Question

find the average rate of change of $f(x) = x^2 - 5x + 2$ from $x = 2$ to $x = 5$. simplify your answer as much as possible.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{f(b)-f(a)}{b - a} \). Here, \( a = 2 \), \( b = 5 \), and \( f(x)=x^{2}-5x + 2 \).

Step2: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x) \):
\( f(2)=(2)^{2}-5(2)+2=4 - 10 + 2=-4 \)

Step3: Calculate \( f(5) \)

Substitute \( x = 5 \) into \( f(x) \):
\( f(5)=(5)^{2}-5(5)+2=25 - 25 + 2=2 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(5)-f(2)}{5 - 2} \), substitute the values of \( f(5) \) and \( f(2) \):
\( \frac{2-(-4)}{5 - 2}=\frac{2 + 4}{3}=\frac{6}{3}=2 \)

Answer:

2