MATH 251 MATH 251: Multivariate Calculus MATH 251 FALL 2006 EXAM-II FALL 2006 EXAM-II EXAMINATION COVER PAGE Professor Moseley PRINT NAME ( ) Last Name, First Name MI (What you wish to be called) ID # EXAM DATE Friday, October 6, 2006 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 1. Besides this cover page, there are 10 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. 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 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. You should explain your solutions fully and carefully. Every thought you have should be expressed in your best mathematics on this paper. No partial credit will be given for multiple choice questions. For multiple choice questions circle the letter or letters that corresponds to your answer. Proof-read 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 1 8 2 6 3 8 4 8 5 14 6 8 7 14 8 17 9 10 10 14 11 12 13 14 15 16 17 18 19 (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 20 21 22 Total 107 Total 100
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 1 On the Cartesian coordinate system below, the point (0,0) is labeled BE. Let P = (1,1), and Q = (3,2). You are to determine the labels for the points P and Q. If the point has no label, circle ABC for none of the labeled points. You should label these points P and Q and draw the vector PQ on the Cartesian coordinate system. Y ABC none of the labeled points. A B C_D_E AB ACADAE BC BD BE CDCE DE X 1. (1 pt.) On the above sketch, the point P is labeled. A B C D E 2. (1 pt.) On the above sketch, the point Q is labeled. A B C D E For questions 3 and 4, 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. 3. (3 pts.) The vector PQ (expressed in terms of i = [1,0] T and j = [0,1] T ) is PQ =. A B C D E 4. (3 pts.) A vector (in terms of î = [1,0] T and ĵ = [0,1] T ) that is in the opposite direction of PQ and is twice its length is. A B C D E Possible answers for questions 3 and 4. A) 3i ˆ ˆj B) 3i ˆ 3j ˆ C) 3i ˆ D) 2i ˆ ˆj E) 4i ˆ ˆj AB) 3i ˆ ˆj AC) 3i ˆ AD) 3i ˆ ˆj AE) 4i ˆ 3j ˆ BC) BD) 2i ˆ ˆj BE) 2i ˆ CD) 4i ˆ ˆj CE) 4i ˆ ˆj DE) 4iˆ ˆj ABC) 4i ˆ 2j ˆ ABD) 4iˆ 2j ˆ ABE) 4i ˆ ACD) 8i ˆ 4j ˆ ACE) 8i ˆ 6j ˆ ADE) 8i ˆ BCD) 3i ˆ ˆj BCE) 6iˆ 2j ˆ BDE) 6iˆ 2j ˆ CDE) 12i ˆ 4j ˆ ABCD) 12i ˆ 3j ˆ ABCE) 8i ˆ 2j ˆ ACDE) 8i ˆ 2j ˆ BCDE) None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE = ˆ 2i
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 3 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). Write your answer in the blank provided and then circle the letter or letters that corresponds to your answer from the possibilities below. 7. (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 8. (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 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 + z 2 = 9 BD) (x +2) 2 + (y 3) 2 + z 2 = 9 BE) (x 2) 2 + (y 3) 2 = 3 CD) (x 2) 2 + (y 3) 2 + z 2 = 9 CE) (x 2) 2 + (y + 3) 2 + z 2 = 3 DE) (x +2) 2 + (y + 3) 2 = 3 ABC) None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 3 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). Write your answer in the blank provided and then circle the letter or letters that corresponds to your answer from the possibilities below. 7. (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 8. (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 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 + z 2 = 9 BD) (x +2) 2 + (y 3) 2 + z 2 = 9 BE) (x 2) 2 + (y 3) 2 = 3 CD) (x 2) 2 + (y 3) 2 + z 2 = 9 CE) (x 2) 2 + (y + 3) 2 + z 2 = 3 DE) (x +2) 2 + (y + 3) 2 = 3 ABC) None of the above Possible points this page = 8. POINTS EARNED THIS PAGE =
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 4 Choose the plane curve that corresponds to the parametric curves given below. Then circle the letter or letters that corresponds to your answer. x t 1 3 y t 9. (4 pts.) The curve for this set of parametric equations is. A B C D E x 3cos(t) y 2sin(t) 10. (4 pts.) The curve for this set of parametric equations is. A B C D E AC. None of the above. Possible points this page = 8. POINTS EARNED THIS PAGE =
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 5 Let C be the curve in the xy-plane described parametrically by: x = t 2, y = t, 0 < t < 2. First eliminate the parameter to determine what kind of curve this is and then find the tangent line to the curve (as in Math 155) at the point where t = 1. Next find a tangent vector to the curve at t = 1 and then find the parametric equations for the tangent line at t = 1. Recall that the parametric equations for a straight line are not unique. To get the set of equations given in the answers below, you should use the procedure illustrated in class (Attendance is mandatory) Write your answer in the blank and then circle the appropriate letter or letters that corresponds to your choice from the possibilities below. (Be careful! If you make a mistake, the rest is wrong.) 11.(3 pts.) The equation of the curve C obtained when the parameter is eliminated is. A B C D E 12. (4 pts.) The tangent line (in the form y = mx +b) to the curve C when t = 1 is. A B C D E 13. (3 pts.) A tangent vector to the curve C at t = 1 is. A B C D E 14. (4 pts.) A set of parametric equations for the tangent line to C at t = 1 is. A B C D E. (To see if you have a correct set of equations, eliminate the parameter and see if you get your answer to question 12 above.) Possible answers this page. A) y = x 2 B) x = y 2 C) y = x 2 +2 D) y = (x+1) 2 E) y = (x1) 2 AB) y = x + 1 AC) y = (½)x + (½) AE) y = x 2 + 1 BC) x = y 2 + 1 BE) ˆi ˆj CD) ˆi 2j ˆ CE) 4i ˆ 2ˆj DE) ABC) 2i ˆ 4j ˆ ABD) x= t, y = t + 1 ABE) x = t + 1, y = 2t + 1 ACD) x = 2t + 1, y = 4t + 1 ACE) x = t + 2, y = t 1 ADE) x = 2t +1, y = t + 1 BCD) x = 2t, y = 2t 1 BCE) x = t +1, y = 2t + 2 CDE) x = 2t + 2, y = 4t + 1 ABCD) None of the above Possible points this page = 14. POINTS EARNED THIS PAGE = 2i ˆ ˆj
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 6 Choose the three-dimensional plot that corresponds to the parametric curves given below. Then circle the letter that corresponds to your answer. 15. (4 pts.) x = t, y = sin 5t, z = cos 5t. The curve for this set of parametric equations is. A B C D E 16. (4 pts.) x = t sin 6t, y = t cos 6t, z = t. The curve for this set of parametric equations is. A B C D E A) B) C) D) E) None of the sketches is the given curve. Possible points this page = 8. POINTS EARNED THIS PAGE =
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 7 Let L 1 and L 2 be the two intersecting lines (check t = 0) whose parametric equations are given by: L 1 : x = 3t + 1, y = 0, z = 2t L 2 : x = 1, y = t, z = t where t R. You are to find an equation of the plane P that contains L 1 and L 2. 17. (2 pts.) The point where the lines intersect is. A B C D E 18. (2 pts.) A vector in the direction of L 1 is. A B C D E (The answer is not unique. Use the procedure given in class. Attendance is mandatory.) 19. (2 pts.) A vector in the direction of L 2 is. A B C D E (The answer is not unique. Use the procedure given in class. Attendance is mandatory.) 20. (4 pts.) A normal to the plane P is. A B C D E (The answer is not unique. Use the procedure given in class. Attendance is mandatory.) 21. (4 pts.) An equation for the plane P is. A B C D E (The answer is not unique. Use the procedure given in class. Attendance is mandatory.) 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 î + 2 ĵ + 3 ˆk AD) 3 î + 3 ˆk AE) 2 ĵ + 3 ˆk BC) 3 î + 2 ˆk BD) 2 î + 2 ĵ + 3 ˆk BE) ĵ ˆk CD) ĵ + 3 ˆk CE) 2 î + 2 ĵ + 3 ˆk DE) 2 î + 2 ĵ + 3 ˆk ABC) 2 î + 2 ĵ + 3 ˆk ABD) 2 î + 3 ĵ + 3 ˆk 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 EXAM II Fall 2006 Prof. Moseley Page 8 Suppose that the position vector for a point mass as a function of the time t is given by: r = 3 i + 4 sin t j + 4 cos t k. (Be careful! If you make a mistake, the rest is wrong.) 22. (3 pts) The velocity v(t) at time t = 0 is v(0) =. A B C D E v(t) v(0) 23. (3 pts) The acceleration a(t) at time t = 0 is a(0) =. A B C D E a(t) a(0) a(0) v(0) 24. (4 pts) is. A B C D E a(0) v(0) 25. (3 pts) v(0) is. A B C D E v(0) 26. (4 pts) The curvature at t = 0 of the curve traced out by the particle is κ =. A B C D E a(0) v(0) κ = A) 3 î B) 4 ˆk C) 4 ĵ 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 ) 12 25 DE) 2 ABC) 3 ABD) 4 ABE) 5 ACD) ACE) ADE) 1/3 BCD) BDE) 1/4 CDE) 1/5 ABCD) None of the above Possible points this page = 17. POINTS EARNED THIS PAGE = 6 25 8 25
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 9 Suppose that the acceleration vector for a point mass M as a function of the time t is given by: a = ( 4 cos (2t) ) i + (12 t 2 + 2) j + (3 e t ) k, the initial position is: r0 = r(0) = ˆi 3kˆ and the initial velocity is: v = v(0) = ˆi 3kˆ 0. You are to compute the velocity and position of M (Be careful. Remember once you make a mistake, the rest is wrong.) 27. ( 5 pts.) The velocity of M is v(t) =. A B C D E v(t) = 28. ( 5 pts.) The position of M is r(t). A B C D E r(t) A) ( 2 sin (2t) 1) î + (4t 3 + 2t ) ĵ + (3 e t ) ˆk B) ( 2 cos (2t) ) î + (t 3 + 2t 2 ) ĵ + (3 e t ) ˆk C) ( sin (2t) 1) î + (2t 2 + t ) ĵ + (3 e t ) ˆk D) ( 2 cos (2t) ) î + (t 4 + 2t 2 ) ĵ + (3 e t ) ) ˆk E) ( 2 sin (2t) 1) î + (4t 3 + 2t 2 ) ĵ + (3 e t ) ˆk AB) ( cos (2t) t ) î + ( t 4 + t 2 ) ĵ + (3 e t ) ˆk AC) ( 2 sin (2t) t) î + (4t 2 + 2t ) ĵ + (3 e t ) ˆk AD) ( cos (2t) t) î + ( t 4 + t ) ĵ + (3 e t ) ˆk AE) ( sin (2t) t) î + (4 t + 2 ) ĵ (3 e t ) ˆk BC) ( cos (2t) t) î + (t 3 + t ) ĵ + (3 e t ) ˆk BD) ( 4 sin (2t) t) î + (4t 2 + 2t ) ĵ + (3 e t ) ˆk BE) None of the above. Possible points this page = 10. POINTS EARNED THIS PAGE =
MATH 251 EXAM II Fall 2006 Prof. Moseley Page 10 Suppose that the position vector for a point mass as a function of the time t is given by: r = 3 i + 4 sin t j + 4 cos t k. (Be careful. If you make a mistake, the rest is wrong.) 31. (3 pts) The speed of the point mass is v(t) =. A B C D E v(t) v(t) 32. ( 3 pts.) The unit tangent vector ˆT to the curve traced out by the point mass at the point (3,0,4) (when t = 0) is. A B C D E ˆT(0) ˆT(t) ˆT(0) 33. ( 5 pts.) The derivative of the unit tangent vector T at the point (3,0,4) when t = 0 is ˆT (0). A B C D E ˆT (t) ˆT (0) 34. ( 3 pts.) The unit normal N at the point (3,0,4) when t = 0 is ˆN(0). A B C D E ˆT (0) ˆN(0) 2 A) 9t 16 B) 9t 2 2 2 16 cos (2t) 16sin (2t) C) 9 16cos(2t) 16sin(2t) D) 9t 16 E) 5 AB) 2 2 9t 16cos (2t) 16sin (2t) AC) 4 AD) 3 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) (3/5) î + (4/5) ĵ ABC) (3/5) î +(4/5) ˆk ABD) (8/5) ĵ ABE) (8/5) ĵ ACD) (8/5) ˆk ACE) (8/5) ˆk ADE) î BCD) ĵ BCE) ĵ BDE) ˆk CDE) None of the above Possible points this page = 14. POINTS EARNED THIS PAGE =