Use the given table of values for g(x) to do the following. x 2 2.3 3.1 4 4.3 5 g(x) 16.2 14.1 11.7 14.2 15.4 20.1 (a) Find the average rate of change of g(x) over [2, 5]. (Round your answer to three decimal places.) (b) Approximate g′(4) as accurately as possible.

Let's solve each part step by step.

### Part (a): Average Rate of Change

The average rate of change of a function \( g(x) \) over an interval \([a, b]\) is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\]

For the interval \([2, 5]\):
- \( a = 2 \), \( g(2) = 16.2 \)
- \( b = 5 \), \( g(5) = 20.1 \)

\[
\text{Average Rate of Change} = \frac{g(5) - g(2)}{5 - 2} = \frac{20.1 - 16.2}{3}
\]

\[
\text{Average Rate of Change} = \frac{3.9}{3} = 1.3
\]

So, the average rate of change of \( g(x) \) over \([2, 5]\) is:
\[
\boxed{1.300}
\]

### Part (b): Approximate \( g'(4) \)

To approximate the derivative \( g'(4) \), we can use the symmetric difference quotient:
\[
g'(4) \approx \frac{g(4.3) - g(3.7)}{4.3 - 3.7}
\]

Since we don't have \( g(3.7) \) from the table, we can instead approximate using points close to 4, such as:
1. The points \( x = 4.3 \) and \( x = 4 \)
2. The points \( x = 4 \) and \( x = 3.1 \)

#### Using \( x = 4 \) and \( x = 4.3 \):
- \( g(4) = 14.2 \)
- \( g(4.3) = 15.4 \)

\[
g'(4) \approx \frac{g(4.3) - g(4)}{4.3 - 4} = \frac{15.4 - 14.2}{0.3}
\]

\[
g'(4) \approx \frac{1.2}{0.3} = 4.0
\]

#### Using \( x = 4 \) and \( x = 3.1 \):
- \( g(4) = 14.2 \)
- \( g(3.1) = 11.7 \)

\[
g'(4) \approx \frac{g(4) - g(3.1)}{4 - 3.1} = \frac{14.2 - 11.7}{0.9}
\]

\[
g'(4) \approx \frac{2.5}{0.9} \approx 2.778
\]

#### Average Approximation:
Taking the average of the two approximations:
\[
g'(4) \approx \frac{4.0 + 2.778}{2} \approx 3.389
\]

Thus, the approximation of \( g'(4) \) is approximately:
\[
\boxed{3.389}
\]

### Summary:
- **(a)** The average rate of change over \([2, 5]\) is \( \boxed{1.300} \).
- **(b)** The approximation for \( g'(4) \) is \( \boxed{3.389} \).

Would you like a more detailed explanation or have any questions about the process?

### Further Questions:
1. What does the average rate of change of a function represent in a real-world scenario?
2. How does the choice of points affect the approximation of \( g'(x) \)?
3. Why is using the symmetric difference quotient often more accurate for derivative approximations?
4. Can you find the average rate of change over other intervals, such as \([3.1, 4.3]\)?
5. What does a positive or negative average rate of change indicate about the function?

### Tip:
When approximating derivatives, using points as close as possible to the point of interest gives a more accurate estimate.

Learn how to find the average rate of change of a function g(x) over the interval [2, 5] and how to approximate the derivative g′(4) using a table of values. This detailed step-by-step solution covers key calculus concepts and is suitable for high school students.

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