What is Tangent line? — Mathematics

Definition 22.2 High School Mathematics · Chapter 22 — Differentiation and Convexity

If $f$ is differentiable at $a$ , the tangent to the curve of $f$ at the point $(a, f(a))$ is the line of equation

y = f(a) + f'(a)(x - a).

As h decreases, the secant line through A = (a, f(a)) and (a+h, f(a+h)) (orange) rotates towards the tangent at A (red): its slope f(a+h)-f(a)/h tends to f'(a). — As $h$ decreases, the secant line through $A = (a, f(a))$ and $(a+h, f(a+h))$ (orange) rotates towards the tangent at $A$ (red): its slope $\frac{f(a+h)-f(a)}{h}$ tends to $f'(a)$ .

Examples

Example 22.14

$f(x) = x^3$ has $f''(x) = 6x$ : $f$ is concave on $\intoc{-\infty}{0}$ , convex on $\intco{0}{+\infty}$ , with an inflection point at the origin, where the curve crosses its tangent (the $x$ -axis).

Example 22.15 (Convexity inequalities)

The exponential is convex on $\R$ (its second derivative is itself, positive). Its tangent at $0$ is $y = 1 + x$ , so

\eu^x \geq 1 + x \quad \text{for all } x \in \R .

Such tangent line inequalities are a standard product of convexity.

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Examples

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