1. 2.1 The Derivative and the Slope of a Graph

2. Rates of Change
Slope of a line - rate at which the line rises or falls
For line, always the same at every point of the line
For graphs other than lines, the rate at which graph rises or falls changes from point to point

4. To find rate of change at a point, find the slope of the tangent line at the point
Can estimate – pg. 122 #5-10
Can find analytically

5. Slope of a Secant Line
Average rate of change of the function:

6. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

7. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

8. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

9. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

10. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

11. Slope of the Tangent Line – As two points of a secant line are brought together, a tangent line is formed.

12. Slope of the Tangent Line

13. Slope of the Tangent Line

14. Slope of a Graph
The slope m of the graph of f at the point is equal to the slope of its tangent line at and is given by

15. NOTE:
Difference in setting up difference quotients between a specific point
and a general formula

16. Examples Find the slope of the graph of at the point (-3, 8)

17. Equation of the Tangent Line at the Point

18. What is the slope? The derivative! The definition of the derivative:
The derivative of f at x is given by
provided that this limit exists.
A function is differentiable at x if its derivative exists at x.
The process of finding the derivatives is called
differentiation.

19. Notation

20. Examples
1. Find the derivative of the function:

21. Examples
2. Find the derivative of the following functions: a) b)

22. Differentiability Function is not differentiable at the point
Vertical Tangent

23. Differentiability
Function is not differentiable at the point 2. Discontinuity

24. Differentiability
Function is not differentiable at the point 3. Cusp

25. Differentiability
Function is not differentiable at the point 4. Node

26. Differentiability & Continuity
If a function f is differentiable at x = c,
then f is continuous at x = c.
This does not work the other way!
If f is continuous at x = c, then a function f
is not necessarily differentiable at x = c.
*** “A B C D” rule