1. What are Tangent and Normal Lines?

A tangent line to a curve at a given point is a straight line that "just touches" the curve at that point and has the same slope as the curve at that point. The normal line is perpendicular to the tangent line at the point of contact.

2. How Does the Calculator Work?

The calculator uses these equations:

Where:

• \( (x_0, y_0) \) — Point of tangency
• \( m \) — Slope of the curve at the point of tangency
• \( -\frac{1}{m} \) — Slope of the normal line (negative reciprocal)

3. Importance of Tangent and Normal Lines

Details: Tangent lines are crucial in calculus for understanding derivatives and instantaneous rates of change. Normal lines are important in physics for calculating forces perpendicular to surfaces and in computer graphics for lighting calculations.

4. Using the Calculator

Tips: Enter the point coordinates (x₀, y₀) and the slope (m) at that point. The slope cannot be zero (for normal line calculation).

5. Frequently Asked Questions (FAQ)

Q1: What if the slope is zero?
A: If the slope is zero, the tangent line is horizontal and the normal line is vertical (undefined slope). Our calculator handles this special case.

Q2: Can I use this for 3D surfaces?
A: No, this calculator is for 2D curves. For 3D surfaces, you would need tangent planes and normal vectors.

Q3: How do I find the slope for a function?
A: The slope at a point is equal to the derivative of the function evaluated at that point.

Q4: What's the difference between normal and perpendicular?
A: In this context, they mean the same thing. The normal line is perpendicular to the tangent line.

Q5: Can the tangent line intersect the curve at other points?
A: Yes, a tangent line can intersect the curve at other points. It only guarantees to touch the curve at the point of tangency.