https://file.aiquickdraw.com/mathbot/file/1725585730716pbycc1rl.jpg

The image you uploaded contains two main tasks:

1. **The point \( P(x, y) = \left( \frac{1}{3}, \ln\left( \frac{1}{3} \right) \right) \)** is to be considered, and you need to use linear approximation to estimate the value of the natural logarithm \( \ln(x) \) at a point close to \( \frac{1}{3} \), which is most likely \( x = 0.4 \).
   
2. **Using the slope table**: You are also asked to find the slope \( m \) of the tangent lines to the graph of \( f(x) = \ln(x) \) near \( x = \frac{1}{3} \) and use that information for linear approximation.

To tackle these tasks:

### Task 1: Use linear approximation formula.

The linear approximation (or tangent line approximation) formula for a function \( f(x) \) at \( x = a \) is:

\[
L(x) = f(a) + f'(a)(x - a)
\]

For \( f(x) = \ln(x) \), we know that \( f'(x) = \frac{1}{x} \). Therefore, you need to calculate \( f\left( \frac{1}{3} \right) = \ln\left( \frac{1}{3} \right) \) and \( f'\left( \frac{1}{3} \right) = \frac{1}{\frac{1}{3}} = 3 \).

Then, using the linear approximation formula to estimate \( \ln(0.4) \), we substitute into the equation.

### Task 2: Slope table.

From the slope table in the image, you can use the slope values for \( x \) close to \( \frac{1}{3} \) and compare them to the theoretical slope of 3, which was calculated earlier.

Would you like me to go through the detailed calculation and provide an estimated value for \( \ln(0.4) \) using this method?

Here are 5 related questions to deepen your understanding:
1. How is the derivative of \( \ln(x) \) generally derived?
2. What is the geometric meaning of linear approximation?
3. Why is it important to choose a point \( x = a \) close to the point you want to approximate?
4. Can you apply linear approximation to other functions, such as trigonometric ones?
5. How does the slope of a tangent line affect the accuracy of an approximation?

**Tip**: Linear approximations are particularly useful for estimating values of functions near points where you know the exact function and its derivative.

The point P(x, y) = (1/3, ln(1/3)) is considered. You are required to use linear approximation to estimate the value of ln(x) at a point near x = 1/3. Additionally, find the slope m of the graph of ln(x) using the slope table provided in the image.

This problem involves using linear approximation to estimate ln(x) at a point near x = 1/3, based on the tangent slope of the function. You will learn how to apply the linear approximation formula and use derivative analysis to find the slopes of the graph. This is ideal for high school or college-level math students.

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