1. Review Topic A toy rocket is launched and has the trajectory that can be modeled by s(t) = 5 + 90t – 16t 2 a)What is the height at 2 seconds? b)What is the initial height? c)When does the rocket reach it’s max height? d)What is the max height? e)When does the rocket hit the ground?

2. AP Calc Unit 2 Day 5 Position -> Velocity -> Acceleration

3. 1. 2. More Multiple Choice Practice

4. 1. 2.

5. MORE Practice Identify the limit statement

6. AND MORE Practice

7. GOALS Define and Calculate: Average Velocity, Instantaneous Velocity, Average Acceleration, Instantaneous Acceleration Specify the relationship between the above values

8. AVERAGE Velocity: change in position over a given interval slope between two points on the position curve Given s(t) is the position function at time t on Average Velocity =

9. INSTANTANEOUS Velocity: derivative of position at specific time value Given s(t) is the position function at time t on Instantaneous Velocity =

10. AVERAGE Acceleration: change in velocity over a given interval slope between two points on the velocity curve Given v(t) is the velocity function at time t on Average Acceleration =

11. Given v(t) is the velocity function at time t on Instantaneous Acceleration = INSTANTANEOUS Acceleration: derivative of velocity at specific time value second derivative of position

12. Position Velocity Acceleration PVAPVA

13. Example The position, in feet, of a ball at some time t is given by the function: The velocity of the ball at t=1 can be found using the following:

14. Example--Continued Find the acceleration function using the power rule. Then find the position, velocity and acceleration at

15. TimePositionVelocityAcceleration 0 1 2 TimePositionVelocityAcceleration 0100 ft20 ft/sec-32 ft/sec 2 1104 ft-12 ft/sec-32 ft/sec 2 276 ft-44 ft/sec-32 ft/sec 2

16. When will the ball hit the ground? What is its velocity when it hits the ground? 3.202 seconds -82.4 ft/sec

17. Vertical position Gravity constants FYI ONLY

18. Calculator Feature nDERIV Tool for finding the derivative of a function at a given x-value.

19. Using nDeriv The height of the pebble above the ground any time t is represented by. Find the velocity at t=4. TI 84 Calculators: ALPHA WINDOW 3: nDeriv( PUT the function in y 1 TI 83 Calculators: ALPHA WINDOW 3: nDeriv( nDeriv( y 1,x,4)

20. #3 On HW tonight A pebble is dropped from a height of 600 feet. The height of the pebble above the ground any time t is represented by. Use your graphing calculator to determine at what time the pebble will hit the ground. Use nDeriv to find the velocity of the pebble when it hits the ground.