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Higher order derivatives
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1. Higher order derivatives

2. Objective: To be able to find higher order derivatives and use them to find velocity and acceleration of objects. TS: Explicitly assess information and draw conclusions.

3. Do you remember your different notations for derivatives?

4. Well these are the same notations for higher power derivatives! Any guesses on what each means?

5. And to find them you just take the derivative again...and again…if necessary! For example to get from f’’(x) to f’’’(x) you just take the derivative of f’’(x). And to get from f’(x) to f(4)(x) you would just take the derivative of f’(x) three times.

6. Example A Find the second derivative of f(x) = x4 – 2x3

7. Example B

8. Example C

9. Position, Velocity & Acceleration Velocity is the rate of change of position with respect to time. Acceleration is the rate of change of velocity with respect to time.

10. Position, Velocity & Acceleration Warning: Professional driver, do not attempt! When you’re driving your car…

11. Position, Velocity & Acceleration squeeeeek! …and you jam on the brakes…

12. Position, Velocity & Acceleration …and you feel the car slowing down…

13. Position, Velocity & Acceleration …what you are really feeling…

14. Position, Velocity & Acceleration …is actually acceleration.

15. Position, Velocity & Acceleration I felt that acceleration.

16. Position, Velocity & Acceleration Example D: A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. A) Where is the crab after 2 seconds? B) How fast is it moving at that instant (2 seconds)?

17. Position, Velocity & Acceleration A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. A) Where is the crab after 2 seconds? feet

18. Position, Velocity & Acceleration A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. B) How fast is it moving at that instant (2 seconds)? Velocity is the rate of change of position. Velocity function feet per second

19. Position, Velocity & Acceleration Example E: A disgruntled calculus student hurls his calculus book in the air.

20. Position, Velocity & Acceleration The position of the calculus book: t is in seconds and p(t) is in feet A) What is the maximum height attained by the book? B) At what time does the book hit the ground? C) How fast is the book moving when it hits the ground?

21. Position, Velocity & Acceleration A) What is the maximum height attained by the book? The book attains its maximum height when its velocity is 0. seconds Velocity function feet

22. Position, Velocity & Acceleration B) At what time does the book hit the ground? The book hits the ground when its position is 0. sec. sec.

23. Position, Velocity & Acceleration C) How fast is the book moving when it hits the ground? This is incorrect. Good guess: 0 ft/sec ft/sec Downward direction

24. Position, Velocity & Acceleration Acceleration: the rate of change of velocity with respect to time. Velocity function Acceleration function ft/sec2 How is the acceleration function related to the position function? Acceleration is the second derivative of position.

25. Position, Velocity & Acceleration Example F: A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t2 – 7t. A) When is the car 30 miles from where it started? B) What is the velocity at the very moment the car is 30 miles away? C) What is the acceleration at the very moment the car is 30 miles away? D) When does the car stop?

26. Position, Velocity & Acceleration A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t2 – 7t. A) When is the car 30 miles from where it started? hours

27. Position, Velocity & Acceleration A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t2 – 7t. B) What is the velocity at the very moment the car is 30 miles away? Miles per hour

28. Position, Velocity & Acceleration A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t2 – 7t. C) What is the acceleration at the very moment the car is 30 miles away? Miles per hour2

29. Position, Velocity & Acceleration A red sports car is traveling, and its position P (in miles) at time t (in hours) is given by P (t ) = t2 – 7t. D) When does the car stop? hours

30. Conclusion • The height/distance of an object can be given by a position function. • Velocity measures the rate of change of position with respect to time. • The velocity function is found by taking the derivative of the position function.

31. Conclusion • In order for an object traveling upward to obtain maximum position, its instantaneous velocity must equal 0. • As an object hits the ground, its velocity is not 0, its height is 0. • Acceleration measures the rate of change of velocity with respect to time. • The acceleration function is found by taking the derivative of the velocity function.