1. What is a Normal Line?

A normal line to a curve at a given point is the line perpendicular to the tangent line at that point. It's calculated using the derivative of the function at that point.

2. How Does the Calculator Work?

The calculator uses the normal line equation:

Where:

• \( f'(x_0) \) — Derivative of the function at point x₀
• \( x_0 \) — The point where the normal line is calculated
• \( f(x_0) \) — Value of the function at point x₀

Explanation: The slope of the normal line is the negative reciprocal of the tangent line's slope at the given point.

3. Importance of Normal Lines

Details: Normal lines are important in physics, engineering, and computer graphics for calculating reflections, surface normals, and optimization problems.

4. Using the Calculator

Tips: Enter a valid mathematical function (like "sin(x)", "x^2", or "3*x+2") and the point x₀ where you want to find the normal line.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between tangent and normal lines?
A: The tangent line touches the curve at one point with the same slope as the curve, while the normal line is perpendicular to the tangent at that point.

Q2: Can I use this for any function?
A: The function must be differentiable at the point x₀ for the normal line to exist.

Q3: What if the derivative is zero?
A: When f'(x₀) = 0, the normal line is vertical (undefined slope) with equation x = x₀.

Q4: Can I graph the results?
A: In a full implementation, this would include graphing capabilities showing both the function and normal line.

Q5: What function formats are supported?
A: Basic operations (+, -, *, /, ^), trigonometric functions (sin, cos, tan), exponential and logarithmic functions.