1. 3.3: Rates of change Objectives: To find the average rate of change over an interval To find the instantaneous rate of change!!

2. For f(x)=x 2 +4x-5, find Warm Up

3. (this is the slope of a line drawn between 2 points on the graph of the function) AVERAGE RATE OF CHANGE OF f(x) WITH RESPECT TO x FOR A FUNCTION f AS x CHANGES FROM a TO b:

4. http://www.coolmath.com/graphit/ Find the average rate of change of f(x)=2x 3 -x over the interval [1,3].

5. What if we wanted to know the exact speed of a car at an instant? Assume the car’s position is given by s(t)=3t 2 for 0< t < 15 What is the car’s speed at EXACTLY 5 seconds?? Take shorter and shorter intervals near t=5, and find avg rate of change over the intervals. This should zoom in (Get it??? Anyone?? Anyone??) the instantaneous rate of change!! t=5 to t=5.1: t=5 to t = 5.01: t=5 to t=5.001: INSTANTANEOUS RATE OF CHANGE (YEAH CALCULUS!!!!!!)

6.  Let’s call this quantity we add to 5 “h” So we are going to find the rate of change from t=5 to t=(5+h): We took smaller and smaller intervals each time. We added a smaller and smaller quantity to 5 each time.

7.  We added smaller and smaller values of h so we have: Bring back the limit!!!!

8.  Let a be a specific x value  Let h be a small number that represents the distance between the 2 values of x PROVIDED THE LIMIT EXISTS!!!!! http://www.ima.umn.edu/~arnold/calculus/secants/seca nts1/secants-g.html DEFINITION: INSTANTANEOUS RATE OF CHANGE

9.  Velocity is the same as instantaneous rate of change of a function that gives the position with respect to time  Velocity has direction, it can be positive or negative  Speed = | velocity | A few notes…..

10. If the position function is s(t)=t 2 +3t-4, find the instantaneous velocity at t=1, 3, and 5.

11. a.) Find the average rate of change from 3 to 5 seconds. b.) Find the instantaneous velocity at any time, t c.) What is the instantaneous velocity at t=7? The distance in feet of an object is given by h(t)=2t 2 -3t+2

12.  Can use if you are given a specific x value Instantaneous rate of change for a function when x=a can be written as: PROVIDED THE LIMIT EXISTS (b is the second x value getting closer and closer to a) ALTERNATE FORM

13.  Find the instantaneous rate of change for s(t)=-4t 2 -6 at t=2. Using Alternate Form

14.  Marginal Cost:  Instantaneous rate of change of the cost function  The rate that the cost is changing when producing one additional item Example: The cost in dollars to manufacture x cases of the DVD “Calculus is my Life” is given by C(x)=100+15x-x 2, 0< x < 7. Find the marginal cost with respect to the number of cases produced when only 2 cases have been manufactured. Business Application