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MATH 241: FINAL EXAM
Name
Instructions and Point Values: Put your name in the space provided ab ove. Check that your test contains 14 di erent pages including one blank page. Work each problem b elow and show ALL of your work. Unless stated otherwise, you do not need to simplify your answers. Do NOT use a calculator.
There are 300 total p oints p ossible on this exam. The p oints for each problem in each part is indicated b elow.
PART I
Problem (1) is worth 20 p oints.
Problem (2) is worth 15 p oints. Problem (3) is worth 15 p oints.
Problem (4) is worth 18 p oints. Problem (5) is worth 18 p oints.
Problem (6) is worth 24 p oints. Problem (7) is worth 20 p oints.
Problem (8) is worth 20 p oints.
PART I I
Problem (1) is worth 30 p oints. Problem (2) is worth 40 p oints.
Problem (3) is worth 40 p oints. Problem (4) is worth 40 p oints.
PART I. Answer each of the following.
(1) Let
! u = h 2 ; 1 ; 2 i and
! v = h 3 ; 1 ; 1 i. Calculate:
(a)
! u 2
! v
(b) the dot pro duct of ! u and ! v
(c)
! u
! v
(d) the length (or magnitude) of
! u
(4) Calculate cylindrical co ordinates (r; ; z ) and spherical co ordinates (; ; ) for the p oint with rectangular co ordinates (x; y ; z ) = (3; 3
p 3 ; 2
p 3).
(r; ; z ):
(; ; ):
(5) Find an equation for the tangent plane to the surface (x + y )(x + y + z ) = 2 z 2 at the p oint (3; 1 ; 2).
(6) Calculate the following integrals. SIMPLIFY your answers.
(a)
Z 1
0
Z x
0
(x y ) dy dx
(b)
Z
0
Z
0
Z
0
d d d
(7) Let
! F = x^2 i + xy j + y 2 k. Calculate the divergence and curl of
! F.
Divergence:
Curl:
PART I I. Answer each of the following. Make sure your work is clear. If you do not know how to answer a problem, tell me what you know that you think is relevant to the problem. If you end up with an answer that you think is incorrect, tell me this as well. Better yet, tell me why you think it is incorrect. In other words, let me know what you know.
(1) The graphs for the equations b elow are similar to 2 graphs on the last page of this test. The orientation and the scaling may b e di erent. For each equation, indicate which graph on the last page b est matches it. For example, if the equation is for a hyp erb olic parab oloid, then the graph you cho ose should b e a hyp erb olic parab oloid. Indicate your choice by putting the corresp onding letter from the last page in the b ox under the equation b elow. Next, read the question on the last page corresp onding to the graph you chose. Then go back to the equation b elow and put the answer to the question for the graph of that equation in the b ox. Do NOT answer the question for the graph on the last page (since it may b e oriented di erently than the graph of the equation b elow).
(a) z 2 + 2 x^2 = 3 y 2
ANSWER:
(b) z 2 + 2 x^2 = 3 y 2 1
ANSWER:
(2) Calculate the following integrals. (They are in the second part of this exam for a reason.)
(a)
Z 2
2
Z p 4 y 2
0
x^2 + y 2
dx dy
(b)
Z 1
1
Z 1
x^2
sin (y 3 =^2 ) dy dx
(3) Let P b e the plane 3 x + y 4 z = 7, let 1 b e the line given by x = 1 + 2 t, y = 1 2 t, and z = 3 + t, and let 2 b e the line given by x = 1 + t, y = 2 + t, and z = 2 + t.
(a) Explain why ` 1 do es not intersect the plane P.
(b) Do es ` 2 intersect the plane P? Justify your answer.
(c) Calculate the minimum distance from 1 to 2. (You should justify your work; in particular, if you happ en to have memorized a formula for such a distance, you should prove that the formula works.)
(4) Find the p oints on the graph of z = x^2 3 y 2 1 which are closest to the origin (0; 0 ; 0). Justify your answer.
