To analyze how the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 20\)[/tex] compares to the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex], let's break down the information and steps involved:
1. Understanding Average Rate of Change:
- The average rate of change of a function [tex]\(f(x)\)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by the formula:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
- This formula essentially calculates the slope of the secant line that connects the points [tex]\((a, f(a))\)[/tex] and [tex]\((b, f(b))\)[/tex] on the graph of the function.
2. Given Information:
- The average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex] is [tex]\(-4.667\)[/tex]. This means that over this interval, the function is decreasing at an average rate of [tex]\(-4.667\)[/tex] units per unit increase in [tex]\(x\)[/tex].
3. Comparing the Average Rate of Change:
- We need to determine how the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 20\)[/tex] compares to the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex].
- The result shows that the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 20\)[/tex] is less than the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex]. In other words, it indicates that the rate of change is becoming more negative or decreasing as we extend the interval to [tex]\(x = 20\)[/tex].
4. Interpretation:
- If the average rate of change is decreasing, it implies the function's slope is getting steeper in the negative direction over the larger interval. This suggests that the function is dropping more steeply or rapidly as we move from [tex]\(x = 15\)[/tex] to [tex]\(x = 20\)[/tex].
- The statement "The average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 20\)[/tex] is decreasing the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex]" confirms this interpretation.
To summarize, the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 20\)[/tex] indicates a steeper decline compared to the average rate of change from [tex]\(x = 0\)[/tex] to [tex]\(x = 15\)[/tex]. This means that as we extend the interval, the function is decreasing at a faster rate.
