MATH 152
Final Exam
Fall 1997
Version B
Student (Print)
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Last,
First
Middle
Student (Sign)
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Student ID
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Instructor
____P. Yasskin__________________________
Section
1-13 |
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14 |
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15 |
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16 |
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17 |
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TOTAL |
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Part I is multiple choice. There is no partial credit. You may not use a calculator.
Part II is work out. Show all your work. Partial credit will be given. You may use your
calculator.
1
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator. You have 1 hour.
1. Find the area between the parabola y x 2 " 2x and the line y x.
a. " 10
3
b. 7
6
c. 13
6
10
d.
3
9
e.
2
2. Find the plane tangent to the hyperbolic paraboloid z x 2 " y 2 at the point Ÿ2, 1, 3 . Its
z-intercept is z a. 1
b. 0
c. "1
d. "2
e. "3
3. In the partial fraction expansion
a.
b.
c.
d.
e.
A
A
A
A
A
"1, B 1, C 2
"1, B 0, C 2
1, B 0, C 4
0, B 1, C "4
1, B "1, C "4
x2 4 A B C
the coefficients are
x
2
2
x
"
2
xŸx " 2 Ÿx " 2 2
4. The area in the first quadrant between the hyperbola y 3
x and the line y 4 " x is
rotated about the y-axis. Find the volume of the solid swept out.
3
2
a. = ; 4 " x " 3
dx
x
1
3
2
b. 2= ; 4 " x " 3
x dx
1
3
c. 2= ; 4 " x " 3
x dx
1
3
d. 2= ; Ÿ4x " x 2 " 3 dx
1
3
e. 2= ; Ÿ4 " x 2 "
1
3
x
2
dx
5. Which limit does not exist?
lim Ÿ2x 2 y 2 a.
Ÿx,y vŸ0,0 lim Ÿ2x 2 " y 2 b.
Ÿx,y vŸ0,0 c.
d.
lim
v
Ÿx,y Ÿ0,0 lim
v
Ÿx,y Ÿ0,0 e.
lim
v
Ÿx,y Ÿ0,0 6. ;
2
1
a.
b.
c.
d.
e.
2x 2 " y 2
x2 y2
2x 4 x 2 y 2
x2 y2
2x 4 " x 2 y 2
x2 y2
x 3 x 2 " 1 dx 2
15
8
15
22 2 " 8
15
15
8 Ÿ2 5/4 2 3/4 15
8 Ÿ2 5/4 2 3/4 " 1 15
7. An airplane is circling above an airport, clockwise as seen from above. Thus its wings
are banked with the right wing lower than the left wing. In what direction does the
binormal B point?
a. horizontally toward the center of the circle
b. vertically up
c. vertically down
d. along the left wing
e. along the right wing
3
8. Solve the differential equation
a.
b.
c.
d.
e.
dy
28 x y . If yŸ0 4, then yŸ1 dx
81
53
49
11
9
9. A 50 ft cable is hanging from the roof down the side of a tall building. If the cable
weighs 3 lb / ft, how much work is done to lift the cable to the roof?
a. 7500 ft-lb
b. 3750 ft-lb
c. 1875 ft-lb
d. 150 ft-lb
e. 75 ft-lb
10. A nuclear power plant is producing a radioactive isotope X at the rate of 75 kg /yr. Let
NŸt kg be the amount of X at the power plant at time t. A known fact is that X
decays at the rate of 50NŸt kg /yr. Write the differential equation to be solved for NŸt .
a. dN 50N " 75
dt
b. dN 75 " 50N
dt
c. dN 50N " 75t
dt
dN
75t " 50N
d.
dt
e. dN "25N
dt
4
11. The function fŸx, y has the values given below. Estimate f Ÿ2.1, 3.0 .
y
fŸ2.0, 3.2 3
fŸ2.1, 3.2 7
fŸ2.2, 3.2 9
fŸ2.0, 3.1 2
fŸ2.1, 3.1 4
fŸ2.2, 3.1 8
fŸ2.0, 3.0 1
fŸ2.1, 3.0 2
fŸ2.2, 3.0 5
a.
b.
c.
d.
e.
2
10
15
20
30
12. Which of the planes described below contain the lines:
a.
b.
c.
d.
e.
x"1 y"2 z"1
2
3
4
x " 2y z "2
x " 2y z 2
2x " y z "1
2x " y z "1
x"yz 0
and
x"1 y"2 z"1 ?
4
3
2
13. The volume of a cone is V 1 =r 2 h. If the radius is measured as r 3 o .2, and the
3
height is measured as h 4 o .1, then the volume is computed as V 12= o V
where the error in the measurement of the volume is V a. 1.1=
b. 1.7=
c. 1.9=
d. 2.2=
e. 2.7=
5
Part II: Work Out
Show all your work. Partial credit will be given.
You may use your calculator but only after 1 hour.
14. (10 points) Consider the curve rŸt Ÿ1 " t 2 , 1 t 2 , t . Compute each of the following:
a. velocity
v
b. speed (Simplify.)
|v | c. acceleration
a
d. curvature
4
e. tangential acceleration
aT f. normal acceleration
aN 6
15. (10 points) Compute
;
3/4
0
1
1 x2
dx
7
16. (10 points) A comet has just passed by the sun.
Its current position is Ÿx, y Ÿ3 • 10 7 mi, 4 • 10 7 mi dy
mi , 7 • 10 7 mi .
4 • 10 7 day
and its current velocity is v dx ,
day
dt dt
2
2
Hence the current distance from the sun is R x y 5 • 10 7 mi.
Find the rate at which the distance from the sun is increasing.
17. (5 points) This is the direction field of a certain differential equation. Draw in the
solution y yŸx which satisfies the initial condition yŸ2 3.
1.
4
y
-8
-6
-4
-2
2
0
2
4
x
6
8
-2
-4
8