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:

• \( f(x) \) — The original function
• \( f'(x) \) — The derivative of the function
• \( x_0 \) — The point where we're finding the lines
• \( f(x_0) \) — The function value at \( x_0 \)
• \( f'(x_0) \) — The derivative value at \( x_0 \)

3. Importance of Tangent and Normal Lines

Details: Tangent lines are fundamental in calculus, representing instantaneous rates of change. Normal lines are important in physics and engineering for determining forces perpendicular to surfaces.

4. Using the Calculator

Tips: Enter a mathematical function (like "x^2" or "sin(x)") and the point x₀ where you want to find the tangent and normal lines. The calculator will compute and display both equations.

5. Frequently Asked Questions (FAQ)

Q1: What functions does this calculator support?
A: Currently supports polynomial functions. Future versions will support trigonometric, exponential, and logarithmic functions.

Q2: Why is the normal line's slope negative reciprocal?
A: Two lines are perpendicular if the product of their slopes is -1, hence the negative reciprocal relationship.

Q3: What if the derivative at x₀ is zero?
A: The tangent line is horizontal, and the normal line is vertical (undefined slope).

Q4: Can I use this for parametric equations?
A: This calculator is for explicit functions y = f(x). Parametric equations require a different approach.

Q5: How accurate are the results?
A: Results are mathematically exact, assuming correct function input and proper differentiation.