1 / 11

3.7 Rates of Change

3.7 Rates of Change. Objectives: Find the average rate of change of a function over an interval. Represent average rate of change geometrically as the slope of a secant line. Use the difference quotient to find a formula for the average rate of change of a function.

Download Presentation

3.7 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. 3.7 Rates of Change Objectives: Find the average rate of change of a function over an interval. Represent average rate of change geometrically as the slope of a secant line. Use the difference quotient to find a formula for the average rate of change of a function.

  2. Distance Traveled by a Falling Object Where d(t) is the distance traveled (in feet) and t is the time in seconds.

  3. Example #1Average Speed Over a Given Interval • Find the average speed of the falling rock • From t = 2 to t = 5 • From t = 0 to t = 3.5

  4. Average Rate of Change of a Function

  5. Example #2Rates of Change of Volume • A cone-shaped tank is being filled with water. The approximate volume of water in the tank in cubic meters is , where x is the height of water in the tank. • Find the average rate of change of the volume of water as the height increases from 1 to 3 meters.

  6. Example #3Manufacturing Costs • A manufacturing company makes toy cars. The cost (in dollars) of producing x cars is given by the function • Find the average rate of change of the cost: • From 0 to 10 cars

  7. Example #3Manufacturing Costs • Find the average rate of change of the cost: • From 10 to 25 cars • From 25 to 50 cars

  8. Example #4Rates of Change from a Graph The graph left shows the weekly sales (in hundreds of dollars) of magazine subscriptions made during a 12-week sales drive. The sales in any single week is s(x), where x is the number of weeks since the sales drive began. What is the average rate of change in sales: From week 2 to week 4 From week 6 to week 11 Sales (Hundreds of Dollars) Sales decrease $150 per week Weeks Sales increase $160 per week

  9. Geometric Interpretation of Average Rate of Change • Using the previous graph and two points located on the curve we can see the geometric interpretation for the average rate of change. The slope of a secant line connecting two points on the curve represents the average rate of change for the interval from weeks 3 to 8.

  10. Example #5Computing Average Speed Using a Formula • The distance traveled by a dropped object (ignoring wind resistance) is given by the function d(t) = 4.9t2, with distance d(t) measured in meters and time t in seconds. Find a formula for the average speed of a falling object from time x to time x + h. Use the formula to find the average speed from 2.8 to 3 seconds.

  11. Example #6Using a Rate of Change Formula • Find the difference quotient of and use it to find the average rate of change of V as h changes from 2 to 2.1 meters.

More Related