Math Camp - Lectures

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Section 1

Lecture Notes - Slides

Practice Problems - Solutions

Practice Problems Archive

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 - Slides

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 - Slides

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 - Slides

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 5

Lecture Notes - Slides

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 6

Lecture Notes - Slides

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 7

Lecture Notes - Slides

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 8

Lecture Notes - Slides

Practice Problems - Solutions

Discrete Distributions
Bernoulli
Binomial
Multinomial
Geometric
Poisson


Section 9

Lecture Notes - Slides

Practice Problems - Solutions

Continuous Distributions
Uniform
Univariate Normal
Bivariate Normal
Chi-square
Exponential
Logistic
Beta
Gamma


Section 10

Lecture Notes - Slides

Practice Problems - Solutions

Introduction to Maximum Likelihood Estimation
The likelihood function: binomial example
Maximizing the likelihood
Computing the margin of error
Computing the margin of error


Additional Topics (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