Definition:Tangent to Curve

Definition
Let $f: \R \to \R$ be a real function.

Let the graph of $f$ be depicted on a Cartesian plane.


 * DerivativeOfCurve.png

Let $A = \tuple {x, \map f x}$ be a point on $G$.

The tangent to $f$ at $A$ is defined as:
 * $\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

Thus the tangent to $f$ at $x$ can be considered as the secant $AB$ to $G$ where:
 * $B = \tuple {x + h, \map f {x + h} }$

as $B$ gets closer and closer to $A$.

By taking $h$ smaller and smaller, the secant approaches more and more closely the tangent to $G$ at $A$.

Hence the tangent to $f$ is a straight line which intersects the graph of $f$ locally at a single point.


 * TangentToCurve.png

In the above diagram, the tangent is the straight line passing through $A$.