c Roberto Barrera, Fall 2015
Math 142 3.2 Rates of Change
Average rate of change
The average rate of change of y = f (x) with respect to x from a to b is the quotient
Secant line:
Average rate of change and secant lines:
1
2
c Roberto Barrera, Fall 2015
Math 142 Example: Calculate the average rate of change of the given function f (x) over the
interval [3,5]:
x
1 2 3 4 5
f (x) 7 8 10 14 22
Example Calculate the average rate of change of f (x) =
1
x
on the interval [2, 4].
Example Between what consecutive points is the average rate of change positive? negative? zero?
c Roberto Barrera, Fall 2015
Math 142 3
Instantaneous rate of change
Instantaneous rate of change: Given a function y = f (x), the instantaneous rate of
change of y with respect to x at x = c is given by
if the limit exists.
Tangent line:
Slope of tangent lines and instantaneous rate of change:
c Roberto Barrera, Fall 2015
Math 142 4
Example A wholesale Christmas tree farm sells evergreen seedlings to growers and
has revenue R(x) in tens of thousands of dollars is given by R(x) = −x2 + 20x, where
x is the number of millions of seedlings sold. Find the instantaneous rate of change of
revenue with respect to number of seedlings sold when 5 million seedlings are sold.
Example Find the slope of the line tangent to the curve in the above example at x = 5.
Example At which points in the graph is the instantaneous rate of change positive?
negative? zero?