Tangent Line to a Graph
The tangent line is one of the most fundamental concepts in calculus — it captures the idea of instantaneous rate of change at a single point on a curve.
🔑 Intuitive Idea
Imagine zooming in on a smooth curve at a specific point. As you zoom in closer and closer, the curve begins to look more and more like a straight line. That straight line is the tangent line at that point.
Think of a car driving along a winding road. At any instant, the car is heading in a specific direction — that direction is given by the tangent line to the road at that point.
📐 Formal Definition
The tangent line to the graph of $f(x)$ at the point $(a, f(a))$ is the line through $(a, f(a))$ with slope:
This limit, when it exists, is called the derivative of $f$ at $a$, denoted $f'(a)$.
The equation of the tangent line is then:
📊 From Secant Line to Tangent Line
The tangent line is the limit of secant lines. A secant line passes through two points on the curve: $(a, f(a))$ and $(a+h, f(a+h))$. As $h \to 0$, the second point slides toward the first, and the secant line approaches the tangent line.
| Concept | Formula | Meaning |
|---|---|---|
| Secant Line Slope | $m_{\sec} = \dfrac{f(a+h) - f(a)}{h}$ | Average rate of change over interval $[a, a+h]$ |
| Tangent Line Slope | $m_{\tan} = \displaystyle\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ | Instantaneous rate of change at $x = a$ |