Question

what is the average rate of change of the function ( b(x)=5 x^{2}+3 ) over the interval ( 2 leq x leq 6 )?
10
20
40
80

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, \(a = 2\), \(b=6\), and \(f(x)=5x^{2}+3\).

Step2: Calculate \(f(6)\) and \(f(2)\)

For \(x = 6\), \(f(6)=5\times6^{2}+3=5\times36 + 3=180 + 3=183\).
For \(x = 2\), \(f(2)=5\times2^{2}+3=5\times4+3=20 + 3=23\).

Step3: Substitute into the formula

\(\frac{f(6)-f(2)}{6 - 2}=\frac{183-23}{4}=\frac{160}{4}=40\).

Answer:

40