This preview shows pages 1–2. Sign up to view the full content.
Math 125  Final(Common)  Fall 2003
1.
[20 points] Find the limits, if they exist, of the following expressions (you may not use
L’Hospital’s rule).
1a.
lim
x
→
0
x
cot
x
1b.
lim
x
→∞
(
√
x

x
)
1c.
lim
x
→
0
1
2+
x

1
2
x
.
2.
[20 points] Consider
f
(
x
) =
x
2

x

x

1

if
x
6
= 1
2
if
x
= 1
.
2a.
Sketch the graph of
f
(
x
).
2b.
Find the numbers at which
f
(
x
) is
discontinuous
. If none, write NONE.
You must justify your answer.
3.
[15 points] If a snowball melts so that its surface area (
S
= 4
πr
2
) decreases at a rate of
1
cm
2
/min
, ﬁnd the rate at which the radius decreases when the radius is 10
cm
.
4a.
[10 points] Find the linear approximation to
f
(
x
) =
e
x
near
x
= 0.
4b.
[5 points] Sketch the graph of
f
and its linear approximation in the
x
interval of [

1
,
1].
5.
Find the derivatives of the following functions:
5a.
[5 points]
f
(
x
) =
x
3
+ 1
x
2
+ 5
5b.
[5 points]
f
(
x
) = 3
e
2
x
2
+1
5c.
[5 points]
f
(
x
) = ln (1 +
x
2
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.
 Fall '07
 Tuffaha
 Math, Calculus, Limits

Click to edit the document details