math 251 old final exam 2006 - west virginia university

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

PRINT NAME ( ) Last Name, First Name MI (What you wish to be called)

ID # EXAM DATE Monday, May 1, 2006

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!!

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MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 1

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

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. Circle the letter or letters that correspondto your answer from those listed below1. ( 2 pt.) If α is a scalar and C = αA, then cij = ____________________. A. B. C. D. E.

2. ( 2 pt.) If D = A + B, then dij = ________________________________. A. B. C. D. E.

3. ( 3 pts.) If E = AB, then eij = _____________________________________.A. B. C. D. E.

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.

4. A)True or B)False, Matrix addition is associative.

5.A)True or B)False, Matrix addition is not commutative.

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

7.A)True or B)False, Multiplication of square matrices is associative.

8. A)True or B)False, Multiplication of square matrices is commutative.

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

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

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

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

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

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



x1 + x2 + x3 - x4 = 1 Solve this system of linear algebraic equations.x1 + 2x2 + x3 = 0 Circle the letter or letters that correspond to your answer from the x3 + x4 = 0 possibilities belowx2 - 2x3 + x4 = 1

14. (3 pts.) x1 = _________.A B C D E

15. (3 pts.) x2 = _________.A B C D E

16. (3 pts.) x3 = __________.A B C D E

17. (3 pts.) x4 = ____________.A B C D E

Possible answers this page.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)3CE) 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 = ________



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

x of (possibly complex) unknowns, and that is an m×1 (possibly complex valued) column

b

vector. Now consider . (*)

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

19. (1 pt.) True or False, The vector equation (*) always has exactly one solution.

20. (1 pt.) True or False, If A is square (n=m) and nonsingular, then (*) always has a unique solution.

21. (1 pt.) True or False, The equation (*) can be considered as a mapping problem from onevector space to another.

22. (1 pt.) 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 write2x2 2x1 2x1A x b

1 iA

i 1

xx

y

1b

i

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

23. (4 pts.) If is reduced to using Gauss elimination, thenA 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 2x2 2x1 2x1A x b

as ____________________________.

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. Circle the letter or letters41 99 2

a bc d

that correspond to your answer. for the values of a, b, c, and d from the possibilities below:

31. (2 pts.) a = ____________.A B C D E .

32. (2 pts.) b = _____________.A B C D E

33. (2 pts.) c = ______________.A B C D E ..

34. (2 pts.) d = _______________.A B C D E ..

Possible answers this page.

A)1 B)2 C) 3 D) 4 E) 9 AB) 10 AC) 20 AD) 30 AE) 41 BC) 50 BD) 82 BE) 1

CD). 2 CE) 3 DE) 4 ABC) 9 ABD) 10 ABE) 20 BCD) 41 BCE) 82

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 {v ,v ,...,v }1 2 n

the 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{v ,v ,...,v }1 2 n

if 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, 1 1 2 2 n nc v +c v +...+c v = 0

c1 = c2 = = cn = 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 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.

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 = Use the back of the previous sheet to compute the determinant of A.

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

Circle the letter or letters that corresponds to your answers to the following questions. 32. ( 3 pts.) The first step of the Laplace Expansion in terms of the first column yields

det(A) =________________________________________________ A B C D E 33. (3 pts.) The first step in using Gauss Elimination to find det(A) yields

det(A) =____________________________________________________.:A B C D E

34. (4 pts.) The numerical value of det(A) is det(A) = ___________________. A B C D E

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 2

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

0 9 2 2 1 4

D) E) AB) 1 0 2 0 3 0

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

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

0 9 2 2 1 4

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

2 1 4 3 0 2

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

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

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

2 1 4 3 0 2

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

0 9 1 3 0 2

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

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

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

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 2

DE) ABC). ABD) ABE) 1 0 3 00 1 0 20 2 1 40 0 0 1

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

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

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

ACD).1 ACE) 2 ADE) 3 BCD) 4 BCE) 5 BDE) 0 CDE) 1 ABCD) 2 ABCE)3

ABDE). 4 ACDE).5 BCDE) None of the above.Possible points this page = 10. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 8

PRINT NAME _________________________(_______________) ID 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 . Circle the letter or letters that corresponds to your b j k

answer for the following:35. (3 pts.) Then the dot product is = ( , ) = , ____________. A B C D E

ab a

b a b

36. (5 pts.) The cross product is × __________________.. A B C D E .a

b

Possible answers this page.A) 1 B) 2 C) 3 D) 4 E).5 AB).1 AC).2 AD).3 AE) 4 BC) , ˆ ˆ ˆ3i 2j k

BD) BE) CD) CE) DE) ABC)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

ABD) ABE) BCD) BCE) ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2j k ˆ ˆ ˆ3i 2 j k ˆ ˆ ˆ3i 2 j k

CDE) ABCD) None of the above.ˆ ˆ ˆ3i 2j k

Possible points this page = 8. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 9


You are to find an equation for a plane and an equation for a sphere. Recall that these equationsare not unique. To get the equations given in the answers below, you should use the proceduresillustrated in class (attendance is mandatory). Choose your answer from the possibilities below. Then circle the letter or letters that corresponds to your answer. 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.

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.

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 = 9BE) (x 2)2 + (y 3)2 = 3 CD).(x 2)2 + (y 3)2 + z2 = 9 CE) (x +2)2 + (y 3)2 + z2 = 3DE) (x +2)2 + (y + 3)2 = 3 ABC) None of the above.



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)

= _________________________________________________. A. B. C. D. E. v(t)

= v(t)

40. ( 5 pts.) Let the acceleration vector for the point mass M be . Then a(t)

= ______________________________________________________. A. B. C. D. E. a(t)

=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

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

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

sin(t) ) kAE) (8e2 t ) + (6 t + 3 ) (3 sin(t) ) BC) (12e3 t ) + (12t2 + 6t ) + (3 sin(t) ) i j k i j k

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

Possible points this page = 10. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 11


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

42. (2 pts.) A vector in the direction of L1 is _______________________________.A B C D E.

43. (2 pts.) A vector in the direction of L2 is _______________________________.A B C D E.

44. (4 pts.) A normal to the plane P is ___________________________________. A B C D E (Recall that a normal vector to a plane is not unique.)

45. (4 pts.) An equation for the plane P is _____________________________. A. B. C. D. E. (Recall that an equation for the plane is not unique.)

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 + i j

3 k

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

3 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 Spring 2006 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 isr i j k

wrong.)46. (3 pts) The velocity at time t = 0 is = _______________________. A. B. C. D. v(t) v(0)

E.

v(t) v(0)

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

E.

a(t) a(0)

48. (4 pts) is _____________________________________________. A. B. C. D. a(0) v(0)

E.

a(0) v(0)

49. (3 pts) is ________________________________________________. A. B. C. D. v(0)

E.

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.

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 ) j k i j i k i j

CD) 8( 4 + 3 ) CE). 8(4 3 ) DE) 2 ABC) 3 ABD) 4 ABE) 5 ACD). j k i k1225

ACE). ADE).1/3 BCD). BCD) 1/4 BDE)1/5 CDE None of the above.625

825

Possible points this page = 17. POINTS EARNED THIS PAGE = ________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 13


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 isi j i j k

f = ________________________________________________________.A. B. C. D. E. 52. ( 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. 53. ( 4 pts.) The set of points on S where the tangent plane to S is horizontal

is __________________________________________________________.A. B. C. D. E.

Possible answersA) 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 + 3i 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 = _________MATH 251 FINAL EXAM Fall 2005 Prof. Moseley Page 14


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 Ewx

(x,y) (1,0)

55. (3 pts.) = __________________________________________________________________. A B C D E wy

(x,y) (1,0)

56. (4 pts.) = _______________________________________________. A B C D E dwdt

t 0

Possible answers.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 = _________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 15


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 the coordinates in the correct form.57. ( 4 pts.) The cylindrical coordinates of P are ___________________________..A B C D E

58. ( 4 pts.) The spherical coordinates of P are _____________________________..A B C D E

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)C2 2 2 2 2 2AB) ( , 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 2BD) ( , π/4,0)S BE) ( , π/4,π/4,)S CD) ( , π/4,1)S CE) ( , π/4, )S 2 2 2 2 2DE) ( , π/4,π/4)S ABC) ( , π/3, )S ABD) ( , π/3,1)S ABE) ( , π/4,2)S 2 2 2 2 2BCD) (1, π/4,1)S BCE)(1, π/4,0)S,0) CDE) None of the above.



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

A B C D E f___________________________________________________.

(x,y) (0,0)

60. (4 pts.) A. B. C. D. ˆ ˆu uD f(P) = D f ________________________________________________. (x,y) (0,0)

E.

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)1ACE). 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



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

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

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 =4ABD) 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.Possible points this page = 8. POINTS EARNED THIS PAGE = _________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 18


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 the gradient is

f = _______________________________________________________. A B C D E

64. ( 4 pts.) The set of critical points of this function

is __________________________________________________________. A B C D E

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.Possible points this page = 8. POINTS EARNED THIS PAGE = _________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 19


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

66. (4pts.) The final numerical value of I is I = __________________________.A. B. C. D. E.

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) 2

3 x

0

(2x e )dx2

3 x

0

(8x 4e )dx1

3 x

0

(6x 2e )dx2

3 x

0

(6x 4e )dx

BC) BD) BE). CD) 0 CE) 1 DE) 5 ABC) 8 2

2

0

(6x 2xe)dy1

3 x

0

(2x e )dy1

3 x

0

(2x 2e )dyABD) 10 ABE) 20 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/5ABCDE). None of the above.Possible points this page = 8. POINTS EARNED THIS PAGE = _________MATH 251 FINAL EXAM Spring 2006 Prof. Moseley Page 20


Assume . You are find g(x,y), α, β, γ, and δ, that is, you are to reverse22 4 x x

3 3

0 2x

4x y dydx g(x, y)dxdy

the order of integration in the integral. DO NOT EVALUATE EITHER INTEGRAL. Beginby drawing an appropriate picture.67. (2 pt.) g(x,y) = _________________________________________________. A B C D E

68. (4 pts.) α = ____________________________________________________. A B C D E

69. (4 pts.) β = _____________________________________________________.A B C D E

70. (2 pts.) γ = ______________________________________________________.A B C D E

71. (2 pts.) δ = ______________________________________________________.A B C D E. ,

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 Spring 2006 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. Determine g(x,y), α, β, γ, and δ. Begin by drawing an appropriate sketch. DO NOT EVALUATE THE INTEGRAL.72. (2 pts.) g(x,y) = __________________________________________________. A B C D E

73. (4 pts.) α = ______________________________________________________. A B C D E

74. (4 pts.) β = _______________________________________________________.A B C D E

75. (2 pts.) γ = _______________________________________________________.A B C D E

76. (2 pts.) δ = _______________________________________________________.A B C D E

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 x2 xBC) 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.

78. (3pts.) Doing the second step in the computation results in the single integral

I = __________________________________________________________. A B C D E

79. (3pts.) After the computation is complete, the numerical value of I is

I = _________________________________________________________.A B C D E

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) 8/51

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 Spring 2006 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 determineg(x,y,z), α, β, γ, δ, µ, and ν (i.e. set up an iterated integral in rectangular coordinates which givesthe value of V). Begin by drawing an appropriate sketch. DO NOT EVALUATE.

80. (2 pts.) g(x,y) = ________________________________________________. A B C D E

81. (1 pts.) α = ____________________________________________________. A B C D E

82. (4 pts.) β = ____________________________________________________.A B C D E

83. (1 pts.) γ = ____________________________________________________.A B C D E

84. (4 pts.) δ = ____________________________________________________.A. B C D E

85. (1 pts.) µ = ___________________________________________________.A B C D E

86. (2 pts.) ν = ___________________________________________________.A B C D E

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).3CDE) 1 ABCD). 2 ABCE) 3 ABDE) None of the above.



Let C be the curve that is the path of a point mass whose position vector is given by: = t + t2 , t [0, 1], (i.e. 0< x<1). Also let (x,y) = xy + . You are to compute

r i jF 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 theabove.

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 +i j i j i j i j i j it3 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) 815

ABD) ABE) ACD) ACE) ADE). BCD) 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.Mx

91. (2 pts.) __________________________________________________. A B C D EMy

92. (2 pts.) ___________________________________________________. A B C D ENx

93. (2 pts.) ___________________________________________________. A B C D ENy

94. (2 pts.) A potential function for (x,y) = M(x,y) + N(x,y) isF

i j

f(x,y) = _____________________________________________________. A B C D E

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 = ________

math 251 old final exam 2006 - west virginia university

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