Question

what is the average rate of change for $f(x)=x^{2}+1$ over $0,2$?
a. 4
b. 3
c. 1
d. 2

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 \(\frac{f(b)-f(a)}{b - a}\).

Step2: Identify \(a\), \(b\), and find \(f(a)\) and \(f(b)\)

Here, \(a = 0\), \(b=2\), \(f(x)=x^{2}+1\).
For \(x = 0\), \(f(0)=0^{2}+1=1\).
For \(x = 2\), \(f(2)=2^{2}+1=4 + 1=5\).

Step3: Calculate the average rate of change

Substitute into the formula: \(\frac{f(2)-f(0)}{2-0}=\frac{5 - 1}{2}=\frac{4}{2}=2\).

Answer:

d. 2