2.1 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
Photo by Vickie Kelly, 1993
Greg Kelly, Hanford High School, Richland, Washington
AP Practice
Find the following limit:
1
lim cos
x 
x
1
AP Practice
Find the following limit:
2 x
lim
x 1 1  x

The slope of a line is given by:
y
m
x
y
x
The slope at (1,1) can be approximated by
the slope of the secant through (4,16).
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
y 16  1 15


5
x
4 1
3
We could get a better approximation if we
move the point closer to (1,1). ie: (3,9)
y
9 1
8


x
3 1
2
4
0 1 2 3 4
f  x  x
Even better would be the point (2,4).
2
y
4 1
3


x
2 1
1
3

The slope of a line is given by:
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
y
m
x
y
x
If we got really close to (1,1), say (1.1,1.21),
the approximation would get better still
y 1.21  1 .21


 2.1
.1
x 1.1  1
How far can we go?
0 1 2 3 4
f  x  x
2

y
slope 
x
f 1  h 
f 1
slope at 1,1
h
1 1 h
f 1  h   f 1

h
1 h

 lim
h 0
2
1
h
h  2  h
1  2h  h 2  1
 lim
 lim
h 0
h 0
h
h

2

The slope of the curve y  f  x  at the point P a, f  a  is:
f  a  h  f  a
m  lim
h 0
h



The slope of the curve y  f  x  at the point P a, f  a  is:
f  a  h  f  a
m  lim
h 0
h
f a  h  f a
h
is called the difference quotient of f at a.
If you are asked to find the slope using the definition or using
the difference quotient, this is the technique you will use.

Recap:
Average rate of change between (a, f(a)) and (b, f(b)
Is also called the average velocity: f b  f a
 
 
ba
Instantaneous rate of change at (a, f(a)) is also called
The actual velocity at that point or the derivative:
f  a  h  f  a
m  lim
h 0
h
Alternatively, you can find the instantaneous rate of
change at (x, f(x)) and evaluate it at x = a.
f  x  h  f  x
m  lim
h 0
h
The slope of a curve at a point is the same as
the slope of the tangent line at that point.
Tangent lines can be found using the point-slope form of a
line:
y  y1  m  x  x.1 
If you want the normal line, use the opposite reciprocal of
the slope.
(The normal line is perpendicular.)

Important example:
Using the limit of the difference quotient,
find the slope of the line tangent to the
graph of the given function at x= -1, then
use the slope to find the equation of the
tangent line:
f  x  x 1
2
Derivatives
lim
h 0
f a  h  f a
h
We write:
is called the derivative of
f   x   lim
h 0
f
at
a.
f  x  h  f  x
h
“The derivative of f with respect to x is …”

You must be able to do this:
Find the equation of a line tangent to the graph
at (2,2)
f  x  x  2
You must be able to do this:
Find the equation of a line tangent to the graph
at (-2,-1)
f  x    2 x  3x  1
2
Homework
• P. 103
1,3, 7, 17, 21, 23, 25, 29, 33, 37-40 all