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3.3: Rates of change

3.3: Rates of change. Objectives: To find the average rate of change over an interval To find the instantaneous rate of change!!. Warm Up. For f(x)=x 2 +4x-5 , find . AVERAGE RATE OF CHANGE OF f(x) WITH RESPECT TO x FOR A FUNCTION f AS x CHANGES FROM a TO b:.

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3.3: Rates of change

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  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. Warm Up For f(x)=x2+4x-5, find

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

  4. Find the average rate of change of f(x)=2x3-x over the interval [1,3]. http://www.coolmath.com/graphit/

  5. INSTANTANEOUS RATE OF CHANGE(YEAH CALCULUS!!!!!!) 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)=3t2 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:

  6. We took smaller and smaller intervals each time. We added a smaller and smaller quantity to 5 each time. 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):

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

  8. DEFINITION:INSTANTANEOUS RATE OF CHANGE 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/secants1/secants-g.html

  9. A few notes….. 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 |

  10. If the position function is s(t)=t2+3t-4, find the instantaneous velocity at t=1, 3, and 5.

  11. The distance in feet of an object is given by h(t)=2t2-3t+2 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?

  12. ALTERNATE FORM • 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)

  13. Using Alternate Form Find the instantaneous rate of change for s(t)=-4t2-6 at t=2.

  14. Business Application • 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-x2, 0< x < 7. Find the marginal cost with respect to the number of cases produced when only 2 cases have been manufactured.

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