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1. 2.2 Position Velocity and AccelerationObjective: Use derivatives to find rates of change Ms. Battaglia AB Calculus

2. Rates of Change • Derivatives can be used to determine the rate of change of one variable with respect to another. • Examples: population growth rates, production rates, water flow rates, velocity, and acceleration.

3. Rates of Change The function s that gives the position (relative to the origin) of an object as a function of time t is called a position function. the average velocity is

4. Finding Average Velocity of a Falling Object If a billiard ball is dropped from a height of 100 ft, its height s at time t is given by the position function where s is measured in ft and t is measured in sec. Find the average velocity over each of the following time intervals. a. [1,2] b. [1,1.5] c. [1,1.1]

5. Suppose you wanted to find the instantaneous velocity of the object at t=1. • The velocity of the object at time t is • The speed of an object is the absolute value of its velocity (can’t be negative). • The position of a free-falling object under the influence of gravity is where is the initial height of the object, is the initial velocity of the object, and g is the acceleration due to gravity.

6. Using the Derivative to Find Velocity At a time t=0, a diver jumps from a platform diving board that is 32 ft above the water. The position of the diver is given by where s is measured in ft and t is measured in sec. • When does the diver hit the water? • What is the diver’s velocity at impact?

7. Example • Find the average rate of change of the function over the given interval. Compare this average rate of change with instantaneous rates of change at the endpoints of the interval. [1,2]

8. Example • Find the average rate of change of the function over the given interval. Compare this average rate of change with instantaneous rates of change at the endpoints of the interval. [3,3.1]

9. Example • Find the average rate of change of the function over the given interval. Compare this average rate of change with instantaneous rates of change at the endpoints of the interval.

10. Classwork/Homework • Page 97-100, worksheet