Question

find the average rate of change of ( k(x)=-2x^{3} ) over the interval ( -1,2 ).
write your answer as an integer, fraction, or decimal rounded to the nearest tenth.
simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The formula for 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, \(a=-1\), \(b = 2\), and \(f(x)=-2x^{3}\).

Step2: Calculate \(f(-1)\) and \(f(2)\)

• For \(x=-1\): \(f(-1)=-2\times(-1)^{3}=-2\times(-1) = 2\).
• For \(x = 2\): \(f(2)=-2\times(2)^{3}=-2\times8=-16\).

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

\(\frac{f(2)-f(-1)}{2-(-1)}=\frac{-16 - 2}{2 + 1}=\frac{-18}{3}\).

Answer:

\(-6\)