Practice Final Exam B for Math 1501, Calculus I
PART ONE – CORRESPONDING TO TEST ONE
(I):
Let
f
be the function given by
f
(
x
)=
x
2
x
+1
for
x
6
=

1.
(a)
Determine all values of
x
for which

f
(
x
)

f
(1)

<
1
/
2
.
(b)
For
±>
0, ﬁnd a number
δ
(
±
)
>
0sothat

x
=1

<δ
(
±
)
⇒
f
(
x
)

f
(1)

<±.
(II):
Determine whether the following limits exist. When they do exist, evaluate the limit.
(a)
lim
x
→
3
√
x

√
3
3

x
(b)
lim
x
→
0
x

2
±
sec
2
(2
1
/
3
x
)

1
²
(c)
lim
x
→
1

x

1

x
1
/
3

1
(III):
At which values of
x
, if any, is the following functions continuous?
f
(
x
)=
n
x
if
x
is irrational,
x
2
if
x
is rational.
Justify your answer.
PART TWO – CORRESPONDING TO TEST TWO
(IV):
Consider the curve given by
y
3

xy
2
=
x.
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Find all points on this curve at which the slope of the tangent line is 1.
(b)
Find the equation for the tangent line to the curve at each of the points found in part
(a).
(V):
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 Fall '08
 N/A
 Calculus, following number, Practice Final Exam, cartesian form, perfectly eﬃcient turbine

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