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*)

11.2

10.1

7.7

10.2

12.6

16.1

(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 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.

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): 11.2, 10.1, 7.7, 10.2, 12.6, 16.1

(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.

Learn how to find the average rate of change of g(x) over the interval [2, 5] and approximate g'(4) using a table of values. This problem involves calculating the average rate of change and using the difference quotient to estimate a derivative. Suitable for students in Grades 10-12.

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