Point-Slope Form

y - y₀ = f'(x₀)(x - x₀)

Where (x₀, y₀) is the point of tangency and f'(x₀) is the derivative evaluated at x₀.

Step-by-Step Process

1. Find y₀: Evaluate f(x₀) to get the y-coordinate
2. Find the slope: Calculate f'(x) and evaluate at x₀
3. Write equation: Substitute into point-slope form
4. Simplify: Convert to slope-intercept form y = mx + b if needed

Example: f(x) = x² at x = 2

• Step 1: f(2) = 2² = 4, so point is (2, 4)
• Step 2: f'(x) = 2x, so f'(2) = 4 (slope = 4)
• Step 3: y - 4 = 4(x - 2)
• Step 4: y = 4x - 4

Normal Line

The normal line is perpendicular to the tangent line at the point of tangency.

Slope of normal = -1 / f'(x₀)

Horizontal Tangent Lines

When f'(x) = 0, the tangent line is horizontal. These points are critical for finding maxima and minima.

Equation: y = y₀ (a horizontal line)

Vertical Tangent Lines

When f'(x) is undefined (approaches ±∞), the tangent line is vertical. This occurs at cusps or vertical inflection points.

Physics

Velocity is the tangent line to a position-time graph. Acceleration is the tangent to velocity-time.

Economics

Marginal cost/revenue is the tangent line to the total cost/revenue curve at a given production level.

Engineering

Design optimal paths, analyze stress distribution, and model instantaneous rates of change.

Optimization

Find maximum profit, minimum cost, or optimal dimensions using horizontal tangent lines.