Question

Find the average rate of change of the function f left parenthesis x right parenthesis equals x squared plus 2 x from x 1 equals 1 to x 2 equals 4.
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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 \( f(x) = x^2 + 2x \) from \( x_1 = 1 \) to \( x_2 = 4 \), we use the formula for the average rate of change:

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

1. First, calculate \( f(x_1) = f(1) \):
\[
f(1) = 1^2 + 2(1) = 1 + 2 = 3
\]

2. Next, calculate \( f(x_2) = f(4) \):
\[
f(4) = 4^2 + 2(4) = 16 + 8 = 24
\]

3. Now, substitute \( f(x_2) \) and \( f(x_1) \) into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1} = \frac{24 - 3}{4 - 1} = \frac{21}{3} = 7
\]

Therefore, the average rate of change of the function from \( x = 1 \) to \( x = 4 \) is \( \boxed{7} \).