COnsider the function f(x) = √x and the point P(4,2). Secant lines Q(x, F(x)) for x-values of 3, 6, and 8. What is the slope of the secant lines? What is the estimate of the tangent line?

To solve the problem, we'll first find the slopes of the secant lines that pass through the point \( P(4, 2) \) and other points on the curve given by \( f(x) = \sqrt{x} \) for \( x \) values of 3, 6, and 8.

### Step 1: Calculate \( f(x) \) at the given points.
Given \( f(x) = \sqrt{x} \), let's calculate the values of \( f(x) \) at \( x = 3 \), \( x = 6 \), and \( x = 8 \):

\[
f(3) = \sqrt{3} \approx 1.732
\]
\[
f(6) = \sqrt{6} \approx 2.449
\]
\[
f(8) = \sqrt{8} \approx 2.828
\]

### Step 2: Calculate the slopes of the secant lines.
The slope of the secant line through the points \( P(4, 2) \) and \( Q(x, f(x)) \) is given by:

\[
\text{slope} = \frac{f(x) - f(4)}{x - 4}
\]

Since \( f(4) = \sqrt{4} = 2 \), we calculate the slopes for the given \( x \) values:

1. **For \( x = 3 \):**
\[
\text{slope} = \frac{f(3) - 2}{3 - 4} = \frac{1.732 - 2}{3 - 4} = \frac{-0.268}{-1} = 0.268
\]

2. **For \( x = 6 \):**
\[
\text{slope} = \frac{f(6) - 2}{6 - 4} = \frac{2.449 - 2}{6 - 4} = \frac{0.449}{2} = 0.2245
\]

3. **For \( x = 8 \):**
\[
\text{slope} = \frac{f(8) - 2}{8 - 4} = \frac{2.828 - 2}{8 - 4} = \frac{0.828}{4} = 0.207
\]

### Step 3: Estimate the slope of the tangent line.
The slope of the tangent line at \( x = 4 \) can be estimated by considering the average of the slopes of the secant lines as \( x \) approaches 4.

Since the secant slopes are 0.268 (at \( x = 3 \)), 0.2245 (at \( x = 6 \)), and 0.207 (at \( x = 8 \)), the tangent slope is approximately the average of these values:

\[
\text{Estimated slope of tangent} \approx \frac{0.268 + 0.2245 + 0.207}{3} \approx 0.2332
\]

This is close to the actual derivative of \( f(x) = \sqrt{x} \) at \( x = 4 \):

\[
f'(x) = \frac{1}{2\sqrt{x}} \quad \text{so} \quad f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4} = 0.25
\]

Thus, the slope of the tangent line at \( x = 4 \) is approximately 0.25.

### Summary
- **Slope of secant lines:**
  - \( x = 3 \): 0.268
  - \( x = 6 \): 0.2245
  - \( x = 8 \): 0.207
- **Estimated slope of the tangent line:** approximately 0.25

Do you want any details or have any questions?

### Related Questions:
1. What is the formal definition of a secant line?
2. How do you calculate the exact derivative of \( f(x) = \sqrt{x} \) using the limit definition?
3. What is the geometric interpretation of the slope of a tangent line?
4. How does the slope of the secant line change as the points get closer to \( x = 4 \)?
5. How can we find the equation of the tangent line at \( x = 4 \)?

### Tip:
Remember, the slope of a secant line is an average rate of change, while the slope of the tangent line represents the instantaneous rate of change at a specific point.

Explore the analysis of secant and tangent lines for the function f(x) = √x at the point P(4, 2). Calculate slopes of secant lines through points x = 3, 6, and 8 and estimate the slope of the tangent line at x = 4. Understand the concepts of secant lines, tangent lines, and rates of change.

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