1 / 14

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:.

drew
Download Presentation

3.3: Rates of change

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related