Find the average rate of change of k(x) = x^3 + 14 over the interval [-4, - 1].
Write your answer as an integer, fraction, or decimal rounded to the nearest tenth. Simplify any fractions,

The average rate of change of a function \( k(x) \) over the interval \([a, b]\) is given by:

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

Given \( k(x) = x^3 + 14 \), \( a = -4 \), and \( b = -1 \), we first need to calculate \( k(-4) \) and \( k(-1) \):

\[
k(-4) = (-4)^3 + 14 = -64 + 14 = -50
\]

\[
k(-1) = (-1)^3 + 14 = -1 + 14 = 13
\]

Now, compute the average rate of change:

\[
\text{Average Rate of Change} = \frac{k(-1) - k(-4)}{-1 - (-4)} = \frac{13 - (-50)}{-1 + 4} = \frac{13 + 50}{3} = \frac{63}{3} = 21
\]

So, the average rate of change of \( k(x) \) over the interval \([-4, -1]\) is **21**.

Would you like more details or have any questions?  
Here are 5 related questions for further practice:
1. What is the average rate of change of \( k(x) = x^3 + 14 \) over the interval \([0, 2]\)?
2. How would the average rate of change change if the interval was \([-4, 0]\)?
3. What is the geometric interpretation of the average rate of change of a function?
4. Can you find the average rate of change of \( f(x) = 2x^2 + 3x \) over the interval \([1, 4]\)?
5. How does the average rate of change relate to the derivative of the function?

**Tip:** The average rate of change is essentially the slope of the secant line connecting two points on the graph of a function.

Learn how to calculate the average rate of change of the function k(x) = x^3 + 14 over the interval [-4, -1]. Follow a step-by-step solution including the evaluation of k(-4) and k(-1), and the application of the Average Rate of Change formula. Suitable for high school students studying functions and calculus.

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