Consider the function g ( x )

We are given the function $g(x) = 13x^2 + 6x - 5$. To find the average rate of change of this function over an interval, we use the formula:

$\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}$ where $a$ and $b$ are the endpoints of the interval.

A) Average rate of change from 14 to 20:

Here, $a = 14$ and $b = 20$.

1. First, calculate $g(14)$: $g(14) = 13(14)^2 + 6(14) - 5 = 13(196) + 84 - 5 = 2548 + 84 - 5 = 2627$

2. Next, calculate $g(20)$: $g(20) = 13(20)^2 + 6(20) - 5 = 13(400) + 120 - 5 = 5200 + 120 - 5 = 5315$

3. Now, apply the formula for the average rate of change: $\frac{g(20) - g(14)}{20 - 14} = \frac{5315 - 2627}{6} = \frac{2688}{6} = 448$

Thus, the average rate of change from 14 to 20 is 448.

B) Average rate of change from 16 to 20:

Here, $a = 16$ and $b = 20$.

1. First, calculate $g(16)$: $g(16) = 13(16)^2 + 6(16) - 5 = 13(256) + 96 - 5 = 3328 + 96 - 5 = 3419$

2. Use the previously calculated $g(20) = 5315$.

3. Now, apply the formula for the average rate of change: $\frac{g(20) - g(16)}{20 - 16} = \frac{5315 - 3419}{4} = \frac{1896}{4} = 474$

Thus, the average rate of change from 16 to 20 is 474.

Do you want more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you interpret the significance of average rate of change in real-world applications?
2. What would the instantaneous rate of change of $g(x)$ be at $x = 14$?
3. How does the average rate of change differ from the derivative of a function?
4. How would the average rate of change change if the interval was from 10 to 14 instead?
5. How do you calculate the rate of change for non-polynomial functions?

Tip: The average rate of change is closely related to the slope of the secant line between two points on a curve.