1. 2.1 Limits, Rates of Change, and Tangent Lines
Devil’s Tower, Wyoming
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

2. Suppose you drive 200 miles, and it takes you 4 hours.
Then your average speed is:
If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

3. A rock falls from a high cliff.
The position of the rock is given by:
After 2 seconds:
average speed:
What is the instantaneous speed at 2 seconds?

4. for some very small change in t
where h = some very small change in t
We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

5. We can see that the velocity approaches 64 ft/sec as h becomes very small.1 80
0.1
65.6
.01
64.16
.001
64.016
.0001
.00001
We say that the velocity has a limiting value of 64 as h approaches zero.
(Note that h never actually becomes zero.)

6. The limit as h approaches zero:

7. Section 2.1, Figure 1
Page 60

8. 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).

9. 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?

10. Section 2.1, Figure 4
Page 62

11. Section 2.1, Figure 5
Page 63

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

13. 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.

14. 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 negative reciprocal of the slope. (in this case, )
(The normal line is perpendicular.)

15. Review:  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 