how to find the average rate of change

Average rate of change over an interval

What is the average rate of change of a function?

When we calculate average rate of change of a function over a given interval, we're calculating the average number of units that the function moves up or down, per unit along the ???x???-axis.

Krista King Math.jpg — Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

We could also say that we're measuring how much change occurs in our function's value per unit on the ???x???-axis.

How do we find the average rate of change? Given the function and the interval we're interested in (???f(x)??? and ???[x_1,x_2]??? respectively), our first step is to calculate the value of our function at both ends of the interval. Then we plug those values and the ends of the interval into our formula to find average rate of change.

The formula for average rate of change is

???\frac{\Delta{f}}{\Delta{x}}=\frac{f(x_2)-f(x_1)}{x_2-x_1}???

over the interval ???[x_1,x_2]???.

How to calculate average rate of change over a particular interval?

Take the course

Want to learn more about Calculus 1? I have a step-by-step course for that. :)

Finding average rate of change of a function on a specific interval

Example

Find the average rate of change over the interval ???[0,4]???.

???f(x)=2x^2-2???

We'll use the formula for average rate of change:

???\frac{\Delta{f}}{\Delta{x}}=\frac{f(x_2)-f(x_1)}{x_2-x_1}???

We already know that ???x_1=0??? and that ???x_2=4???. We'll find ???f(x_1)??? and ???f(x_2)??? by plugging ???0??? and ???4??? into the function we've been given, ???f(x)=2x^2-2???.

???f(0)??? is

???f(0)=2(0)^2-2???

???f(0)=-2???

???f(4)??? is

???f(4)=2(4)^2-2???

???f(4)=2(16)-2???

???f(4)=30???

Average rate of change for Calculus 1.jpg — When we calculate average rate of change of a function over a given interval, we're calculating the average number of units that the function moves up or down, per unit along the x-axis.

Plugging these values into the formula for average rate of change, we get

???\frac{\Delta{f}}{\Delta{x}}=\frac{f(x_2)-f(x_1)}{x_2-x_1}???

???\frac{\Delta{f}}{\Delta{x}}=\frac{f(4)-f(0)}{4-0}???

???\frac{\Delta{f}}{\Delta{x}}=\frac{30-(-2)}{4}???

???\frac{\Delta{f}}{\Delta{x}}=\frac{32}{4}???

???\frac{\Delta{f}}{\Delta{x}}=8???

The average rate of change of the function on ???[0,4]??? is ???8???.

Get access to the complete Calculus 1 course

Learn math

February 24, 2021

math, learn online, online course, online math, calculus 1, calculus i, calc 1, calc i, average rate of change, single variable calc, single variable calculus, rate of change, differentiation, applications of differentiation, applications of derivatives, secant line