2.4 Rates of Change and Tangent Lines

Devil’s Tower, WyomingGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

The slope of a line is given by:y

mx

x

y

The slope at (1,1) can be approximated by the slope of the secant through (4,16).

y

x

16 1

4 1

15

3 5

We could get a better approximation if we move the point closer to (1,1). ie: (3,9)

y

x

9 1

3 1

8

2 4

Even better would be the point (2,4).

y

x

4 1

2 1

3

1 3

2f x x

The slope of a line is given by:y

mx

x

y

If we got really close to (1,1), say (1.1,1.21), the approximation would get better still

y

x

1.21 1

1.1 1

.21

.1 2.1

How far can we go?

2f x x

1f

1 1 h

1f h

h

slopey

x

1 1f h f

h

slope at 1,1 2

0

1 1limh

h

h

2

0

1 2 1limh

h h

h

0

2limh

h h

h

2

The slope of the curve at the point is: y f x ,P a f a

0

lim h

f a h f am

h

The slope of the curve at the point is: y f x ,P a f a

0

lim h

f a h f am

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.

In the previous example, the tangent line could be found

using . 1 1y y m x x

The slope of a curve at a point is the same as the slope of

the tangent line at that point.

If you want the normal line, use the opposite signed

reciprocal of the slope. (in this case, )1

2

(The normal line is perpendicular.)

Example 4:

a Find the slope at .x a

0

lim h

f a h f am

h

0

1 1

lim h

a h ah

0

1lim

hh

a a h

a a h

0lim h

a a h

h a a h

2

1

a

Let 1f x

x

On the TI-89:

limit ((1/(a + h) – 1/ a) / h, h, 0)

F3 CalcNote:If it says “Find the limit” on a test, you must show your work!

a a h

a a h

a a h

0

Example 4:

b Where is the slope ?1

4

Let 1f x

x

2

1 1

4 a

2 4a

2a

Example 4:

c What are the tangent line equations when and ?

2x

2x

2 :x 1

2y

1 1y y m x x

1 12

2 4y x

1 1 1

2 4 2y x

11

4y x

2 :x 1

2y

1 1y y m x x

1 12

2 4y x

1 1 1

2 4 2y x

11

4y x

Example 4:Graph the curve and the tangents on theTI-89:

Y= y = 1 / x

WINDOW

6 6

3 3

scl 1

scl 1

x

y

x

y

GRAPH

Graph the curve and the tangents on theTI-89:

Example 4:

Y= y = 1 / x

WINDOW

6 6

3 3

scl 1

scl 1

x

y

x

y

GRAPH

F5 Math

A: Tangent ENTER

2 ENTER

Repeat for x = -2

tangent equation

We can let the calculator plot the tangents:

Review:

average slope:y

mx

slope at a point:

0lim h

f a h f am

h

average velocity: ave

total distance

total timeV

instantaneous velocity:

0

lim h

f t h f tV

h

If is the position function: f t

These are often mixed up by Calculus students!

So are these!

velocity = slope