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Here are the slopes of the secant lines passing through the point \((1, 2)\) and \((x, f(x))\), rounded to 4 decimal places for each given value of \(x\):

- For \(x = 0.8\), slope = 2.1115
- For \(x = 0.9\), slope = 2.0527
- For \(x = 0.99\), slope = 2.0050
- For \(x = 0.999\), slope = 2.0005
- For \(x = 1.001\), slope = 1.9995
- For \(x = 1.01\), slope = 1.9950
- For \(x = 1.1\), slope = 1.9524
- For \(x = 1.2\), slope = 1.9089

For part (b), based on these slopes, the apparent slope of the tangent line at \(x = 1\) appears to be approximately **2.0000**.

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How do secant lines differ from tangent lines in terms of slope?
2. What happens to the secant line as the second point approaches \((1, 2)\)?
3. How can you calculate the derivative of the function \(f(x) = 4\sqrt{x} - 2\) at any point?
4. What is the geometric interpretation of the derivative at \(x = 1\)?
5. How do different values of \(x\) affect the accuracy of the slope approximation?

**Tip:** The closer the points on the secant line are to each other, the closer the secant slope is to the tangent slope.

Estimate the slope of the tangent line at (1, 2) for the function f(x) = 4√x - 2, given the graph and table.

This problem involves estimating the slope of the tangent line at the point (1, 2) on the graph of f(x) = 4√x - 2. The solution uses secant lines and limits to approach the derivative, providing a detailed understanding of the tangent's slope. Suitable for high school students in Grades 11-12 or Calculus 1.

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