Let's address the two parts of the question step by step.

(a) Finding the Average Rate of Change of $g(x)$ over the interval $[2, 5]$:

The formula for the average rate of change of a function $g(x)$ over the interval $[a, b]$ is:

$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$

In this case:

• $a = 2$
• $b = 5$
• From the table, $g(2) = 11.2$ and $g(5) = 16.1$

Now, calculate the average rate of change:

$\text{Average rate of change} = \frac{g(5) - g(2)}{5 - 2} = \frac{16.1 - 11.2}{5 - 2} = \frac{4.9}{3} = 1.633$

Thus, the average rate of change of $g(x)$ over $[2, 5]$ is approximately:

$\boxed{1.633}$

(b) Approximating $g'(4)$:

To approximate the derivative $g'(4)$, we can use a symmetric difference quotient based on values around $x = 4$. The symmetric difference quotient is given by:

$g'(4) \approx \frac{g(x_2) - g(x_1)}{x_2 - x_1}$

We will choose values of $x_1$ and $x_2$ around 4. From the table, the closest values are:

• $x_1 = 3.1$, $g(3.1) = 7.7$
• $x_2 = 4.3$, $g(4.3) = 12.6$

Now, we can calculate:

$g'(4) \approx \frac{g(4.3) - g(3.1)}{4.3 - 3.1} = \frac{12.6 - 7.7}{4.3 - 3.1} = \frac{4.9}{1.2} = 4.083$

Thus, $g'(4)$ is approximately:

$\boxed{4.083}$

Summary:

• (a) The average rate of change of $g(x)$ over $[2, 5]$ is approximately $1.633$.
• (b) The approximation of $g'(4)$ is approximately $4.083$.

Let me know if you need more details on any step or further clarification!

Related Questions:

1. What is the geometric interpretation of the average rate of change?
2. How would the result change if we used different points to approximate $g'(4)$?
3. What is the relationship between the average rate of change and the derivative?
4. How does the approximation of $g'(4)$ improve with smaller intervals?
5. Can we use a similar approach to approximate higher-order derivatives?

Tip:

For a better approximation of derivatives, using points very close to the target value gives more accurate results when using difference quotients.