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2.4 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly,
1993
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View Full Document The slope of a line is given by:
y
m
x
∆
=
∆
x
∆
y
∆
The slope at (1,1) can be approximated by the
slope of the secant through (4,16).
15
3
5
=
We could get a better approximation if we
move the point closer to (1,1).
ie: (3,9)
8
2
=
4
=
Even better would be the point (2,4).
y
x
∆
∆
4 1
2 1

=

3
1
=
3
=
( 29
2
f x
x
=
→
The slope of a line is given by:
y
m
x
∆
=
∆
x
∆
y
∆
If we got really close to (1,1), say (1.1,1.21),
the approximation would get better still
.21
.1
=
2.1
=
How far can we go?
( 29
2
f x
x
=
→
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View Full Document ( 29
1
f
1
1
h
+
( 29
1
f
h
+
h
slope
y
x
∆
=
∆
( 29 ( 29
1
1
f
h
f
h
+

=
slope at
( 29
1,1
( 29
2
0
1
1
lim
h
h
h
→
+

=
2
0
1
2
1
lim
h
h
h
h
→
+
+

=
( 29
0
2
lim
h
h
h
h
→
+
=
2
=
The slope of the curve
at the point
is:
( 29
y
f x
=
( 29
( 29
,
P a f a
( 29 ( 29
0
lim
h
f a
h
f a
m
h
→
+

=
→
The slope of the curve
at the point
is:
( 29
y
f x
=
( 29
( 29
,
P a f a
( 29 ( 29
0
lim
h
f a
h
f a
m
h
→
+

=
( 29 ( 29
f a
h
f a
h
+

is called the difference quotient of
f
at
a
.
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This note was uploaded on 02/11/2011 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.
 Spring '08
 Smith
 Slope

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