Question

what is the average rate of change for the function ( f(x) = -2x^{3}+x - 1 ) over the interval from ( x = 1 ) to ( x = 3 )? -27 -25 25 27

Explanation:

Step1: Recall the average rate of change formula

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 \(a = 1\), \(b=3\).

Step2: Calculate \(f(1)\)

Substitute \(x = 1\) into \(f(x)=-2x^{3}+x - 1\).
\(f(1)=-2(1)^{3}+1 - 1=-2\).

Step3: Calculate \(f(3)\)

Substitute \(x = 3\) into \(f(x)=-2x^{3}+x - 1\).
\(f(3)=-2(3)^{3}+3 - 1=-2\times27 + 2=-54 + 2=-52\).

Step4: Compute the average rate of change

Using the formula \(\frac{f(3)-f(1)}{3 - 1}\), substitute \(f(1)=-2\) and \(f(3)=-52\).
\(\frac{-52-(-2)}{3 - 1}=\frac{-52 + 2}{2}=\frac{-50}{2}，=-25，\).

Answer:

，，-，25，