g(x,y), α, β, γ, and δ. Begin by drawingan appropriate sketch. DO NOT EVALUATE THE INTEGRAL.

72. (2 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. (2 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. (2 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 x3y3, B. x3y3, C. 4x-x2, D. 2 x, E. , AB. , AC. , AD. , AE. , 3 4 x 3 2x 4 x x 2 x

BC. 2x, BD. x/2, BE. x/3, CD. x/4, CE. x/5, DE.3x, ABC.x2, ABD. 0, ABE. 1, ACD.2, ACE. 3, ADE. 1, BCD. 2, BCE.3, ABDE. None of the above.



On the back of the previous sheet, you are to evaluate the iterated integral .xy1 2x

2

0 0 0

I 15xyz 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.

A. B. C. D. E. 1 2x

0 0

5xydydx 1 2x

3 3

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx

AB. AC. , AD. , AE. , BC. , 1 2x

4 4

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx 1 2x

4 4

0 0

5x y dydx 1

9

0

8x dx

BD. , BE. , CD. , CE. , DE. , ABC. , 1

8

0

16x dx1

6

0

16x dx1

7

0

16x dx1

8

0

32x dx1

9

0

32x dx1

9

0

24x dx

ABD. , ABE. , ACD. 0, ACE. 1, ADE. 2, BCD. 1, BCE.2, BDE. 8/5, 1

9

0

16y dy2

10

0

(16x dxCDE.16/5, ABCD.18/5, ABCE.5/3, ABDE.4/3, ACDE. 5/4, BCDE. None of the above.

Possible points this page = 9. POINTS EARNED THIS PAGE = _________MATH 251 EXAM IV Fall 2005 Prof. Moseley Page 23


ν δ βLet V = g(x,y,z) dz dy dx be the volume of the solid in the first octant bounded by µ γ αthe planes 2x + y +3 z = 6, x = 0, y = 0, and z = 0. 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.

80. (2 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

81. (1 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

82. (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. (1 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. (1 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. (2 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 x3y3, B. x3y3, C. 4x-x2, D. 3 (x/3) (y/6), E. 2 (x/3), AB. 2 (y/6), AC 2 x (y/6) AD. 2 (x/3) y, AE. 2 (x/3) (y/6), BC. 1 (x/3) (y/6), BD. 2 (x/2) (y/6),

BE. 2 (x/3) (y/6), CD. 2x, CE. 2y, DE. x/2, ABC. y/2, ABD. 6 2x, ABE. y/3, ACD. 3x, ACE. 3y, ADE. 0, BCD. 1, BCE. 2, BDE. 3, CDE. 1, ABCD. 2, ABCE. 3,

ABDE. None of the above.Possible points this page = 14. POINTS EARNED THIS PAGE = _________



Let C be the curve that is the path of a point mass whose position vector is given by: = t + t2 , r i j

t [0, 1], (i.e. 0< x<1). Also let (x,y) = xy + . You are to compute .F i j

C

I F(x, y) dr

87. (2 pts.) A parameterization of the curve is x(t) =________and y(t) =____.

A. x(t) = t2, y(t) = t2, B. x(t) = 1, y(t) = 2t, C. x(t) = t, y(t) = t2, D. x(t) = t2/2, y(t) = t3/3,

E. x(t) = t2, y(t) = t, AB. x(t) = 2, y(t) = 3t2, AC. x(t) = 2t2, y(t) = 3t2, AD. None of the above.

88. (3 pts.) With the above parameterization, along the curve C is = _________.F(x(t), y(t))

F(x(t), y(t))

A. t3 + , B. t + t2 , C. t2 + t3 , D. (t2/2) +( t3/3) , E. 2t +3t2 , AB. t3 + t3 ,i j i j i j i j i j i j

AC. 2t2 + 3t3 , AD. t2 + 2t3 , AE. None of the above. i j i j

89. (3 pts.) The numerical value for I is I = _______. A. 0, B. 1, C. 3, D. 4, E. 5, AB. 6, AC. 7,

AD. 8, AE. 1, ABC. , ABD. , ABE. , ACD. , ACE. , ADE. , BCD. ,815

715

35

23

13

1115

45

BCE. , BDE. , CDE. , ABCD. None of the above.1315

1415

115



( 10 pts.) Let (x,y) = M(x,y) + N(x,y) where M(x,y) = 6xy3 and N(x,y) = 9x2y2F i j

90. (2 pts.) ___________. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC.Mx

ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD.

91. (2 pts.) ___________. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC.My

ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD

92. (2 pts.) ___________. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC.Nx

ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD93. (2 pts.) ___________. A. B. C. D. E. AB. AC. AD. AE. BC. BD. BE. CD. CE. DE. ABC.N

y

ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD94. (2 pts.) A potential function for (x,y) = M(x,y) + N(x,y) is f(x,y) = ________________. F

i j


ABD. ABE. ACD. ACE. ADE. BCD. BCE. BDE. CDE. ABCD

Possible answers this pageA. 18y3, B. 12xy3, C. 18xy2 D. 18xy3, E. 12xy3, AB. 3x2y3 AC. 9x2y2, AD. 18x2y, AE. 18x2y2

ABC. 3x3y2, ABD. 18x2y3, ABE. 3x2y2, ACD. 18x2y2, ACE.18x3y2, ADE. 18x3y3, BCD. 3xy2,

BDE. None of the above could possibly be a potential function as (x,y) is not conservative. F

CDE. (x,y) is conservative but none of the above functions is a potential function for (x,y). F

F

ABCD. None of the above.Possible points this page = 10. POINTS EARNED THIS PAGE = ________

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