Topic Index

Find which class sessions cover specific topics. Use Ctrl+F to search for your topic.

TipQuick Navigation

Click on class numbers to jump directly to that day’s content.

A-D

Chain Rule

Differentiating composite functions: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Classes: 12, 18, 19, 20, 21, 22, 23

Conditional Probability

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

Classes: 35, 37

Critical Points

Finding where \(f'(x) = 0\) or \(f'(x)\) is undefined

Classes: 24, 26, 27, 31

Cumulative Distribution Function (CDF)

\(F(x) = P(X \leq x)\)

Classes: 40, 41, 42, 44

Derivatives (Basic Rules)

Power rule, sum rule, constant multiple

Classes: 11, 16, 17, 18, 19

E-L

Expected Value

  • Discrete: \(E[X] = \sum x_i p(x_i)\)
  • Continuous: \(E[X] = \int x \cdot f(x) \, dx\)

Classes: 35, 38, 39, 40, 41

Exponential Functions

\(f(x) = ae^{bx}\) and properties

Classes: 3, 9, 11, 13

Functions

Evaluation, composition, domain, range

Classes: 1, 2, 6, 7, 12

Integration

Definite and indefinite integrals, Riemann sums

Classes: 36, 37, 38, 43, 44

Intermediate Value Theorem (IVT)

Existence of zeros for continuous functions

Classes: 11

Least Squares Regression

Minimizing squared residuals

Classes: 31, 32

Linearization

\(L(x) = f(a) + f'(a)(x-a)\)

Classes: 16

Logarithms

Properties and solving logarithmic equations

Classes: 9, 10, 12, 26

M-P

Maximum Likelihood

Finding parameters that maximize likelihood/loglikelihood

Classes: 14, 15, 24, 25, 26, 27, 28, 29, 30

Optimization

Finding maximum/minimum values using derivatives

Classes: 24, 26, 27, 31

Partial Derivatives

\(\frac{\partial f}{\partial x}\) for multivariable functions

Classes: 23, 28, 31

Percentiles

Finding \(x_p\) such that \(P(X \leq x_p) = p/100\)

Classes: 40, 41, 42

Plotting (in R)

Using plot(), lines(), abline()

Classes: 3, 4, 11, 26

Probability

Basic rules, sample spaces, events

Classes: 7, 8, 13, 14, 34, 37, 39

Probability Density Function (PDF)

\(f(x)\) such that \(\int f(x) \, dx = 1\)

Classes: 38, 39, 40, 41, 42

Product Rule

\(\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)

Classes: 17, 18

Q-Z

Quotient Rule

\(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\)

Classes: 17, 18

Random Variables

Discrete and continuous random variables

Classes: 34, 38, 39, 42

Second Derivative Test

Classifying critical points as maxima or minima using \(f''(x)\)

Classes: 23, 26, 27

Solving Equations (uniroot)

Using uniroot() in R to solve equations numerically

Classes: 10, 11, 12, 17

Variance

  • Discrete: \(\text{Var}(X) = E[X^2] - (E[X])^2\)
  • Continuous: Same formula with integrals

Classes: 39, 40, 41, 42

Visual Model Fitting

Selecting parameters to match data visually

Classes: 6, 11, 12

By Project

Project 1: LED Bulb Lifetime Modeling

Relevant Classes: 5, 6, 8, 11, 12, 13

Key Topics: - Function modeling and evaluation - Solving equations with uniroot() - Visual parameter fitting - Plotting in R

Project 2: Maximum Likelihood

Relevant Classes: 14-15, 24-30

Key Topics: - Likelihood and loglikelihood - Derivatives (chain rule, product rule) - Partial derivatives - Optimization (finding maxima) - Second derivative test

Project 3: Probability Distributions

Relevant Classes: 34-45

Key Topics: - PDF and CDF - Expected value and variance - Percentiles - Integration (definite integrals) - Probability calculations

By Week

Weeks 1-2: Functions and Modeling

Classes: 1-8 Topics: Function evaluation, plotting, visual fitting, probability basics

Weeks 3-4: Solving Equations

Classes: 9-15 Topics: Logarithms, uniroot(), IVT, likelihood, exponential functions

Weeks 5-6: Derivatives

Classes: 16-22 Topics: Linearization, power rule, product rule, quotient rule, chain rule

Weeks 7-8: Optimization

Classes: 23-30 Topics: Partial derivatives, critical points, second derivative test, maximum likelihood, least squares

Weeks 9-10: Integration Intro

Classes: 31-37 Topics: Riemann sums, definite integrals, probability rules, conditional probability

Weeks 11-12: Continuous Distributions

Classes: 38-43 Topics: PDF, CDF, expected value, variance, percentiles, integration with Mathematica

Week 13: Synthesis

Classes: 44-46 Topics: Connecting CDFs and PDFs, change of variables, review

R Functions Reference

By First Use

R Function First Used Purpose
plot() Class 3 Basic plotting
lines() Class 3 Add curves to plots
uniroot() Class 10 Solve equations numerically
exp() Class 11 Exponential function
log(), log2() Class 9 Logarithm functions
abline() Class 11 Add reference lines
sum() Class 26 Sum vectors

Mathematica Functions Reference

By First Use

Mathematica Function First Used Purpose
Integrate[] Class 36 Symbolic integration
NIntegrate[] Class 37 Numerical integration
Solve[] Class 40 Solve equations symbolically
NSolve[] Class 42 Solve equations numerically