## Section 1 |
## 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

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

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

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

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

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

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

Lecture Notes -
Slides

Practice Problems - Solutions

- Discrete Distributions
- Bernoulli
- Binomial
- Multinomial
- Geometric
- Poisson

Lecture Notes -
Slides

Practice Problems - Solutions

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

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

- 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