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.