2.2 Basic Differentiation Rules and Rates of Change (Part 2)

1. 2.2 Basic Differentiation Rules and Rates of Change (Part 2) Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Colorado National Monument

2. Objectives • Use derivatives to find rates of change.

3. Rate of Change Derivative: rate of change of one variable with respect to another

4. B distance (miles) A time (hours) (The velocity at one moment in time.) Consider a graph of displacement (distance traveled) vs. time. Average velocity can be found by taking: The speedometer in your car does not measure average velocity, but instantaneous velocity.

5. Velocity is the first derivative of position.

6. Average and Instantaneous Velocity If you're given a position function at time t, f(t), you can calculate the average and instantaneous velocities.

7. Example If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function Find the average velocity on [1,2]. Find the average velocity on [1, 1.1].

8. Example (continued) Average velocity on [1,2] = - 48 ft/sec Average velocity on [1, 1.1] = - 33.5 ft/sec Why is the velocity negative? Find the instantaneous velocity at 2 seconds:

9. Position Function The position function of a free falling object under the influence of gravity (ignoring air resistance) is initial height initial velocity

10. Example At time t=0, a diver jumps from a diving board that is 32 feet above the water. The position of the diver is given by When does the diver hit the water?

11. Example (continued) At time t=0, a diver jumps from a diving board that is 32 feet above the water. The position of the diver is given by (Part a: the diver hits the water after 2 seconds.) What is the diver's velocity at impact?

12. Homework 2.2 (page 116) #65, 72, 73, 87, 93, 95, 97-99 all