How do you find the average rate of change for the function #y= 4x²# on the indicated intervals [1,5]?

Question

Cameron Alexander · Accepted Answer

The average rate of change is #(Deltay)/(Deltax) = 24#

The average rate of change of function #f# on interval #[a,b]# is:

#(Deltaf)/(Deltax) = (f(b)-f(a))/(b-a)#

In this case, we have #a=1#, and #b = 5#,

and #f(x) = 4x^2#.

Thus, we obtain:

#f(5) = 4(5)^2 = 100# and #f(1) = 4#, thus

#(Deltaf)/(Deltax) = (f(5)-f(1))/(5-1) = (100-4)/4 = 96/4 = 24#

Notice that the average rate of change is the same as the slope of the line connecting the points: #(1, f(1))# and #(5, f(5))#.

Christopher Agresta · Answer

undefined To find the average rate of change of a function over an interval, you use the formula:

$$
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
$$

where $ f(a) $ and $ f(b) $ are the values of the function at the endpoints of the interval, and $ a $ and $ b $ are the corresponding x-values.

For the function $ y = 4x^2 $ on the interval [1, 5], the average rate of change is:

$$
\frac{f(5) - f(1)}{5 - 1} = \frac{4(5)^2 - 4(1)^2}{5 - 1} = \frac{4(25) - 4(1)}{4} = \frac{100 - 4}{4} = \frac{96}{4} = 24
$$

So, the average rate of change for the function $ y = 4x^2 $ on the interval [1, 5] is 24.