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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

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.

distance traveled by a falling object
Distance Traveled by a Falling Object

Where d(t) is the distance traveled (in feet) and t is the time in seconds.

example 1 average speed over a given interval
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
example 2 rates of change of volume
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.
example 3 manufacturing costs
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
example 3 manufacturing costs1
Example #3Manufacturing Costs
  • Find the average rate of change of the cost:
  • From 10 to 25 cars
  • From 25 to 50 cars
example 4 rates of change from a graph
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

geometric interpretation of average rate of change
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.

example 5 computing average speed using a formula
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.
example 6 using a rate of change formula
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.