Question

find the average rate of change of f(x)=x² + 3x + 2 from x=4 to x=7. 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, \( a = 4 \), \( b = 7 \), and \( f(x)=x^{2}+3x + 2 \).

Step2: Calculate \( f(7) \)

Substitute \( x = 7 \) into \( f(x) \):
\( f(7)=7^{2}+3\times7 + 2=49 + 21+2=72 \)

Step3: Calculate \( f(4) \)

Substitute \( x = 4 \) into \( f(x) \):
\( f(4)=4^{2}+3\times4 + 2=16+12 + 2=30 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(7)-f(4)}{7 - 4} \), substitute the values of \( f(7) \) and \( f(4) \):
\( \frac{72-30}{7 - 4}=\frac{42}{3}=14 \)

Answer:

14