Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam 2
Fall 2000
Test Form A
Solutions
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:
Mn
x f x0
2
x1
f x1
2
x2
C
f x n"1 x n
2
x ¡fŸx o 2fŸx C 2fŸx n fŸx n ¢
"1
1
2
S n x ¡fŸx o 4fŸx 1 2fŸx 2 C 2fŸx n"2 4fŸx n"1 fŸx n ¢
3
KŸb " a 3
where K f UU Ÿx for all x in ¡a, b ¢
|E M | 24n 2
KŸb " a 3
where K f UU Ÿx for all x in ¡a, b ¢
|E T | 12n 2
KŸb " a 5
where K f Ÿ4 Ÿx for all x in ¡a, b ¢
|E S | 180n 4
Tn
; lnx dx
x lnx " x C
; tan x dx
" ln|cos x| C
; sec x dx
ln|sec x tan x| C
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. =
M
correctchoice
3
; > 9 " x 2 dx
0
My
x
> 1 =r 2
4
9= >
4
3
3/2
; >x 9 " x 2 dx "> 1 Ÿ9 " x 2 3
0
My
9> 4
4
=
M
9=>
3
0
> 1 9 3/2
3
9>
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
correctchoice
5
7
3
4
4 2
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.
x
T4
T4
4"2 1
4
2
x fŸ2 2f
2
1 1 4 5
4 2
5 2fŸ3 2f 7
2
2
2 4 1
7
3
4
fŸ4 2
4
; 1x dx, which of the following is
2
the smallest upper bound on the error |E T | in the approximation?
a. 1
12
b. 1
24
c. 1
48
correctchoice
d. 1
96
e. 1
192
3. If you use the Trapezoid Rule with n
f UU Ÿx |E T |
2
x3
4 to approximate
On ¡2, 4 ¢ the maximum is K
KŸb " a 3
12n 2
1 Ÿ4 " 2 3
4
12Ÿ4 2
f UU Ÿ2 2
8
dy
dx
x sin 2x y tan x is IŸx 1
96
4. An integrating factor for the differential equation
correctchoice
a. |cos x|
b.
c.
d.
e.
1 . So the error is bounded by
4
"sec 2 x
e
e "|tan x|
"|sec x|
e |cos x|
dy
" Ÿtan x y x sin 2x
dx
sin x dx ln|cos x|
" ; tan x dx " ; cos
x
Standard form:
; Pdx
IŸ x e
; P dx
e ln|cos x|
P
" tan x
|cos 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.
5. The parametric curve x
2
a. ; 2=Ÿe t " t Ÿe 2t
b. ; 2=Ÿe t " t Ÿe t
0
2
c. ; 8=e t/2 Ÿe 2t
d. ; 8=e t/2 Ÿe t
0
2
e. ; 8=e t/2 e t
1 dt
correctchoice
1 dt
1 dt
1 dt
2e t
0
2
2e t
0
2
1 dt
0
dx
dt
A
; 2=r ds
2
dy
dt
et " 1
2
; 8=e
0
; 2=y
0
t/2
Ÿe
2t
2e t/2
dx
dt
" 2e
t
2
1 Ÿ
dy
dt
4e t
2
2
dt
; 2=4e t/2
0
Ÿe
2
dt
; 8=e t/2 e 2t
0
t
" 1 2 Ÿ2e t/2 2 dt
2e t
1 dt
2
; 8=e t/2 Ÿe t
0
1 dt
3
A tank is completely filled with water. The end is
6.
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
m3
5
a. 9800 ; Ÿ5 " y 2 25 " y 2 dy
9.8 m 2 .
sec
0
-5
5
correctchoice
0
5
b. 9800 ; Ÿ5 y 25 " y 2 dy
0
5
c. 9800 ; Ÿ5 " y Ÿ25 " y 2 dy
0
5
d. 9800 ; Ÿ5 y 2 25 " y 2 dy
0
5
e. 9800 ; Ÿ5 y Ÿ25 " y 2 dy
0
The slice at height y is at a distance h
x 25 " y 2
F
; >ghwdy
7. Compute ;
.
e
5 " y below the surface. Its width is w
2x where
5
9800 ; Ÿ5 " y 2 25 " y 2 dy
0
1 dx.
x lnx
a. 0
b. 1
e
c. 1
d. e
e. .
u
;
.
e
correctchoice
ln x
du 1x dx
1 dx ; 1 du ln|u|
u
x lnx
ln|lnx|
.
e
lim ln|ln b| " ln|lne|
bv.
.
4
8. Compute nlim
v.
lnŸ2 e n 3n
a. 0
b. 1
correctchoice
3
c. 2
3
d. ln 2
3
e. .
en
lnŸ2 e n 2 en
Use l’Hopital’s Rule twice:
lim
lim
nv.
n
v.
3
3n
lnŸ2 e n n
OR:
Since e is much greater than 2,
lim
nv.
3n
1 lim e n 1 lim e n
3 nv. 2 e n
3 nv. e n
n
lnŸe n 1
lim
lim
nv.
nv. 3n
3
3n
9. After separating variables, we can solve the differential equation
a. ; 1
y
y dy
y
dy
y2 1
c. ;Ÿy 2 y dy
b. ;
dy
dt
ty yt
1
3
by solving
; t dt
; t dt
;Ÿt 1 dt
correctchoice
2
1 1
t
;
y
dt
dy
e. None of the above, the differential equation is not separable.
d. ;
y
dy
dt
t y 1y
t
y2
1
®
y
y
y2
1
10. Find the arc length of the parametric curve x
a.
4 cos t,
y
y
;
®
t dt
y2
4 sin t for
1
dy
; t dt
= t t t =.
3
6
=
3
2
b. =
3
c.
dy
correctchoice
=
2
d. 4=
3
e. 3=
4
dx
dt
L
OR:
;
"4 sin t
=/3
=/6
dy
dt
4 cos t
2
dx 2 dy
dt
dt
dt
=/3 " =/6 2=4 2=
2=
3
;
=/3
=/6
16 sin 2 t 16 cos 2 t dt
;
=/3
=/6
4 dt
4 = " =
6
3
2=
3
5
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.
Circle one: Converges Diverges
2
Explain:
sin n=
2
takes the values 1, 0, "1, 0 repeatedly. So it can never have a limit.
cos 2 n
b. lim
n
nv.
Circle one: Converges Diverges
2
Explain:
2
0 t cosn n t 1n
2
2
1
and lim
nv. 2 n
cos 2 n
0. By the sandwich theorem, lim
nv.
2n
0.
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. ;
Circle one: Converges Diverges
1
1 " 2
1
2
x
1 2x
x 2x
1 " 2
dx lnx " lnŸ1 2x x
1 2x
.
1 " ln 1 ln 3
ln
ln x
2
2
1 2x 1
3
Partial fractions gives
;
.
1
1
dx
x 2x 2
;
.
1
1
1
dx
x 2x 2
Explain:
b. ;
;
.
1
Circle one: Converges Diverges
0
1
0
1
dx
x 2x 2
;
1
0
1 " 1
x
1 2x
dx
lnx " lnŸ1 2x 1
0
¡0
" ln 3 ¢ " ¡". " 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.
a. Write out the differential equation and the initial condition for SŸt , the number of lbs of salt
in the tank at time t.
dS lb
dt min
2 lb 5 gal
S t lb 5 gal
" Ÿ gal min
50 gal min
dS
dt
10 " 1 S
10
SŸ0 20
" 10 ln 10 " 1 S
b. Solve the initial value problem.
dS
dt
10 " 1 S
10
dS
;
10 " 1 S
10
ln 10 " 1 S " t " C
10
10
10
SŸ0 20 100 " 10A
A8
; dt
10
10 " 1 S Ae "t/10
10
S 100 " 80e "t/10
S
tC
100 " 10Ae "t/10
c. How much salt is in the tank after 10 hours?
SŸ600 100 " 80e "600/10 100 " 80e "60
The problem should have said 10 min which gives
SŸ10 100 " 80e "10/10 100 " 80
e
14. Compute the surface area of the surface obtained by rotating the curve x
1 t y t 2 about the x-axis?
A
u
; 2=r ds
2
1 16y
2
; 2=y 1 1
du
2
dx
dy
dy
1 2y 2 for
2
; 2=y 1 16y 2 dy
1
du
32
32y dy
1 ; 2= u du = 2u 3/2
16 3
32 y1
3/2
3/2
=
=
"
Ÿ65 Ÿ17 24
24
2
A
y dy
= Ÿ1 16y 2 3/2
2
y1
24
2
1
15. Solve the initial value problem
x
dy
2 y cos x
x
dx
x2
dy
2xy cos x
x2
dx
x = when y 1
P
2
x
dy
dx
I
2y
e
; Pdx
cos x
x
e
2
; x dx
with
e 2 ln x
yŸ= 1.
x2
d Ÿx 2 y cos x
x 2 y ; cos x dx sin x C
dx
C =2
x 2 y sin x = 2
y
= 2 sin = C
sin x = 2
x2
7