write the equation of the normal line as well

To find the equation of the normal line to the function \( f(x) = x^2 + 2 \) at the point where \( x = -1 \), follow these steps:

### 1. Recap the Tangent Line Slope
We've already calculated the slope of the tangent line at \( x = -1 \), which is \(-2\).

### 2. Find the Slope of the Normal Line
The normal line is perpendicular to the tangent line. The slope of the normal line is the negative reciprocal of the slope of the tangent line. 

If the slope of the tangent line is \( m = -2 \), then the slope of the normal line \( m_{\text{normal}} \) is:

\[
m_{\text{normal}} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2}
\]

### 3. Write the Equation of the Normal Line
Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope of the normal line and \( (x_1, y_1) \) is the point of tangency:

The point on the curve is \( (-1, 3) \) (from previous calculations).

\[
y - 3 = \frac{1}{2}(x + 1)
\]

Simplifying:

\[
y - 3 = \frac{1}{2}x + \frac{1}{2}
\]
\[
y = \frac{1}{2}x + \frac{1}{2} + 3
\]
\[
y = \frac{1}{2}x + \frac{7}{2}
\]

### Final Answer
The equation of the normal line at \( x = -1 \) is:

\[
y = \frac{1}{2}x + \frac{7}{2}
\]

Would you like more details or have any questions?

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**Relative Questions:**

1. How do you find the normal line for the function \( f(x) = \sin(x) \) at \( x = \frac{\pi}{4} \)?
2. What is the significance of the normal line in geometric problems?
3. How does the equation of the normal line change if the function is \( f(x) = \ln(x) \)?
4. How do you determine the angle between the tangent line and the normal line at a given point?
5. What happens to the equation of the normal line if the curve is a circle centered at the origin?

**Tip:** The normal line at a point on a curve is always perpendicular to the tangent line at that point, making it an essential tool in analyzing the geometry of curves.

Learn how to find the equation of the normal line to the function f(x) = x^2 + 2 at the point x = -1. This problem involves calculus and the concept of perpendicular slopes. Suitable for advanced high school students.

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