Tangents in Calculus
Tangents are straight lines that touch a curve at a given point without crossing over. They represent the instant direction in which a point on the curve is headed. This is essential in understand many physical phenomena! 🐸
Equation of a Tangent
The equation of the tangent to a curve \( y = f(x) \) at a point \( x = a \) is given by:
\[ y - f(a) = f'(a)(x - a) \]
Normals in Calculus
Normals are perpendicular to tangents at the point of contact. This right angle relationship helps in many fields like optics and architecture.
Equation of a Normal
The equation of the normal to a curve \( y = f(x) \) at the point \( x = a \) is:
\[ y - f(a) = -\frac{1}{f'(a)}(x - a) \]