1. Higher Order Derivatives

2. Objectives Students will be able to Calculate higher order derivatives Apply higher order derivatives in application problems

3. Symbol Representations First Derivati ve Second Derivati ve

4. Symbol Representations Third Derivati ve Fourth Derivati ve

5. Symbol Representations nth Derivati ve

6. Example 1 Calculate the second derivative of the function

7. Example 2 For the function find

8. Example 3 Calculate the second derivative of the function

9. Example 4 For the function find

10. Example 5 Calculate the second derivative of the function

11. Example 6 For the function find

12. Example 7 Calculate the second derivative of the function

13. Example 8 For the function find

14. Example 9 Calculate the third and fourth derivative of the function

15. Example 10 Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

16. Example 11 Find the open interval(s) where the function is concave up or concave down. Find any points of inflection.

17. Example 12-1 For an original function f(x) being a distance function with respect to time, the first derivative of f(x) is the velocity (instantaneous rate of change of distance) and the second derivative of f(x) is called acceleration (instantaneous rate of change of velocity). In terms of the demand

18. Example 12-2 A car rolls down a hill. Its distance (in feet) from its starting point is given by where t is in seconds. a.How far will the car move in 10 seconds? b.What is the velocity at 5 seconds? At 10 seconds? c.How can you tell from v(t) that the car will not stop?

19. Example 12-3 A car rolls down a hill. Its distance (in feet) from its starting point is given by where t is in seconds. d.What is the acceleration at 5 seconds? At 10 seconds? e.What is happening to the velocity and the acceleration as t increases?