Example
Let
f
(
x
) =
1
2
x
+ 1
.
Use the definition of the derivative to find
f
0
(1) = lim
h
→
0
f
(1 +
h
)

f
(1)
h
,
(1)
and then write the equation of the tangent line to the curve
y
=
f
(
x
) at the point
(1
, f
(1)).
Solution:
Making the substitution
f
(1 +
h
) = 1
/
[2(1 +
h
) + 1] into (1), it follows that
f
0
(1)
=
lim
h
→
0
1
2(1 +
h
) + 1

1
2
·
1 + 1
h
=
lim
h
→
0
1
3 + 2
h

1
3
h
=
lim
h
→
0
3

(3 + 2
h
)
3(3 + 2
h
)
h
=
lim
h
→
0
1
h
·

2
h
3(3 + 2
h
)
=
lim
h
→
0

2
3(3 + 2
h
)
=

2
9
.
Thus the slope of the tangent line to the curve
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 KIHYUNHYUN
 Calculus, Derivative, Slope, Line segment

Click to edit the document details