Question

find the average rate of change of the function ( f(x)=x^{2}+5x ) from ( x_{1}=7 ) to ( x_{2}=9 ).
the average rate of change is ( square ). (simplify your answer.)

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( x_1 \) to \( x_2 \) is given by \( \frac{f(x_2)-f(x_1)}{x_2 - x_1} \).

Step2: Calculate \( f(x_1) \) when \( x_1 = 7 \)

Substitute \( x = 7 \) into \( f(x)=x^2 + 5x \):
\( f(7)=7^2+5\times7=49 + 35=84 \)

Step3: Calculate \( f(x_2) \) when \( x_2 = 9 \)

Substitute \( x = 9 \) into \( f(x)=x^2 + 5x \):
\( f(9)=9^2+5\times9=81 + 45=126 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(x_2)-f(x_1)}{x_2 - x_1} \), substitute \( f(x_2)=126 \), \( f(x_1)=84 \), \( x_2 = 9 \) and \( x_1 = 7 \):
\( \frac{126 - 84}{9 - 7}=\frac{42}{2}=21 \)

Answer:

21