1. What is the Normal Line Equation?

The 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
• \( x_0 \) — x-coordinate of the point
• \( f(x_0) \) — y-coordinate of the point

Explanation: The normal line has a slope that is the negative reciprocal of the tangent line's slope (which is f'(x₀)).

3. Importance of Normal Line Calculation

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 the derivative at the point, the x-coordinate of the point, and the function value at that point. The derivative cannot be zero (as the normal line would be vertical).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between tangent and normal lines?
A: The tangent line has slope f'(x₀) while the normal line has slope -1/f'(x₀). They are perpendicular to each other.

Q2: What if the derivative is zero?
A: If f'(x₀) = 0, the tangent line is horizontal and the normal line is vertical (x = x₀). Our calculator can't handle this case.

Q3: Can I use this for 3D surfaces?
A: No, this is for 2D curves. For 3D surfaces, you need to calculate the normal vector using partial derivatives.

Q4: How accurate is the calculator?
A: The accuracy depends on the precision of your input values. The calculator uses double-precision floating point arithmetic.

Q5: What applications use normal lines?
A: Computer graphics (lighting calculations), physics (reflection laws), engineering (stress analysis), and optimization problems.