Before My First Lecture

A summary of the basic mathematical concepts and properties that are typically covered in secondary education.

This poster is a quick reference for the transition from secondary school mathematics to introductory statistics. It summarises the basic mathematical concepts and properties a student should be comfortable with before stepping into a first statistics lecture. The full content of the poster is reproduced below.

Basic Identities

Algebraic

{(a + b)}^{2} = a^{2} + 2 a b + b^{2}

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

a^{2} - b^{2} = (a - b) (a + b)

{(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

{(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

Trigonometric

\sin^{2} x + \cos^{2} x = 1

\tan x = \frac{\sin x}{\cos x}

\cot x = \frac{\cos x}{\sin x}

\sin^{2} x = \frac{\tan^{2} x}{1 + \tan^{2} x}

- 1 \leq \sin x \leq 1

- 1 \leq \cos x \leq 1

The graphs of sine and cosine are periodic, with values ranging between −1 and 1:

Logarithmic & Exponential

\log (x y) = \log (x) + \log (y)

\log (\frac{x}{y}) = \log (x) - \log (y)

\log x^{k} = k \cdot \log x

The logarithm and the exponential are inverse functions. The logarithm is defined only for positive $x$ and tends to $- \infty$ near zero, while the exponential always takes positive values:

Limits

Basic limits

For the natural logarithm:

\lim_{x \to 0^{+}} \ln x = - \infty

\lim_{x \to + \infty} \ln x = + \infty

For the exponential function $a^{x}$ with $a > 1$ :

\lim_{x \to + \infty} a^{x} = + \infty

\lim_{x \to - \infty} a^{x} = 0

Specifically for base $e$ :

\lim_{x \to + \infty} e^{x} = + \infty

\lim_{x \to - \infty} e^{x} = 0

Two fundamental trigonometric limits:

\lim_{x \to 0} \frac{\sin x}{x} = 1

\lim_{x \to 0} \frac{\cos x - 1}{x} = 0

Indeterminate forms

Some limits cannot be evaluated directly:

Form	Solution approach
$\frac{0}{0}$	L’Hôpital’s rule
$\frac{\infty}{\infty}$	L’Hôpital’s rule
$0 \cdot \infty$	Non-standard
$\infty - \infty$	Non-standard
$0^{\infty}$	Non-standard
$1^{\infty}$	$e^{\ln (\dots)}$
$\infty^{0}$	Non-standard

L’Hôpital’s rule applies when a limit has the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . In that case it allows the limit to be computed through the derivatives of the numerator and the denominator:

\lim_{x \to x_{0}} \frac{f (x)}{g (x)} = \lim_{x \to x_{0}} \frac{f^{'} (x)}{g^{'} (x)}

Set Operations

Venn diagrams

Union ( $A \cup B$ ): all elements that belong to $A$ or to $B$ (or to both).

Intersection ( $A \cap B$ ): the elements that belong to both sets at the same time.

Difference ( $A - B$ ): the elements of $A$ that do not belong to $B$ .

Difference ( $B - A$ ): the elements of $B$ that do not belong to $A$ .

Probability identities

$\Pr (A \cup B) = \Pr (A) + \Pr (B) - \Pr (A \cap B)$
$\Pr (A) \leq \Pr (B), if A \subseteq B$
$\Pr (A^{'}) = 1 - \Pr (A)$

Derivatives

Notation

There are three ways to denote derivatives:

Lagrange notation: $f^{'} (x), f^{''} (x), \dots, f^{(n)} (x)$
Leibniz notation: $\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}, \dots, \frac{d^{n} y}{d x^{n}}$
Newton notation: $\dot{y}, \ddot{y}, \dots, f^{(n)} (x)$

Basic derivatives

{(c)}^{'} = 0

{(x^{n})}^{'} = n \cdot x^{n - 1}

{(\sin x)}^{'} = \cos x

{(\cos x)}^{'} = - \sin x

{(\tan x)}^{'} = \frac{1}{\cos^{2} x}

{(\cot x)}^{'} = - \frac{1}{\sin^{2} x}

{(\ln x)}^{'} = \frac{1}{x}

{(e^{x})}^{'} = e^{x}

{(a^{x})}^{'} = a^{x} \cdot \ln a

Logarithmic differentiation

Purpose: a differentiation method that is useful for complicated expressions involving products or powers. Let $y = f (x)$ :

Step A: Take the logarithm of both sides.

\ln (y) = \ln f (x)

Step B: Differentiate both sides.

\frac{d}{d x} \ln (y) = \frac{d}{d x} \ln (f (x))

Step C: Carry out the differentiation.

\frac{1}{y} \frac{d y}{d x} = \frac{f^{'} (x)}{f (x)}

Step D: Solve for $\frac{d y}{d x}$ .

Integration

By parts

\int f (x) g^{'} (x) d x = f (x) \cdot g (x) - \int g (x) f^{'} (x) d x

A general rule of thumb for choosing $f (x)$ is the LIATE approach. The options are ranked in decreasing order of priority:

Logarithmic function
Inverse trigonometric function
Algebraic function
Trigonometric function
Exponential function

When the integral contains several functions of the same category, the LIATE approach does not help. A few practical tips:

Set $d v$ to be the most complicated part.
Set $u$ to be the function whose derivative $d u$ will be simpler.

By substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

where $u = g (x)$ and $d u = g^{'} (x) d x$ .

Appendix

Constants

$π \approx 3.1415 \dots$
$e \approx 2.7182 \dots$

Common misconceptions

Logarithms

$\ln$ : natural (Napierian) logarithm, $\ln (x) = \log_{e} (x)$
$\log$ : common logarithm, $\log (x) = \log_{10} (x)$

Factorials

$k! = 1 \cdot 2 \dots k$
$1! = 1$
$0! = 1$