1. Derivative at a point

3. Average Rate of Change of A Continuous Function on a Closed Interval

4. Exercise 1 Let f(x)=x 2 1. Calculate the average rate of change of the function y=f(x) on each of the intervals below. Interpret the answers geometrically. a. [2,2.05] b. [1.93, 2]

5. Exercise 1 Let f(x)=x 2 2. Write a mathematical expression to represent the average rate of change of the function y=f(x) on each of the intervals below. a. [2, b] b. [2,2+h], h>0 c. [c,2] d. [2+h,2], h<0

6. In this example the equations of the right and left tangent line to f(x)=x 2 at a=2 will be calculated algebraically.

7. Average Rate of Change VS Instantaneous Rate of Change

8. Right Average Rate of Change This expression represents the slope of the secant line through the points (2,f(2)), and (2+h, f(2+h)).

9. Instantaneous Rate of Change (Equivalent to Right Velocity)

10. Geometrically f + '(2) represents the slope of the right tangent line to the graph at x=2.

11. Left Average Rate of Change This expression represents the slope of the secant line through the points (2,f(2)), and (2+h, f(2+h)).

12. Instantaneous Rate of Change (right velocity)

13. Geometrically f _ '(2) represents the slope of the left tangent line to the graph at x=2.

14. Derivative of f(x)=x 2 at x=2 is a limit of the form

15. Derivative of f(x)=x 2 at x=2 instantaneous rate of change of the function f(x)=x 2 at x=2 is the slope of the tangent line to the graph at the point where x=2.

16. SIDED DERIVATIVES OF A CONTINUOUS FUNCTION AT A POINT For f + '(a) to exist there must be points in the domain to the right of x=a. For f _ '(a) to exist there must be points in the domain to the left x=a.

17. Derivative At A Point Inside Open Interval When f + '(a) =f _ '(a) this value is called THE DERIVATIVE of the function f(x) at x=a, and it is denoted, Click hereClick here to see once again the relation between secant lines through a point and the tangent line at that point or

18. Derivative at x=1?

19. Conditions to Check For Derivative For the derivative of a function y=f(x) to exist at a point x=a the following two conditions are required: I. The function has to be defined at x=a II. exists. Graphically it means that the right and left tangent lines have the same slope.

20. EXERCISE 3 Consider the graphs of the functions below to answer the questions that follow. a.For each function identify the discontinuity point, and indicate the type of discontinuity. b. Indicate which one of the conditions for the existence of the derivative at a point is not satisfied at the discontinuity point.

21. Discussion Discuss the following claims: 1. If a function is discontinuous at x=a, the function does not have derivative at the point where x=a. 2. If a function has derivative at the point x=a, the function is continuous at the point where x=a. 3. If a function is continuous at a point x=a, then the derivative of the function exists at that point

22. SO FAR WE LEARNED THAT TO HAVE ANY HOPES FOR THE DERIVATIVE OF A FUNCTION TO EXIST AT A POINT, ONE HAS TO GUARANTEE THAT THE FUNCTION IS CONTINUOUS AT THAT POINT

23. EXPLORATION The given functions are all continuous at the given point. Justify why the derivative of the function does not exist at that point. Based your arguments in left and right tangent lines.

24. EXERCISE 5

25. One more

26. Practice Make sure you finish all the problems in Workbook #9. Practice, practice, practice