Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam 2
Fall 2000
Test Form A
1-10
/50
11
/10
12
/10
Part II is work out. Show all your work. Partial credit will be given.
13
/10
You may not use a calculator.
14
/10
15
/10
Part I is multiple choice. There is no partial credit.
TOTAL
Formulas:
C
x 1 f x 1 x 2 f x n"1 x n
Mn x f x0
2
2
2
x
f x o 2f x 1 2f x n"1 f x n
Tn 2
2f x n"2 4f x n"1 f x n
S n x f x o 4f x 1 2f x 2 3
K b"a 3
where K f x for all x in a, b
|E M | 24n 2
K b"a 3
where K f x for all x in a, b
|E T | 12n 2
K b"a 5
where K f Ÿ4 x for all x in a, b
|E S | 180n 4
¡Ÿ
¡Ÿ
Ÿ
Ÿ
Ÿ
Ÿ C Ÿ Ÿ ¢
Ÿ Ÿ C Ÿ Ÿ Ÿ ¢
Ÿ ¡ ¢
Ÿ ¡ ¢
Ÿ ¡ ¢
; lnx dx x lnx " x C
; tan x dx " ln|cos x| C
; sec x dx ln|sec x tan x| C
UU
UU
1
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator.
1. Calculate the x-component (or coordinate) of the center of mass of a plate with uniform density
> and whose shape is the quarter circle of radius 3 given by 0 t x t 3 and 0 t y t 9 " x 2 .
a.
=
4
9
b. = >
4
c. 9>
d. 8
=
4
e. =
4
2. Which of the following gives the Trapezoid Rule approximation to ; 1
x dx with n 4?
2
1 2 1 2 1
5
7
3
8
4 4
1 1 2 1 2 1
5
7
3
4
4 2
1 1 4 2 4 1
5
7
4 2
3
4
1 ln 2 ln 3 ln 3 ln 7 ln 4
2
2
2
1 ln 2 2 ln 3 2 ln 3 2 ln 7 ln 4
2
4
2
a. 1
b.
c.
d.
e.
4
; 1x dx, which of the following is
2
the smallest upper bound on the error |E T | in the approximation?
3. If you use the Trapezoid Rule with n 4 to approximate
a.
b.
c.
d.
e.
1
12
1
24
1
48
1
96
1
192
2
4. An integrating factor for the differential equation
a.
b.
c.
d.
e.
Ÿ dy
x sin 2x y tan x is I x dx
|cos x|
2
e "sec x
e "|tan x|
"|sec x|
e |cos x|
5. The parametric curve x e t " t,
y 4e t/2 for 0 t t t 2 is rotated about the x-axis. Which
integral gives the area of the surface of revolution?
HINT: Look for a perfect square.
2
Ÿ Ÿe 2e 1 dt
2=Ÿe " t Ÿe 1 dt
8=e Ÿe 2e 1 dt
8=e Ÿe 1 dt
a. ; 2= e t " t
2t
b. ;
t
c. ;
d. ;
0
2
0
2
0
2
0
2
t
t/2
2t
t/2
t
t
t
e. ; 8=e t/2 e t 1 dt
0
6.
A tank is completely filled with water. The end is
5
a vertical semicircle with radius 5 m as shown.
Which integral gives the hydrostatic force on this
end of the tank?
The density of water is 1000
kg
and g 9.8 m 2 .
m3
sec
-5
0
5
Ÿ b. 9800 ; Ÿ5 y 25 " y dy
c. 9800 ; Ÿ5 " y Ÿ25 " y dy
d. 9800 ; Ÿ5 y 2 25 " y dy
e. 9800 ; Ÿ5 y Ÿ25 " y dy
5
a. 9800 ; 5 " y 2 25 " y 2 dy
0
5
0
5
2
2
0
5
0
5
2
2
0
3
7. Compute ;
.
e
1 dx.
x lnx
a. 0
b. 1
e
c. 1
d. e
e. .
8. Compute nlim
v.
Ÿ
ln 2 e n
3n
a. 0
b. 1
3
c. 2
3
d. ln 2
3
e. .
9. After separating variables, we can solve the differential equation
dy
ty yt
dt
by solving
a. ; 1
y y dy ; t dt
y
dy ; t dt
y2 1
c. ; y 2 y dy ; t 1 dt
b. ;
Ÿ
Ÿ y2 1 1
; t
y
dy
dt
e. None of the above, the differential equation is not separable.
d. ;
10. Find the arc length of the parametric curve x 4 cos t,
a.
=
c.
=
y 4 sin t for
= t t t =.
6
3
3
2
b. =
3
2
d. 4=
3
e. 3=
4
4
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may not use a calculator.
11. Determine whether each of the following sequences converges or diverges. If it converges,
find the limit. Fully justify your answers.
a. lim
sin n=
nv.
2
Circle one: Converges Diverges
Explain:
cos 2 n
b. lim
n
nv.
2
Explain:
Circle one: Converges Diverges
5
12. Determine whether each of the following integrals converges or diverges. If it converges, find
its value.
Hint: Partial fractions
.
1
dx
x 2x 2
Explain:
a. ;
1
1
dx
x 2x 2
Explain:
b. ;
Circle one: Converges Diverges
1
Circle one: Converges Diverges
0
6
13. A tank contains 20 lb of salt mixed with 50 gal of water. Salt water containing 2 lb of salt per
gal is added to the tank at the rate of 5 gal per min. The tank is kept thoroughly mixed and
drains at the same rate.
Ÿ a. Write out the differential equation and the initial condition for S t , the number of lbs of salt
in the tank at time t.
b. Solve the initial value problem.
c. How much salt is in the tank after 10 hours?
7
14. Compute the surface area of the surface obtained by rotating the curve x 1 2y 2 for
1 t y t 2 about the x-axis?
15. Solve the initial value problem
x
dy
x
2y cos
x
dx
with
Ÿ y = 1.
8