Find the average rate of change of the function over the given intervals.
​f(x)equals7 x cubed plus 7​;     a​) ​[6​,8​],   b​) ​[negative 4​,4​]
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Part 1
a​)  The average rate of change of the function ​f(x)equals7 x cubed plus 7 over the interval ​[6​,8​] is 
  
enter your response here.
​(Simplify your​ answer.)

To find the average rate of change of the function \( f(x) = 7x^3 + 7 \) over an interval \([a, b]\), we use the formula for the average rate of change:

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

### Part a) Interval \([6, 8]\):

1. **Step 1: Evaluate \( f(6) \) and \( f(8) \)**

\[
f(6) = 7(6^3) + 7 = 7(216) + 7 = 1512 + 7 = 1519
\]
\[
f(8) = 7(8^3) + 7 = 7(512) + 7 = 3584 + 7 = 3591
\]

2. **Step 2: Apply the formula for the average rate of change**

\[
\text{Average rate of change} = \frac{f(8) - f(6)}{8 - 6} = \frac{3591 - 1519}{2} = \frac{2072}{2} = 1036
\]

Thus, the average rate of change over the interval \([6, 8]\) is **1036**.

Would you like the details for part b or more questions on related topics? Here are some you might consider:

1. What is the geometric meaning of the average rate of change of a function?
2. How does the average rate of change relate to the derivative of a function?
3. Can the average rate of change be negative? If so, under what conditions?
4. How does the average rate of change vary for quadratic or linear functions?
5. What is the average rate of change of a constant function?

**Tip:** The average rate of change over an interval is essentially the slope of the secant line between two points on the curve.

Find the average rate of change of the function over the given intervals. f(x) = 7x^3 + 7; a) [6, 8], b) [negative 4, 4]

Learn how to find the average rate of change of the function f(x) = 7x^3 + 7 over the intervals [6, 8] and [-4, 4]. This problem covers key calculus concepts like the average rate of change and secant line slope for polynomial functions. Suitable for students in Grades 10-12.

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