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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
a a h
a a h
a a h
0
The box above the zero is for a + or – sign to indicate direction.
Example 4:
Note:If it says “Find the limit” on a test, you must show your work!
On the TI-nspire:
menu 4
Calculus
4
Limit
enter
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
(Find the tangent line to the function y=1/x when x is 2.)
On the TI-nspire:
T tangentLine(
tangentLine(1/x, x, 2)
tangentLine(1/x, x, -2)Hint: Instead of re-entering the formula, use the up arrow to highlight the first formula and then press enter.Then insert the negative sign.
Now we will graph the function and the tangent lines.
1 1/f x x
menu enter3
enter
Use Zoom – Standard if you don’t have this window.
Use to toggle to the graph screen.
Now go back to the calculate screen, use the up arrow to highlight the first tangent equation, and use .ctrl C
Go to the graph screen and input the first tangent equation using . ctrl V
Repeat the process to input the second tangent equation.
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
