MIDTERM 2
MAT 142
11/11/05
Name
Se.
ID number
TA's name
1
2
3
4
5
total
THIS EXAM IS WORTH 50 POINTS. PUT ALL ANSWERS IN THE SPACE
PROVIDED. NO NOTES OR CALCULATORS ALLOWED.
1.
Put the letter of the slope eld in the box of the orresponding equation.
All slope elds are graphed on [ 2; 2℄ [ 2; 2℄.
(Part A, 6 pts)
A
B
C
D
E
F
i.
ii.
(Part
i.
ii.
y
0 = x2 + y 2
y
0 = y (1
B, 4 pts)
x
2
e
x2
y
)
iii.
y
0=x
iv.
y
0 = ex
y
2
v.
y
vi.
y
0 = ( 2xy )=(1 + x2 )
y
0 = sin(3x)
Math eah dierential equation to the orresponding solution.
iii.
iv.
sin(x)
osh(x)
1
0 + y = ex
C
xy
00 = (2 + 4x2 )y
D
y
A
y
B
y
0 = 2y
00 =
y
2.
Put a `C' (for onverges) or `D' (for diverges) in the box next
to eah innite series and explain why this is orret using tests from the textbook.
(2 pts eah, 10 pts total)
(a)
P1
(b)
P1
()
P1
(d)
P1
(e)
P1
1
n=1 n+10
n=1
n
1+n3
sin2 (n)
p
n=1 n n
2n
n=1 nn
n=1 an ,
where a1 = 1 and an+1 = 21 (an + 1) for n > 1.
2
3.
Evaluate eah of the following innite series. Put the nal answer in
the box and show your work below.
(10 pts, 2 pts eah)
(a)
P1
(b)
P1
n=1 3
n
41
n
2n+1
n=1 n2 (n+1)2
=
=
() 1 + 12 x + 14 x2 + 81 x3 + : : : =
(d)
P1
(e)
P1
n
n=0 sin (x)
n=1 (1
=
+ ( 1)n )2
n
=
3
4.
Solve eah of the following dierential equations.
(10pts; 2pts eah)
(a)
dy
dx
= ex
(b)
dy
dx
= 2x 1
()
dy
dx
= (1 + y 2 )ex ,
y
p
y
2
,
jj
y
y
<
1
(0) = 0
4
5.
x dy
dx
(d)
e
(e)
x
dy
dx
+ 2ex y = 1
= x2 + 3y with y (1) = 1, x > 0
Do TWO of the following problems (your hoie). Put a mark
in the box next to the two problems you want to be graded.
(5 pts eah, 10 pts total)
(a)
A tank ontains 100 gallons of fresh water. A solution ontaining 1 lb/gal of
soluble fertilizer runs into the tank at a rate of 1 gal/min and the well stirred mixture
is drawn from the tank at a rate of 3 gal/min. Find the amount of fertilizer in the tank
as a funtion of time.
(b)
Suppose Euler's method with n = 100 is applied to the dierential equation
= x + y with initial ondition y (0) = 1 to estimate y (1). Is the resulting estimate
y100 larger or smaller than the atual value y (1) of the real solution? Explain why.
()
(d)
dy
dx
P an onverges then prove that P1 (an)2
If fan g is a positive sequene and if 1
n=1
n=1
also onverges.
R
Does the improper integral 11 j sinx xj dx diverge or onverge? Justify your answer.
5