Question

f(x) = x³ - 9x
what is the average rate of change of f over the interval 1,6?

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, \(a = 1\), \(b=6\).

Step2: Calculate \(f(1)\) and \(f(6)\)

For \(x = 1\), \(f(1)=1^{3}-9\times1=1 - 9=-8\).
For \(x = 6\), \(f(6)=6^{3}-9\times6=216-54 = 162\).

Step3: Substitute into the formula

\(\frac{f(6)-f(1)}{6 - 1}=\frac{162-(-8)}{6 - 1}=\frac{162 + 8}{5}=\frac{170}{5}\)

Answer:

\(34\)