determine the equation of the tangent line of the function sqrt(x) at x=2

To find the equation of the tangent line to the function \( f(x) = \sqrt{x} \) at \( x = 2 \), we follow these steps:

### Step 1: Compute the derivative of \( f(x) \)
The derivative of \( f(x) = \sqrt{x} \) can be computed using the power rule. Rewriting \( \sqrt{x} \) as \( x^{1/2} \), we get:

\[
f'(x) = \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}
\]

### Step 2: Find the slope of the tangent line at \( x = 2 \)
The slope of the tangent line is given by the value of the derivative at \( x = 2 \):

\[
f'(2) = \frac{1}{2\sqrt{2}} = \frac{1}{2\cdot 1.414} \approx 0.353
\]

### Step 3: Find the point on the curve at \( x = 2 \)
The point on the curve at \( x = 2 \) is \( f(2) = \sqrt{2} \approx 1.414 \). So the point is \( (2, 1.414) \).

### Step 4: Use the point-slope form of the equation of a line
The point-slope form of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Where:
- \( m \) is the slope of the tangent line,
- \( (x_1, y_1) \) is the point of tangency.

Substitute \( m = 0.353 \), \( x_1 = 2 \), and \( y_1 = 1.414 \):

\[
y - 1.414 = 0.353(x - 2)
\]

### Step 5: Simplify the equation
Now, distribute and simplify:

\[
y - 1.414 = 0.353x - 0.706
\]

\[
y = 0.353x + 0.708
\]

Thus, the equation of the tangent line to \( f(x) = \sqrt{x} \) at \( x = 2 \) is approximately:

\[
y = 0.353x + 0.708
\]

### Summary
The equation of the tangent line is \( y = 0.353x + 0.708 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the derivative of a square root function generally computed?
2. What is the geometric interpretation of a tangent line?
3. How can the point-slope form of a line be derived?
4. What are other methods to find the equation of a tangent line?
5. Can we use the same process for functions that involve trigonometric or exponential terms?

**Tip**: The derivative of \( \sqrt{x} \) can always be expressed as \( \frac{1}{2\sqrt{x}} \), which is helpful for finding slopes of tangent lines for square root functions.

Find the equation of the tangent line to the function sqrt(x) at x = 2 using calculus. This problem involves computing derivatives, applying the power rule, and using the point-slope form to derive the equation of the tangent line. Suitable for students in Grades 11-12 and college-level calculus.

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