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

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1 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, Apri4,, :3 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. page Scores points score SIGNATURE DATE INSTRUCTIONS: Besides this cover page, there are pages of problems with questions 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. Pages - are Fillin-the Blank/Multiple Choice or True/False. Expect no part credit on these pages. For each Fill-in-the Blank/Multiple Choice question write your answer in the blank provided. There is a point penalty for not writing your answer. Next find your answer from the list given and write the corresponding letter or letters for your answer in the blank provided. Then circle this letter or letters. There are no free response pages. However, to obtain credit on any regrade, you must explain your solutions fully and carefully. For regrades, your entire solution may be graded, not just your final answer. SHOW YOUR WORK! Every thought you have should be expressed in your best mathematics on this paper. For regrades, partial credit will be given as deemed appropriate. Generally, unless you have a perfect solution, you will receive no credit. 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.) (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 3 Total

2 MATH 5 EXAM 4-B3 Prof. Moseley Page PRINT NAME ( ) ID No. = f(x,y) = 5e 3x sin(3y) 5x, P be the point P = (,) in the xy plane, and û be a unit vector in the direction of v 4i ˆ 3j ˆ. You are to compute the directional derivative of f at the point P in the direction of v. 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) (,) Let z Possible answers. A) B) C) D)3 E)4 AB)5 AC) AD) AE)3 BC) 4 BD) 5 BC) 6 BD) 7 BE) CD) 6 CE) 4/(5) DE) 8/5 ABC) 4/(5) ABD) 8/ 5 ACD) 8/ 5 ACE)5 î +5 ĵ ˆk ADE)5 î 3 ĵ ˆk BCD) 5 î +3 ĵ ˆk BCE) 5 î 3 ĵ ˆk BDE) 5 î +5 ĵ CDE) 5 î 5 ĵ ABCD) 5 î +3 ĵ ABCE) 5 î +45 ĵ ABDE) 5 î +6 ˆk ABCDE) None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =

3 MATH 5 EXAM 4-B3 Prof. Moseley Page PRINT NAME ( ) ID No. be the surface that is the graph of the function z = f(x,y). Let P be the point on S where x = and y =. Suppose that f 3i ˆ 7j ˆ (x,y) (,) use geometric notation. 3. ( pts.) A normal, call it, to the surface S at the point P is n Let S and that f(,) = 6. Find the following. For vectors, n =. A B C D E (The answer may not be unique. Use the procedure given in class. Attendance is mandatory.) 4. ( pts.) The point P on the surface S is P = A B C D E 5. (4pts.) An equation for the tangent plane to S at the point P on the surface S is. A B C D E (The answer may not be unique. Use the procedure given in class. Attendance is mandatory.) Possible answers this page. A) î +7 ĵ + ˆk B) î +7 ĵ ˆk C) î 7 ĵ + ˆk D) î +7 ĵ ˆk E)3 î +7 ĵ + ˆk AB) 3 î +7 ĵ ˆk AC) 4 î +7 ĵ + ˆk E)4 î +7 ĵ ˆk AB) 5 î +7 ĵ + ˆk BC) (,,) BD) (,,) BE)(,,4) CD) (,,6) CE) (,,8) DE) x +7y z = 6 ABC) x + 7y z = 5 ABD) 3x +7y z = 4 ABE) 4x +7y z = 3 ACD) 6x + 4y z = 5 ACE) 6x + 4y z = 5 ADE) 6x + 4y z = 5 BCD) 3x + 7y z = BCE) 3x 7y z = BDE) 3x 7y z = 4 CDE) 3x + 7y z = 5 ABCD)3x + 7y z = 6 ABCE) 3x + 7y z = 3 ABDE) None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE =

4 MATH 5 EXAM 4-B3 Prof. Moseley Page 3 PRINT NAME ( ) ID No. Consider the function f:r R defined by z = f(x,y) = x 3 + (9/)x + y (4 pts.) Using geometric notation, the gradient of f is f =. 7. (4 pts.) The set of critical points of this function, call it C, is A B C D E C =. A B C D E Possible answers this page. A)(3x +3x) î + y ĵ B)(3x 3) î +y ĵ C)(3x +6x) î +y ĵ D)(3x 6x) î +y ĵ E)(3x +9x) î +y ĵ AB)(3x 9x) î y ĵ AC)(3x +x) î +y ĵ 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) R ABC){(,),(,)} ABD){(,),(,)} ABE) {(,),(,)} ACD){(,),(,)} ACE) {(,), (3,)} ADE){(,),(3,)} BCD) {(3,),(3,)} BCE) {(3,),(3,)} BDE) {(,),(3,)} CDE) {(,),(4,)} ABCD) {(4,)} ABCE) {(4,)} ABCDE) None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =

5 MATH 5 EXAM 4-B3 Prof. Moseley Page 4 PRINT NAME ( ) ID No. Assume that z = f(x,y)a (R,R) and that the only point where f = is P = (,) and that f xx (,) =, f yy (,) =, and f xy (,) =. 8. (4 pts.) Given the data above, the value of the discriminant D (as defined in the text) for this function at the point P is D =. A B C D E 9. (4 pts.) Using the Second Derivative Test the behavior of f at the critical point P may be classified as follows:. A B C D E Possible answers this page. A) B) C) D) 3 E) 4 AB) 5 AC) 6 AD) 7 AE) 8 BC) 9 BD) BE) CD) 3 CE) 4 DE) 5 ABC) 6 ABD) 7 ABE) 8 ACD) 9 ACE) f(,) is a local maximum ADE) f(,) is a local minimum BCD) (,) is a saddle point BCE) the second derivative test is indeterminate CDE)(,) is a horseshoe point BDE) f(,) is both a local maximum and a local mimimum ABCD) (,) is a bridle point ABCE)(,) is a rainbow point ABCDE) None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =

6 MATH 5 EXAM 4-B3 Prof. Moseley Page 5 PRINT NAME ( ) ID No. For each question write your answer in the blank provided. Next find your answer from the list of possible answers listed below and write the corresponding letter or letters for your answer in the blank provided. Finally, circle this letter or letters. Evaluate the iterated integral the numerical value of the double integral. 3 x I (6x y 8ye )dydx in steps by first finding a single integral and then. (4pts.) Doing the first step in the evaluation of this double integral results in the single integral I =. A B C D E. (4pts.) The final numerical value of I is I =. A B C D E Possible answers A) (x e )dy B) (x e )dx C) (x e )dy D) (x 4e )dy E) AB) (x 8e )dx AC) (x 4e )dx AD) (x 4e )dx AE) (4x e )dx BC) (x 6e )dx ( 4 x x e )d y BD) ( 4x e )dy BE) ( 4x e )dy CD) CE) DE) ABC)3 ABD)4 ABE) 5 ACD) 6 ACE) e ADE) 4e BCD) 4 6e BCE)6 8e BDE)3e CDE) 4e ABCD)8+e ABCE) 8e ABDE)9+e ACDE)9e BCDE)9+e ABCDE)None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =

7 MATH 5 EXAM 4-B3 Prof. Moseley Page 6 PRINT NAME ( ) ID No.. Assume 4 6 x 3 3x y dydx 3x 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. Do not evaluate either integral. Begin by drawing an appropriate picture.. ( pt.) g(x,y) =. A B C D E 3. (4 pts.) α =. A B C D E 4. (4 pts.) β =. A B C D E 5. ( pts.) γ =. A B C D E 6. ( pts.) δ =. A B C D E Possible answers this page. A) B) C) D)3 E)4 AB)5 AC)6 AD)7 AE)8 BC)9 BD) BE)6 CD)x CE)y DE)x/ ABC)y/ ABD)y/3 ABE)y/4 ACD)x ACE)y ADE)x /4 BCD)y /4 BCE)x /(6) BDE) y /(6) CDE) y /(36) ABCD) y /(64) ABCE)x y 3 ABDE)x y 3 ACDE)3x y 3 BCDE)4x y 3 ABCDE)None of the above

8 Possible points this page = 3. POINTS EARNED THIS PAGE = MATH 5 EXAM 4-B3 Prof. Moseley Page 7 PRINT NAME ( ) ID No. Let A g(x, y) dydx be the area of the region in the first quadrant bounded by the curves y = x /3, x + y = 9 and x=. 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 8. (4 pts.) α =. A B C D E 9. (4 pts.) β =. A B C D E. ( pts.) γ =. A B C D E. ( pts.) δ =. A B C D E Possible answers. A) B) C) D) 3 E) 4 AB) 5 AC) 6 AD) 7 AE) 8 BC) 9 BD) 3 x BE) 6 x CD) 9 x CE) x DE) 3 4 x ABC) ABD) 4x ABE) x ACD) x ACE) x ADE) x / BCD) x /3 BCE) x /4 BDE) x /5 CDE)x ABCD) 3x ABCE) None of the above

9 Possible points this page = 3. POINTS EARNED THIS PAGE = MATH 5 EXAM 4-B3 Prof. Moseley Page 8 PRINT NAME ( ) ID No. Sketch the cardioid C whose equation in polar coordinates is given by r = + sin θ. Suppose A g(r, )drd is the double integral in polar coordinates that gives the area in the third quadrant that is inside the cardioid C. Determine g(r,θ), α, β, γ, and δ. Do not evaluate the integral.. ( pts.) g(r,θ) =. A B C D E 3. ( pts.) α =. A B C D E 4. ( 3 pts.) β =. A B C D E 5. ( pts.) γ =. A B C D E 6. ( pts.) δ =. A B C D E Possible answers this page. A) B) C) D)3 E) 4 AB) 5 AC)6 AD)7 AE)8 BC) 9 BD)π/6 BE)π/4 CD)π/3 CE)π/ DE) π ABC) 3π/ ABD)π ABE)r ACD) r ACE) 3θ + 3 ADE) cos θ BCD) sin θ BDE)+sin θ CDE)+ cos θ ABCD) sin θ ABCE) cos θ ABCDE) None of the above

10 Possible points this page =. POINTS EARNED THIS PAGE = MATH 5 EXAM 4-B3 Prof. Moseley Page 9 PRINT NAME ( ) ID No. Evaluate the iterated integral I x xy 35x 3yzdzdydx in steps. 7. (3pts.) Doing the first step in the computation results in the double integral I =. A B C D E 8. (3pts.) Doing the second step in the computation results in the single integral I =. A B C D E 9. (3pts.) The final numerical value of I is I = A B C D E Possible answers. A) B) C) D) 3 E)4 AB)6 AC)3 AD) 48 AE) 64 BC) 8 BD) 96 BE) CD) 9x dx CE) 88x dx DE) 384x dx ABC) 96x dx ABD) 9x dx ABE) x ACD) 384x dx ACE) 96x dx ADE) 5x ydydx BCD) 5x y dydx BCE) x BDE) 3x y dydx CDE) 45x ydydx ABCD) 45x y dydx ABCE) x x ABDE) 6x y dydx ACDE) 5x y dydx ABCDE) None of the above x x x 88x dx x 3 3x ydydx x 3 6x ydydx 96x dx

11 Possible points this page = 9. POINTS EARNED THIS PAGE = MATH 5 EXAM 4-B3 Prof. Moseley Page PRINT NAME ( ) ID No. V = g(x, y, z)dzdydx be the volume of the solid in the first octant bounded by the ellipsoid 4x + y +4 z = 4, and the planes 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). Begin by drawing an appropriate sketch. Do not evaluate the integral. 3. ( pts.) g(x,y,z) =. A B C D E 3. ( pt.) α =. A B C D E 3. (4 pts.) β =. A B C D E 33. ( pt.) γ =. A B C D E 34. (4 pts.) δ =. A B C D E 35. ( pt.) µ =. A B C D E 36. ( pts.) ν =. A B C D E Let Possible answers. A) B) C) D) 3 E) 4 AB)5 AC) 6 AD)7 AE)8 BC) 4x BD) x/ BE) x/4 CD) 6 x CE) x/3 DE) 3x ABC) x y 6 x y ) ABD) 6 4x y ABE) 4 ACD) x y 4 4 y ADE) x BCD) BCE) 4 x (y / 4) BDE) 4 (x / 4) y CDE) x y 4 4 x y ABCD) 6 x ABCE) 4 x ABDE) 4 4x ACDE) 4 ( x / 4 ) ABCDE)None of the above

12 Possible points this page = 4. POINTS EARNED THIS PAGE =

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

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