Question

if ( f(x)=2x^{3}-x + 4 ), what is the average rate of change over (0,2)?
a. 6
b. 10
c. 7
d. 8

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 given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 0\), \(b=2\), and \(f(x)=2x^{3}-x + 4\).

Step2: Calculate \(f(0)\) and \(f(2)\)

• For \(x = 0\):

Substitute \(x = 0\) into \(f(x)\): \(f(0)=2(0)^{3}-0 + 4=4\).

• For \(x = 2\):

Substitute \(x = 2\) into \(f(x)\): \(f(2)=2(2)^{3}-2 + 4\).
First, calculate \(2(2)^{3}=2\times8 = 16\). Then \(f(2)=16-2 + 4=18\).

Step3: Apply the average - rate - of - change formula

\(\frac{f(2)-f(0)}{2-0}=\frac{18 - 4}{2}\).
Since \(18−4 = 14\), then \(\frac{14}{2}=7\).

Answer:

c. 7