Find the average rate of change for the given function.
​f(x)equalsx squared plus 7 x between xequals0 and xequals6

To find the average rate of change of the function \( f(x) = x^2 + 7x \) between \( x = 0 \) and \( x = 6 \), we use the formula:

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

where:
- \( f(x) = x^2 + 7x \)
- \( a = 0 \)
- \( b = 6 \)

First, calculate \( f(0) \):

\[
f(0) = 0^2 + 7(0) = 0
\]

Next, calculate \( f(6) \):

\[
f(6) = 6^2 + 7(6) = 36 + 42 = 78
\]

Now, use the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(6) - f(0)}{6 - 0} = \frac{78 - 0}{6 - 0} = \frac{78}{6} = 13
\]

So, the average rate of change of the function \( f(x) = x^2 + 7x \) between \( x = 0 \) and \( x = 6 \) is **13**.

Would you like further details or have any questions about this? Here are some related questions you might find helpful:

1. What is the average rate of change of \( f(x) = x^2 + 7x \) between \( x = 2 \) and \( x = 5 \)?
2. How does the average rate of change compare to the instantaneous rate of change?
3. How do you interpret the average rate of change in a real-world context?
4. What does the derivative of \( f(x) = x^2 + 7x \) tell us about the rate of change?
5. Can you find the average rate of change for a function over a non-linear interval?

**Tip:** The average rate of change of a function over an interval gives you the slope of the secant line that passes through the points on the function corresponding to the endpoints of the interval.

Find the average rate of change for the given function f(x) = x^2 + 7x between x = 0 and x = 6.

Learn how to calculate the average rate of change for the function f(x) = x^2 + 7x between x = 0 and x = 6. This solution demonstrates the use of the average rate of change formula and includes a step-by-step breakdown. Suitable for students in Grades 9-11.

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