1. 2.4 Rates of Change and Tangent Lines
Devil’s Tower, Wyoming
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

2. The slope of a line is given by:
The slope at (1,1) can be approximated by the slope of the secant through (4,16).
We could get a better approximation if we move the point closer to (1,1). ie: (3,9)
Even better would be the point (2,4).

3. The slope of a line is given by:
If we got really close to (1,1), say (1.1,1.21), the approximation would get better still
How far can we go?

4. slope
slope at
The slope of the curve at the point is:

5. The slope of the curve at the point is:
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.

6. 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
If you want the normal line, use the opposite signed reciprocal of the slope. (in this case, )
(The normal line is perpendicular.)

7. limit ((1/(a + h) – 1/ a) / h, h, 0)
Example 4:
Let a
Find the slope at
On the TI-89:
limit ((1/(a + h) – 1/ a) / h, h, 0)
Note:
If it says “Find the limit” on a test, you must show your work! F3
Calc

8. Example 4:
Let b
Where is the slope ?

9. What are the tangent line equations when and ?
Example 4: c
What are the tangent line equations when and ?

10. Graph the curve and the tangents on theTI-89:
Example 4:
Graph the curve and the tangents on theTI-89: Y=
y = 1 / x
WINDOW
GRAPH

11. Graph the curve and the tangents on theTI-89:
Example 4:
Graph the curve and the tangents on theTI-89:
We can let the calculator plot the tangents: F5
Math Y=
y = 1 / x
A: Tangent
ENTER
WINDOW 2
ENTER
Repeat for x = -2
tangent equation
GRAPH

12. Review: p velocity = slope
These are often mixed up by Calculus students!
average slope:
slope at a point:
average velocity:
So are these!
instantaneous velocity:
If is the position function:
velocity = slope p