Lecture Note: §1 Functions

Last Update: 2025-10-28

Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. If necessary, refresh your skills by referring to the review materials that are provided.

Common Inequlity

• The arithmetic mean-geometric mean (AM-GM) inequality
  • For a collection of \( n \) non-negative real numbers \( x_1, x_2, \ldots, x_n \), we have \(\dfrac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1x_2\cdots x_n}\), with equality if and only if \( x_1 = x_2 = \cdots = x_n \).
  • For \(n = 2\), we have \( x_1 + x_2 \geq 2\sqrt{x_1 x_2}\) with \(x_1, x_2 \geq 0\).
• The Triangle Inequality:
  • If \(a\) and \(b\) are any real numbers, then \( \lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert\).

Trigonometric Identities

• The Six Trigonometric Functions

  • \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \cot\theta = \dfrac{\cos\theta}{\sin\theta}\)

  • \(\csc\theta = \dfrac{1}{\sin\theta}, \sec\theta = \dfrac{1}{\cos\theta}\),

  • \(\cot\theta = \dfrac{1}{\tan\theta}\)

  • \(\sin(x + \pi/2) = \cos x, \cos(x+ \pi/2) = -\sin x\)

• The Basic Identities

  • \(\sin^{2}\theta + \cos^{2}\theta = 1\)

  • \(\tan^{2}\theta + 1 = \sec^{2}\theta\)

• Addition Formulas

  • \(\sin(x + y) = \sin x \cos y + \cos x \sin y \)

  • \(\cos(x + y) = \cos x \cos y - \sin x \sin y \)

  • \(\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y} \)

• Subtraction Formulas

  • \(\sin(x - y) = \sin x \cos y - \cos x \sin y \)

  • \(\cos(x - y) = \cos x \cos y + \sin x \sin y \)

  • \(\tan(x - y) = \dfrac{\tan x - \tan y}{1 + \tan x \tan y} \)

• Double-Angle Formulas

  • \(\sin 2x = 2 \sin x\cos x\)

  • \(\cos 2x = \cos^{2} x - \sin^{2} x = 2\cos^{2} x - 1 = 1 - 2\sin^{2} x\)

• Half-Angle Formulas

  • \(\cos^{2} x = \dfrac{1 + \cos 2x}{x}\)

  • \(\sin^{2} x = \dfrac{1 - \cos 2x}{x}\)

• Product Identities

  • \(\sin x \cos y = \dfrac{1}{2}[\sin(x + y) + \sin(x - y)]\)

  • \(\cos x \sin y = \dfrac{1}{2}[\sin(x + y) - \sin(x - y)]\)

  • \(\cos x \cos y = \dfrac{1}{2}[\cos(x + y) + \cos(x - y)] \)

  • \(\sin x \sin y = -\dfrac{1}{2}[\cos(x + y) - \cos(x - y)] \)

• Law of Sines: In any triangle \(ABC\),

  • \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\)

• Law of Cosines: In any triangle \(ABC\),

  • \(a^{2} = b^{2} + c^{2} - 2bc\cos A \)

  • \(b^{2} = a^{2} + c^{2} - 2ac\cos B \)

  • \(c^{2} = a^{2} + b^{2} - 2ab\cos C \)