Math Camp - Lectures

Section 1 ( Lecture Notes, 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, 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