92.131
Calculus 1
Differentiability and Continuity
1)
a)
Write down the limit definition of the derivative of a function
)
(
x
f
′
.
b)
Using part (a) calculate
)
(
x
f
′
, for
2
6
)
(
+
=
x
x
f
.
How does
the result compare to
what you already know about linear functions?
2)
a)
Write down the limit definition of the derivative of a function
)
(
x
f
′
.
b)
Using part (a) calculate
)
(
x
f
′
, for
3
4
2
)
(
2
−
+
=
x
x
x
f
.
c)
Write down the power rule, and use it to check your answer in part(b).
3)
a)
Write down the limit definition of the derivative of a function
)
(
x
f
′
.
b)
Using part (a) calculate
)
(
x
f
′
, for
2
3
3
)
(
2
+
−
=
x
x
x
f
.
c)
Write down the power rule, and use it to check your answer in part(b).
4)
a)
Using the limit definition of the derivative, calculate
)
(
x
f
′
, for
3
1
)
(
3
x
x
x
f
+
=
.
b)
Use the power rule to check your answer.
5)
a)
Using the limit definition of the derivative, calculate
)
(
x
f
′
, for
2
1
)
(
2
2
x
x
x
f
+
=
.
b)
Use the power rule to check your answer.
6)
Consider the function
3
2
)
(


2
)
(
π
+
−
=
x
x
x
f
.
a)
Write down the limit definition for the derivative of a function.
b)
Using the definition of the derivative, determine whether or not the function
)
(
x
f
differentiable at
0
=
x
.
Justify your steps using limit laws or theorems.
c)
Is
)
(
x
f
continuous at at
0
=
x
?
Explain your reasoning.
7)
Consider the function
1


)
(
2
)
(
+
+
=
x
x
x
f
.
a)
Using the definition of the derivative, determine whether or not the function
)
(
x
f
differentiable at
0
=
x
.
b)
Is
)
(
x
f
continuous at at
0
=
x
?
Explain your reasoning.
8)
Suppose that the function
)
(
x
f
is defined by the rule
⎩
⎨
⎧
>
+
≤
+
=
2
2
1
)
(
3
2
x
a
x
x
x
a
x
f
,
where
a
is a real constant. What must the value of
a
be in order for the function to be
continuous at
2
=
x
?
Justify your reasoning.
9)
Suppose that the function
)
(
x
f
is defined by the rule
⎩
⎨
⎧
≥
+
<
+
=
1
1
1
)
(
2
x
x
b
x
a
x
x
f
.
What relationship must
a
and
b
satisfy for the function to be continuous at
1
=
x
?
10)
Using the limit definition of the derivative, calculate
)
(
x
f
′
, for
2
3
)
(
x
x
f
=
.
Use the
power rule to check your answer.
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View Full Document92.131
Calculus 1
Differentiability and Continuity
Solutions
1)
a)
h
x
f
h
x
f
x
f
h
)
(
)
(
lim
)
(
0
−
+
=
′
→
b)
[]
[
]
h
x
h
x
h
x
f
h
x
f
x
f
h
h
2
6
2
)
(
6
lim
)
(
)
(
lim
)
(
0
0
+
−
+
+
=
−
+
=
′
→
→
6
6
lim
2
6
2
6
6
lim
0
0
=
=
−
−
+
+
=
→
→
h
h
h
x
h
x
h
h
Since
2
6
)
(
+
=
x
x
f
is a linear function its graph is a line whose constant slope is
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 Fall '09
 Staff
 Calculus, Continuity, Derivative, Power Rule, lim, 1 K

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