Equation of the Tangent Line at Point T

Introduction to Tangent Lines on Curves

A tangent line to a curve at a given point is the line that touches the curve at that point and has the same slope there as the curve. The tangent equation is a key tool in differential calculus, with many applications in geometry, analysis, and physics.

Basic Principles for Determining the Tangent Equation

To determine the equation of the tangent at point T, you need:

• The equation of the curve, usually y = f(x).
• The coordinates of the point T(x0, y0) through which the tangent passes. The point T must lie on the curve.
• The derivative f'(x) at x0, which gives the slope of the tangent.

Procedure to Compute the Tangent Equation

The tangent to y = f(x) at T(x0, y0) is:
y - y0 = f'(x0) * (x - x0).

Example: f(x) = x^2 and T(2, 4)

1. Derivative: f'(x) = 2 * x.
2. At x0 = 2: f'(2) = 4.
3. Formula: y - 4 = 4 * (x - 2).
   Simplified: y = 4 * x - 4.

Key Points

• The tangent at T touches the curve at T and has the same slope there.
• Finding its equation requires the derivative at the contact point.
• Use y - y0 = f'(x0) * (x - x0).