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Math 114, MAKEUP
FINAL EXAM
December 16, 2002
INSTRUCTIONS
OFFICIAL USE ONLY
INSTRUCTIONS: 1. Please complete the information requested below. There are 14 multiple choice problems and 6 open answer problems. No partial credit will be given on the multiple choice questions. 2. Please show all your work on the exam itself. Correct answers with little or no supporting work will not be given credit. 3. Only 18 of the 20 problems on the exam will count towards your final grade. We will automatically drop the worst score answer from each part (multiple choice and open answer) of the exam. 4. You are allowed to use one hand-written sheet of paper with formulas. No calculators, books or other aids are allowed. Please turn in your crib sheet together with your exam. Problem 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Points
o Dr.
.I
. Name . Name
(please- print): of your professor:
Hughes
o Dr.
0
Pantev
o Dr.
Sabitova
Preston
o Madeeha
certify
. Name
of your TA:
Khalid that
o Maria
Ian Blumenfeld
o Christine Devena o Bill Semus
is my own.
all of the work on this test
Signature:
14. 15. 16. 17. 18. 19. 20. Total
. Recitation
section:
1
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Part I: Multiple
choice questions
1. What is the x-intercept of the line tangent to the curve x (t)
when t = I?
(A) 3 (D) 1 (B) 2 (E) 3 (C) does not exist 5 (F) 4
= 3 + cos(1ft),Y(t) = t2+ t + 1
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2.
Find the angle between the curves y
= x2 and x = y3 at
the point (1,1).
(A) 0
1f' (D) "3
(B) ~ 6 (E) ~ 2
(C) ~ 4 (F) undefined
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3.
Find the area inside the lemniscater = ~ cos(B), -~ < B < ~. 3 2-2 4 (A) 3 2 (D) 9 27r (B) "3 1 (E) V3 1 (0) 3V3 7r (F) "9
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4.
Find the integral that is equivalent to
2 2x
1 1X2 f(x, o
2 2Y 4
y) dydx
..fY
1
(A)
1o 1~
2
y2
f(x, y) dx dy
(B)
10 1Y~ f(x,
2 v'2Y
y) dx dy
(C)
1o
4
/
Y/2
..fY
f(x, y) dx dy
VY72
f(x,y)dydx
(D)
1o 1 2y
f(x,y)dxdy
(E)
10 1y2/2
f(x,y)dydx
(F)
10 1y
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5.
Use polar coordinates to evaluate the integral
J L Jx2
where D = {(x, y)lx2 + y2 :::;1,Y 2::Ixl}.
+ y2 + 3dxdy,
(A) 0
(B) -1 (E) ~(8 - 3V3) 6
(C) ~(9 - h) (F) ~ 3
(D) ~ 2
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6. Find the distance from the point P = (1,1,1) to the line passing through A = (1,0,-1) and perpendicular to the plane x - y + z = 1.
(A) y'2 (D) 14 3
(B) J3
(C) 2 (F) v7
(E) I
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7. Let z = eXy + cos(x) and let x and y be functions of t satisfying x(O) and y(O) = 2, y'(O)= -1. Find z'(O).
= 0, x'(O) = 3
(A) 5 (D) 1/2
(B) e (E) -1
(C) 0
(F) undefined
8. Find the point of intersection of the line x tangent to X2+ 2y = z2 at (2,0, -2). (A) (3,3,0) (D) (-1, -3, -2)
= 1 + 2t, y = 3t, z = t - 1 and the plane
(B) (0,-~, -~) (E) (~,~, -~)
(C) (1,0, -1) (F) (7,9,2)
-
- -
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-
9. the Use linear approximation of the function f(x, y) = sin(ln(l + x)y) at (0,11")to estimate f( -0.1,11"). (A) -0.111"
(D) 0.3511"
(B) 0.911" (E) 0.3
(C) 0 (F) -0.2
10.
Suppose y(x) satisfies the differential equation
y' equation. (A) -2 (D) 1 (B) -1 (E) 2 (C) 0
= x2
+ y2
with initial condition y(O) = -1. Find y"(O). Hint: you don't need to solve the differential
(F) does not exist
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11.
Which of these functions is continuous at (O,O)?
(I) f (x, y) = ;J;- ;-+ Y x2 x+y (II) f(x, y) = X2 +y
(III) f(x, y) = xy2 2 X +y
(A) (I) only (D) (I) and (II)
(B) (II) only (E) (I) and (III)
(C) (III) only (F) (II) and (III)
12.
The function f(x, y) = 8x3 + y3 + 6xy has
(A) one local maximum and one local minimum (B) one saddle point and one local maximum (C) one saddle point and one local minimum (D) two saddle points (E) two local maxima (F) two local minima
-
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13.
Which of these differential equations is exact as written?
(I) (x + y2)dx + (2xy + y3)dy = 0
(II) (X + t) dx+ (y + ~) dy= 0 (III) (1 - :2) dx+ (y + ~) dy= 0
(A) (I) only (D) (I) and (II)
(B) (II) only (E) (I) and (III) (C) (III) only (F) (II) and (III)
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14.
Find the difference of the absolute
maximum
and the absolute
minimum
of the
function f(x, y) = 7x - 4y + 3 on the curve (x - 1)2+ (x - y)2 = 25. (A) 1 (D) 12 (B) -6 (E) -7 (C) 50 (F) 4
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Part II: Open answer questions
15.
True or false. Explain your reasoning.
(a) The surface given in cylindrical coordinates by the equation r = 1rZCsc(e)is a plane.
(b) The spherical equations <p = 1r/3,lei = 1r/4describe a single ray.
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16.
Use polar coordinates...