Question

for ( f(x)=4 x^{2}-5 x + 1 ), what is the average rate of change over (0,3)?

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, \(a = 0\), \(b=3\), and \(f(x)=4x^{2}-5x + 1\).

Step2: Calculate \(f(0)\) and \(f(3)\)

• For \(x = 0\):

Substitute \(x = 0\) into \(f(x)\): \(f(0)=4\times0^{2}-5\times0 + 1=1\).

• For \(x = 3\):

Substitute \(x = 3\) into \(f(x)\): \(f(3)=4\times3^{2}-5\times3 + 1=4\times9-15 + 1=36-15 + 1=22\).

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

Substitute \(f(0)=1\), \(f(3)=22\), \(a = 0\), and \(b = 3\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(3)-f(0)}{3-0}=\frac{22 - 1}{3}=\frac{21}{3}=7\).

Answer:

d. 7