Final Exam Key, 2 hours and 45 minutes
April 24, 2003
The average was about 60/100 with the lowest scores on the recent material
 power series and polar coordinates. Warning  this key was written days
after the exam was graded, so I don’t expect too much interest in it, and I
haven’t checked it carefully. The basic methods given are all correct.
1)
1
2
x
4
dx
= 32
/
10 = 16
/
5.
2A) From memory, ln(1 +
x
) =
x

x
2
/
2 +
x
3
/
3

. . .
. So, ln(2)
≈
1

1
/
2 +
1
/
3 = 5
/
6. Since this is a decreasing alternating series, this answer is too
big, but the error is less than the next term, 1/4.
2B) ln(2) =
2
1
dt
t
≈
2

1
2
·
3
(1 + 4
·
2
3
+ 1
/
2) = 25
/
36.
Here, for example,
y
1
=
f
(
x
1
) =
f
(1
.
5) = 1
/
1
.
5 = 2
/
3.
3a) Set
u
= tan
x
and get (tan
3
x
)
/
3 +
C
.
3b) Do IBP twice to get
e
x
(
x
2

2
x
+ 2) +
C
.
3c) This is improper b/c it has an asymptote at
x
= 2. A graph is recom
mended. So, it should be split in two (this also helps remove the absolute
value signs).
The first part is
2
0

x

2


1
/
2
= lim
t
→
2

t
0
(2

x
)

1
/
2
=
lim
t
→
2


2(2

x
)
1
/
2

t
0
= 2
·
2
1
/
2
. You can get the right number without
using the ”lim” but you must include that for full credit. The other part
gives the same answer, which we add in to get 4
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 Storfer
 Calculus, Power Series, Polar Coordinates, Mathematical Series, Mathematical analysis, A.S. Test

Click to edit the document details