Algebra Review

Common Math Notation

Fractions

Exponents

Logarithms

Order of Operations

Coordinate Geometry

Equation of a line

Functions and Their Limits

General Definition

Domain and Range

Examples

Linear Equations

Quadratic Equations

Exponential/Logarithmic

Continuous Functions

Piecewise Functions

Limits

Definition

Limit from Above/Approaching from Right

Limit from Below/Approaching from Left

Section 2 (Lecture Notes, Practice Problems, Solutions)

Differentiation of Functions

Defining the Derivative

Finding the Derivative the "Long Way"

Basic Differentiation Rules

Derivative of a Constant

Derivative of a Power of x

Derivative of an Exponential Function

Derivative of a Logarithmic Function

Derivative of Trigonometric Functions

Derivative of a Function Multiple

Derivative of a Sum of Functions

Product Rule

Quotient Rule

Combining the Rules - CHAIN RULE

Second Derivative (Third and so on)

Critical Points of Functions

What is a critical point?

Why are we interested?

Using the first derivative

Maximum

Minimum

Using the second derivative to tell the difference

Global vs. Local

Using Numerical Methods

Newton-Raphson

Taylor Series

Section 3 (Lecture Notes, Practice Problems, Solutions)

Integration

Finding the Area under a curve using calculus

Why do we do it?

Indefinite Integrals

Definite Integrals - Fundamental Theorem of Calculus: F(b) - F(a)

Integration Rules

Integrating a Constant

Integrating a Power of x

Integrating an Exponential Function

Integrating a Logarithmic Function

Integrating Trigonometric Functions

Integrating a Function Multiple

Integrating a Sum of Functions

Integrating a Product - Integration by Parts

Check by differentiating

Methods of Integration

Numerical Integration - Quadrature

Trapezoidal Rule

Monte Carlo Integration

Section 4 (Lecture Notes, Practice Problems, Solutions)

Differential Equations

Definition

Time Rate of Change

Specific Solution for Initial Value

Solution by Direct Integration

General Solution

Particular Solution for a Given Initial Value

Separable Equations

Examples

Section 5 (Lecture Notes, Practice Problems, Solutions)

Matrix Algebra

Definition/Dimensions

Special Cases

Vector

Square

Symmetric

Diagonal

Matrix Arithmetic

Matrix Transpose

Determinants - Existence of Inverse

Matrix Inverse

2 by 2 Shortcut

Cofactor Expansion

Linear Equations and Least Squares

Why do we use this?

Setting up a system of linear equations

Example of 2 linear equations

Regression with Matrices

Setting up the X,B,y matrices

Section 6 (Lecture Notes, Practice Problems, Solutions)

Set Theory

Notation

Define set, elements

Empty Set, Universal Set

Unions

Intersections

Containment/Subsets

Mutually Exclusive/Complements

Experiments

Definition

Outcomes/Sample Space

Finding Your Sample Space

Venn Diagram

Finding Which Outcomes Belong to Each Event

Examples

Probability

Definition as a function

How do we find the Probability of an Event?

Union of Two Events

Intersection of Two Events

Conditional Probability

Testing for Independence - Multiplicative Rule

Bayes' Rule

Section 7 (Lecture Notes, Practice Problems, Solutions)

Random Variables

Definition

A Value for Every Outcome

Discrete R.V. Example

Find Distribution of X

Find Mean

Find Variance

Discrete or Continuous

Expectation of a function of X

Variance of a function of X

Introduce Idea of Multiple Random Variables X, Y

Independent, Dependent

Section 8 (Lecture Notes, Practice Problems, Solutions)

Cumulative Distribution Function

non-decreasing function, between 0 and 1

discrete R.V. examples

Probability Density Function

f(x)

Marginal distribution of 1 R.V.

Joint distribution of 2 R.V.

Conditional Distribution of one R.V. given another R.V.

Discrete Examples

Continuous Examples

Section 9 (Lecture Notes)

Discrete Distributions

Bernoulli

Binomial

Multinomial

Geometric

Poisson

Section 10 (Lecture Notes)

Continuous Distributions

Uniform

Univariate Normal

Bivariate Normal

Chi-square

Exponential

Logistic

Beta

Gamma