Question

find the average rate of change of ( f(x)=x^{3}-4x + 5 ) over the following intervals.
(a) from -5 to -2
(b) from -1 to 4
(c) from 4 to 6
(a) the average rate of change from -5 to -2 is 35.
(b) the average rate of change from -1 to 4 is 9.
(c) the average rate of change from 4 to 6 is \\( \square \\).

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here \(a = 4\), \(b=6\), and \(f(x)=x^{3}-4x + 5\).

Step2: Calculate \(f(4)\) and \(f(6)\)

• For \(x = 4\):

\(f(4)=4^{3}-4\times4 + 5=64-16 + 5=53\)

• For \(x = 6\):

\(f(6)=6^{3}-4\times6+5=216-24 + 5=197\)

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(6)-f(4)}{6 - 4}=\frac{197 - 53}{2}\)
First, calculate the numerator: \(197-53 = 144\). Then divide by \(2\): \(\frac{144}{2}=72\)

Answer:

\(72\)