Question

Find the average rate of change of the function f left parenthesis x right parenthesis equals x squared plus 6 x from x 1 equals 5 to x 2 equals 8.
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Part 1
The average rate of change is
  
enter your response here. ​(Simplify your​ answer.)

Answer 1

To find the average rate of change of the function \( f(x) = x^2 + 6x \) from \( x_1 = 5 \) to \( x_2 = 8 \), we use the formula for average rate of change:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

1. First, we calculate \( f(x_1) \) and \( f(x_2) \):

- For \( x_1 = 5 \):
\[
f(5) = 5^2 + 6(5) = 25 + 30 = 55
\]

- For \( x_2 = 8 \):
\[
f(8) = 8^2 + 6(8) = 64 + 48 = 112
\]

2. Now we substitute the values into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(8) - f(5)}{8 - 5} = \frac{112 - 55}{8 - 5} = \frac{57}{3} = 19
\]

Thus, the average rate of change of the function \( f(x) \) from \( x = 5 \) to \( x = 8 \) is \( 19 \).