1. Lesson 2-4: Rates of Change
AP Calculus
Mrs. Mongold

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

7. If it says “Find the limit” on a test, you must show your work!
Example 4:
Let a
Find the slope at
Note:
If it says “Find the limit” on a test, you must show your work!

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

9. 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

10. Example
Find an equation of the tangent line to the hyperbola at P(3, 1)

11. Example

12. Example

13. Yes…. Complex fractions!!
Example
Yes…. Complex fractions!!

14. Example

15. Example

16. Example

17. Example

18. Example

22. Example
Find slope of the curve at x=a and describe what happens to tangent at x=a as a increases. For y=x2 + 2.

23. Example

24. Example

25. Example

26. Example

27. Example

28. Example

29. Example
As a increases the slope will increase!

30. Your Turn
Write an equation for the normal line to the curve f(x)=4-x2 and x=1.

31. Your Turn

32. Your Turn
Write equation for the tangent line and normal line to at x=2

33. Your Turn
Write equation for the tangent line and normal line to at x=2
Answer: Tangent y = -x + 3
Normal y = x - 1

34. Homework
Pgs 87-89/ 2-32 Even