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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.

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3.7 Rates of Change

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  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.

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