2.4 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
Photo by Vickie Kelly, 1993
Greg Kelly, Hanford High School, Richland, Washington
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.

The slope of a curve at a point is the same as the slope of
the tangent line at that point.
In the previous example, the tangent line could be found
using y  y1  m  x  x1  .
If you want the normal line, use the negative reciprocal of
the slope. (in this case, 
1
)
2
(The normal line is perpendicular.)

Example 4:
1
Let f  x  
x
a Find the slope at x  a .
On the TI-89:
limit ((1/(a + h) – 1/ a) / h, h, 0)
F3
Calc
f  a  h  f  a
m  lim
h 0
h
1 a  a  h 1 a  a  h

a
 lim a  h
h 0
h a  a  h
1 a  a  h
 lim 
h 0 h
a a  h
aah
 lim
0
h 0 h  a  a  h 
Note:
If it says “Find the limit”
on a test, you must
show your work!
1
 2
a

Example 4:
On the TI-89:
1
Let f  x  
x
b Where is the slope  1 ?
4
1
1
  2
4
a
a2  4
a  2
Y=
y=1/x
WINDOW
6  x  6
3  y  3
x scl  1
y scl  1
GRAPH

Example 4:
1
Let f  x  
x
b Where is the slope  1 ?
4
We can let the calculator
Ontangent:
the TI-89:
plot the
F5
Math
y=1/x
Y=
A: Tangent ENTER
WINDOW
2
tangent equation
ENTER
6  x  6
3  y  3
x scl for
 1 x = -2
Repeat
y scl  1
GRAPH

Review:
y
average slope: m 
x
f  a  h  f  a
slope at a point: m  lim
h 0
h
average velocity:
Vave
instantaneous velocity:
velocity = slope
total distance

total time
These are
often
mixed up
by
Calculus
students!
So are these!
If f  t  is the position function:
f t  h   f t 
V  lim
h 0
h
p