Use the graph to estimate the slope of the tangent line to y = xn at the given point. Verify your answer analytically. (Round your answers to one decimal place.)
(a)
    y = x1/2,    (1, 1)

To solve the problem of finding the slope of the tangent line to the curve \( y = x^{1/2} \) at the point \( (1, 1) \), we will first estimate the slope graphically and then verify the result analytically.

### Step 1: Estimation using the graph
We know that the graph of \( y = x^{1/2} \) is the same as \( y = \sqrt{x} \), which is a familiar shape — a curve that increases more slowly as \( x \) grows.

At the point \( (1, 1) \), the graph is smooth, and the slope of the tangent line represents the instantaneous rate of change of the function at that point. From visual inspection of such a graph (without an actual visual provided here), we expect the tangent line at \( (1, 1) \) to have a slope around 0.5 to 1, as the curve is less steep near \( x = 1 \).

However, to be precise, we will verify this using calculus.

### Step 2: Verifying analytically
The slope of the tangent line at any point on a curve is given by the derivative of the function at that point.

The function we are given is:

\[
y = x^{1/2}
\]

Now, let's differentiate this function using the power rule. The derivative of \( x^n \) is \( n \cdot x^{n-1} \).

\[
\frac{dy}{dx} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}
\]

Now, substitute \( x = 1 \) to find the slope at the point \( (1, 1) \):

\[
\frac{dy}{dx} \Bigg|_{x=1} = \frac{1}{2\sqrt{1}} = \frac{1}{2} = 0.5
\]

### Step 3: Conclusion
Thus, the slope of the tangent line to the curve \( y = x^{1/2} \) at the point \( (1, 1) \) is **0.5**, both by estimation and by analytical verification.

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Would you like any further clarification or details?

Here are five related questions to deepen understanding:
1. What is the derivative of \( y = x^n \) for any \( n \)?
2. How does the slope of the tangent line change as you move along the curve \( y = x^{1/2} \)?
3. What is the geometric interpretation of the derivative at a point?
4. How do you find the equation of the tangent line once you have the slope?
5. What is the slope of the tangent line to \( y = x^{1/3} \) at \( (1,1) \)?

**Tip:** Always check the units of the derivative to ensure you're interpreting the slope correctly.

Use the graph to estimate the slope of the tangent line to y = x^(1/2) at the given point. Verify your answer analytically. (Round your answers to one decimal place.) (a) y = x^(1/2), (1, 1)

Learn how to estimate and verify the slope of the tangent line to y = x^(1/2) at the point (1, 1). This problem covers key calculus concepts such as derivatives, tangent lines, and the power rule. Step-by-step solution included, suitable for high school calculus students.

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