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Lab Section:
MATH 215 – Winter 2006
Final Examination
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This exam contains eight problems, worth a total of 100 points. The first four questions are multiple choice. Your answers for these four questions are to be entered in the table below. No partial credit will be given for the first four problems, so double check your work. Please do not cheat. The use of books, calculators, cell phones, computers, notes, cheat sheets, and all similar aids is strictly prohibited. Good luck. SHOW YOUR WORK.
Problem Indicate your answer Points
1a ©A ©B ©C ©D ©E 5
1b ©A ©B ©C ©D ©E 10
2 ©A ©B ©C ©D ©E 5
3a ©A ©B ©C ©D ©E 5
3b ©A ©B ©C ©D ©E 5
4a ©A ©B ©C ©D ©E 5
4b ©A ©B ©C ©D ©E 5
0–4 TOTAL
Problem Points Score
TOTAL 100
Problem 1. (15 = 5 + 10 points)
(a) (5 points) If ~u = 〈 3 , 1 , 2 〉 and ~v = 〈− 1 , 2 , 2 〉, then ~u · ~v equals
A. 1 B. − 1 C. 3 D. 13 E. None of the above.
(b) (10 points) If ~u = 〈 3 , 1 , 2 〉 and ~v = 〈− 1 , 2 , 2 〉, then ~u × ~v equals
A. 〈− 3 , 2 , 4 〉 B. 〈− 2 , − 8 , 7 〉 C. 〈− 3 , − 2 , 4 〉 D. 〈− 2 , 8 , 7 〉 E. None of the above
Problem 3. (10 = 5 + 5 points)
(a) ( 5 points) The direction (expressed as a unit vector) of steepest ascent for the function f (x, y) = x^2 y + y^2 x + x at the point (3, −1) is
A. 〈− 4 , 3 〉 B. 〈 4 , − 3 〉 C. 〈 4 / 5 , − 3 / 5 〉 D. 〈− 4 / 5 , 3 / 5 〉 E. None of the above.
(b) ( 5 points) The critical point (− 2 /
- for the function f (x, y) = x^2 y + y^2 x + x is A. a local minimum for f. B. a local maximum for f. C. a saddle point for f. D. All of the above. E. None of the above.
Problem 4. (10 = 5 + 5 points)
(a) (5 points) Suppose C is the boundary of a domain D with C oriented as in the statement of Green’s theorem. We have that∮
C
(3x^2 y^2 + exy^ + 1) dx + (ey^ + xy^2 + 2yx^3 ) dy
is equal to
A. (^) ∫ ∫
D
(− 6 x^2 y^2 − yexy^ + ey^ + 2xy + 2x^3 ) dA
B. ∫ ∫
D
(y^2 − xexy) dA
C. ∫ ∫
D
(−y^2 + xexy) dA
D. ∫ ∫
D
(6x^2 y^2 + yexy^ − ey^ − 2 xy − 2 x^3 ) dA
E. None of the above.
Problem 5. (15 = 5 + 10 points) Suppose S is that part of the cylinder of radius 9 centered about the z-axis which lies above the plane described by the equation z = −17, below the plane described by the equation z = 17, and in the half space defined by the equation y ≥ 0.
(a) (5 points) Sketch S.
(b) (10 points) Evaluate (^) ∫ ∫
S
f dS
where f (x, y, z) = xy^2 z^2.
Continue your work for Problem 5 here.
(b) (5 points) Give a parametrization of the tangent line at the point (18, 6 , 12) to the parametric curve C given by r(t) = 〈 2 t^2 , 2 t, 4 t〉
where 0 ≤ t ≤ 5.
(c) (5 points) Show that the line found in part (b) belongs to the plane found in part (a). Also, in fifteen words or less, explain why the line found in part (b) belongs to the plane found in part (a).
Problem 7. (10 points) Suppose C is the curve parametrized by r(t) = 〈3 cos(t), 3 sin(t), 2 〉 for 0 ≤ t ≤ 2 π. Compute the line integral of F(x, y, z) = 〈−x^2 yz, xy^2 z, exy〉 along the curve C. (Hint: Stokes’ Theorem may be useful.)
Problem 8. (15 = 5 + 10 points) Let P be the prism bounded below by the plane with equation z = 0, on the sides by the planes with equations y = 0, y = 5, and x = 0, and “above” by the plane with equation z = 2 − x.
(a) (5 points) Sketch P.
(b) (10 points) Let S′^ be the boundary of P and let S be the surface obtained by removing the bottom face (i.e., the face in the xy-plane) from S′. We suppose that S has “outward” orien- tation. Note that S has four sides. Compute the flux across S of the vector field F(x, y, z) = 〈x^3 y, x^2 y^2 , x^2 y(z + 1)〉. (Hint: the divergence theorem may be useful.)
Continue your work for Problem 8 here.
END OF EXAM
Before leaving the examination room: (1) Make sure you have transferred your answers for problems 0 – 3 to the chart on page one. (2) Make sure you have placed your name and section number on page one. (3) Make sure you turn in your exam to the proctor.
