1. Sec 2.8: THE DERIVATIVE AS A FUNCTION
replace a by x

2. Sec 2.8: THE DERIVATIVE AS A FUNCTION
Slopes :

3. Sec 2.8: THE DERIVATIVE AS A FUNCTION

4. Sec 2.8: THE DERIVATIVE AS A FUNCTION

5. Sec 2.8: THE DERIVATIVE AS A FUNCTION

6. Sec 2.8: THE DERIVATIVE AS A FUNCTION

7. Sec 2.8: THE DERIVATIVE AS A FUNCTION

8. Sec 2.8: THE DERIVATIVE AS A FUNCTION

9. Sec 2.8: THE DERIVATIVE AS A FUNCTION
Definition:
A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval.

10. Sec 2.8: THE DERIVATIVE AS A FUNCTION
Continuity VS Differentiability
Continuity: Measure if the function continues
Differentiability: Measure if the function smooth
Example:

11. Sec 2.8: THE DERIVATIVE AS A FUNCTION
continuity
2 properties
differentiability
Proof:
Remark:
f cont. at a
f diff. at a
Remark:
f discont. at a
f not diff. at a
Remark:
f not diff. at a
f discont. at a

12. Sec 2.8: THE DERIVATIVE AS A FUNCTION
f cont. at a
f diff. at a
f discont. at a
f not diff. at a
f not diff. at a
f discont. at a
Example:

13. HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?
Sec 2.8: THE DERIVATIVE AS A FUNCTION
HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?

14. Higher Derivative Sec 2.8: THE DERIVATIVE AS A FUNCTION Note: velocity
acceleration
jerk

15. R

16. H R

18. H

19. H H

20. R R H

21. H H R H