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

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1 MATH 5 MATH 5: Multivariate Calculus MATH 5 FALL 5 EXAM-IV FALL 5 EXAM-IV EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Thursday, April 3,, 6 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 DATE INSTRUCTIONS. Besides this cover page, there are 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!! REQUEST FOR REGRADE Please regrade the following problems for the reasons I have indicated: (e.g., I do not understand what I did wrong on page.) Scores page points score (Regrades should be requested within a week of the date the exam is returned. Attach additional sheets as necessary to explain your reasons.) I swear and/or affirm that upon the return of this exam I have written nothing on this exam except on this REGRADE FORM. (Writing or changing anything is considered to be cheating.) Date Signature 9 3 Total

2 MATH 5 EXAM IV Spring 6 Prof. Moseley Page Let w = f(x,y) = 5 e 3x cos(y), P be the point (,), and û be a unit vector in the direction of ˆ ˆ. Compute the following. Then circle the letter or letters that corresponds to your v 3i 4j answers from the list below. f. (4 pts.). A B C D E (x,y) (,) D f(p) = D f. uˆ uˆ. (4 pts.) A B C D E. (x, y) (,) 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). 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.

3 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page Let S be the surface which is defined by the graph of the function z = f(x,y). Suppose that f 6iˆ 4ˆj. (x, y) (,) and that f(,) =. Compute the following. Then circle the letter or letters that correspond to your answers. 3. (4 pts.) Using geometric notation (i.e. î and ĵ or î, ĵ, and ˆk ),, a normal to the surface S when x = and y = is. A B C D E 4. (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 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 = 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.

4 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 3 Consider the quardic surface S which is the graph of the equation x + y - 6x + y z = 4. Compute the following. Then circle the letter or letters that correspond to your answers. 5. ( 4 pts.) Using geometric notation, a formula for the normal to S at the point (x,y,z) on S is n =. A B C D E 6. ( 4 pts.) The set of points on S at which the tangent plane to S is horizontal is.a B C D E A) (x 6) î + (y+) ĵ + z ˆk B) (x+6) î + (y+) ĵ + z ˆk C) (x 6) î + (y+) ĵ z ˆk D) (x+6) î + (y ) ĵ + z ˆk E) (x+6) î + (y+) ĵ z ˆk AB) (x 3) î + (y+) ĵ z ˆk AC) (x 6) î + (y+) ĵ + z ˆk AD) (x 3) î + (y+) ĵ + z ˆk AE) (x 6) î + (y ) ĵ + z ˆk BC) (x 6) î + (y ) ĵ z ˆk BD) (x 3) î + (y+) ĵ + z ˆk BE) (x+6) î + (y+) ĵ + z ˆk CD) (x+6) î + (y+) ĵ + z ˆk CE) DE) {( 3,,),(3,, )} ABC) R ABD) {( 3,,),(3,, )} ABE) {(3,,),( 3,, )} ACD) {( 3,,),( 3,, )} ACE) {(3,,),(3,, )} ADE) {(3,,),(3,, )} BCD) {(3,,),(3,, )} BCE) {( 3,,),(3,, )} BDE) {(3,,),( 3,, )}, CDE) {( 3,,),( 3,, )} ABCD). {(3,, )} ABCE) {(3,,)} ABDE) None of the above.

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

6 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 5 Assume that the only critical point for the function z = f(x,y) is (,) and that f x (,) = 5, f y (,) =, f xx (,) = 3, f yy (,) =, and f xy (,) =. Fill in the following blanks. Then circle the letter or letters that corespond to your answers. 9. (4 pts.) The discriminant D (as defined in the text) at the point (,) is D =. A B C D E.. (4 pts.) The critical point (,) of this function may be classified using the Second. Derivative Test as follows.a B C D E A) B) C) D) 3 E) 4 AB) 5 AC). 6 AD) 7 AE) 8 BC) BD) BE) 3 CD) 45 CE) 6 DE) ABC) 5 ABD) 8 ABE) ACD) 45 ACE) f has a local maximum at (,) ADE) f has a local minimum at (,) BCD) f has a saddle point at (,) BCE) The second derivative test is indeterminate at (,) BDE) f has both a local maximum and a local mimimum at (,). CDE) f has a horseshoe point at (,). ABCD). f has a bridle point at (,). ABCE) f has a rainbow point at (,) ABDE) None of the above.

7 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 6 You are to evaluate the iterated integral 3 x I (6x y ye )dydx circle the letter or letters the correspond to your answers from the list below.. (4pts.) Doing the first step in the evaluation results in the single integral. Fill in the blanks below. Then I =.A B C D E.. (4pts.) The final numerical value of I is I =.A B C D E A) (x e )dy B) (3x e )dx C) (6x 4e )dy D) (6x e )dy E) AB) (x e )dx AC) (8x 4e )dx AD) (6x e )dx AE). (6x 4e )dx (3x e )dx BC) (6x xe)dy BD). (x e )dy BE) (x e )dy CD) CE) DE) 5 ABC) 8 ABD) ABE) ACD) 7 e ACE) 7 e ADE) 7 3e BCD). BCE). 5 BDE) 8 CDE) 4/(5) ABCD) ABCE) 7 e ABDE) 7 e ACDE) 7 3e BCDE) 7 8e/5 ABCDE) None of the above

8 Possible points this page = 8. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 7 Assume 4 x x 3 3 4x y dydx x g(x, y)dxdy. Find g(x,y), α, β, γ, and δ, that is, you are to reverse the order of integration in the integral. Fill in the blanks below. Then circle the letter or letters that correspond to your answers. DO NOT EVALUATE EITHER INTEGRAL. Begin by drawing an appropriate picture. 3. ( pt.) g(x,y) =. A B C D E 4. (4 pts.) α =. A B C D E. 5. (4 pts.) β =.A B C D E 6. ( pts.) γ =.A B C D E 7. ( pts.) δ =.A B C D E A) 4 x 3 y 3 B) x 3 y 3 C) 4x-x D) x 4 6 4y 4 6 4y E) AB) AC) AD) AE) BC) BD) BE) CD) 4 6 4y 4 6 4y 4 6 4y 6 4y 4 6 4y 4 6 4y 4 6 4y 4 4 CE) x DE) y ABC) x/ ABD) y/ AB). x/3 ACD) y/3 ACE)3x ADE) 3y BCD) BCE) BDE) CDE) 3 ABCD) ABCE) ABDE) 3 ACDE) None of the above 4

9 Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 8 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 =. Determine g(x,y), α, β, γ, and δ. Then circle the letter or letters that correspond to your answers. Begin by drawing an appropriate sketch. DO NOT EVALUATE THE INTEGRAL. 8. ( pts.) g(x,y) =. A B C D E 9. (4 pts.) α =. A B C D E. (4 pts.) β =.A B C D E. ( pts.) γ =.A B C D E. ( pts.) δ =.A B C D E A) 4 x 3 y 3 B) x 3 y 3 C) 4x-x D) x E) 3 4 x AB) AC) 4 x AD) x AE) x 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.

10 Possible points this page = 4. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 9 Consider the cardioid C whose equation in polar coordinates is given by r = + cos Θ. Sketch the cardioid C. Let A g(r, )drd be the area in the first quadrant that is inside the cardioid C. Set up the double integral in polar coordinates that gives the value of A. Circle the letter or letters that correspond to your answers. DO NOT EVALUATE. 3. ( pts.) g(r,θ) =.A B C D E 4. ( pts.) α =. A B C D E. 5. ( 3 pts.) β =.A B C D E. 6. ( pts.) γ = `.A B C D E. 7. ( 3 pts.) δ =.A B C D E A) θ B) r θ C) r D) r + θ E) 3r + 3 AB) cos θ AC) θ AD) 3r+ θ AE) + cos θ BC) BD) BE) 5 CD) 8 CE) DE) ABC) 3 ABD) 45 ABE) 6 ACD) ACE) 5 ADE) 8 BCD) BCE). 45 BDE) π/5 CDE) π/4 ABCD) π/5 ABCE) π/ ABDE) π ACDE) π BCDE) None of the above.

11 Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page PRINT NAME ( ) ID No. You are to evaluated the iterated integral I x xy 5xyz dzdydx. Fill in the blanks. Then circle the letter or letters that corresponds to your answers. 8. (3pts.) Doing the first step in in the computation results in the double integral I =. A B C D E 9. (3pts.) Doing the second step in the computation results in the single integral is I =. A B C D E 3. (3pts.) After the computation is complete, the numerical value of I is E. I =. A. B. C. D. x x x 3 3 A) 5xydydx B) 5x y dydx C) 5x y dydx D.) 5x y dydx E) x x x 5x y dydx 9 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). x x 9 4x dx 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.

12 Possible points this page = 9. POINTS EARNED THIS PAGE = MATH 5 EXAM IV Spring 6 Prof. Moseley Page 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 =. 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). Then circle the letter or letters that correspond to your answers. Begin by drawing an appropriate sketch. DO NOT EVALUATE. 3. ( pts.) g(x,y) =. A B C D E 3. ( pts.) α =. A B C D E 33. (4 pts.) β =.A B C D E 34. ( pts.) γ =.A B C D E 35. (4 pts.) δ =.A B C D E 36. ( pts.) µ =.A B C D E 37. ( pts.) ν =.A B C D E 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) 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

13 ABDE) None of the above Possible points this page = 5. POINTS EARNED THIS PAGE =

MATH 251 MATH 251: Multivariate Calculus MATH 251 SPRING 2012 EXAM-4 SPRING 2012 EXAM-4-B3 EXAMINATION COVER PAGE Professor Moseley

MATH 251 MATH 251: Multivariate Calculus MATH 251 SPRING 2012 EXAM-4 SPRING 2012 EXAM-4-B3 EXAMINATION COVER PAGE Professor Moseley MATH 5 MATH 5: Multivariate Calculus MATH 5 SPRING EXAM-4 SPRING EXAM-4-B3 EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Friday,