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Rates of Change
Objectives: Students will be able to
- Calculate the average rate of change over an interval.
- Calculate the instantaneous rate of change at a given point.
We mentioned in the section on derivates that one main focus is rates of change. We will be looking at the average rate of change of a function over an interval and also the instantaneous rate of change of a function a specific point.
In the case of the average rate of change, we will be finding the slope of the secant line through the end points of the interval.
In the case of the instantaneous rate of change, we will once again be finding the derivative of the function at a point using the Newton’s Quotient and limits process.
Example 1: Find the average rate of change between x = 1 and x = 3 for the function f ( x )= x^2 + 2 x
Example 2: Find the instantaneous rate of change at x = 1 for the function f ( x )= x^2 + 2 x
Example 3: Find the average rate of change between x = –2 and x = 1 for the function f ( x )= − 3 x^3 + 2 x^2 − 4 x + 1
Example 4: Find the instantaneous rate of change at x = –2 for the function f ( x )= − 3 x^3 + 2 x^2 − 4 x + 1
Example 5: Find the instantaneous rate of change at x = 2 for the function f ( x )= x ln( x )
Example 6: The graph to the right shows the money remaining in the Medicare Trust Fund at the end of the calendar year, adjusted for inflation in the year 2000 dollars.
a. Find the approximate average rate of change in the trust fund from 1994 (high point) to 1998 (low point). b. Find the approximate average rate of change in the trust fund from 1998 to 2010. c. Find the approximate average rate of change in the trust fund from 1990 to 1998.
Example 7: The revenue (in thousands of dollars) from producing x units of an item is R ( x )= 10 x − 0. 002 x^2 a. Find the average rate of change of revenue when production is increased from 1000 to 1001 units. b. Find the instantaneous rate of change of revenue with respect to the number of items produced when 1000 units are produced. (marginal revenue)
Example 8: Suppose customers in a hardware store are willing to buy N(p) boxes of nails at p dollars per box, as given by N ( p )= 80 − 5 p^2 , 1 ≤ p ≤ 4 a. Find the average rate of change of demand for a change in price from $2 to $3. b. Find the instantaneous rate of change of demand when the price is $2. c. Find the instantaneous rate of change of demand when the price is $3. d. As the price is increased from $2 to $3, how is demand changing? Is this change to be expected?
