...for Honors Calculus III/Differential Equations

• Geogebra 3D graphing
  
• Ti-89 Calculator emulator
  
• Partial Fraction Expansion Calculator
  
• Cos+Sin=phase shifted Sin
  
• Ti-83/84 calculator emulator
  
•  How Multivariable Optimization is used to compute a Least-Squares Regression Line to fit a line to a linear data set in statistics

• Calculus III topics...
  • This video explains what a vector field is and gives good intuitive understanding of the means of Divergence and Curl of a vector field:
      3Blue1Brown: Divergence and Curl
  • This video is the very best explanation I've seen yet about how Electric fields work and how light propogates...highly recommended! (the first 11 minutes contain the key part):
      3Blue1Brown: This equation explains (nearly) all of optics
  • This video does a great job of explaining the physics meaning of how electric and magnetic fields interact and how Maxwell's Equations consolidates multiple, earlier, physics laws. She uses the integral forms of Maxwell's equations instead of Divergence and Curl, but everywhere you see a closed path line integral or a closed surface integral, that term could instead be written using a dot or cross product - the 'divergence and curl form' of Maxwell's Equations:
      Up and Atom: Maxwell's Equations: The Ultimate Beginner's Guide
  • This longer video is also excellent for a complete, detailed mathematical understanding of how electromagnetic waves are solutions to differential equations developed from Maxwell's Equations (expressed using Divergence and Curl):
      MIT lecture on Electromagnetic Waves
  • A great video about how resistance, inductance, and capacitance are related and affect the flow of electrical power in a circuit:
      Practical Engineering: Power Factor
• Differential Equations topics...
  • Intro to Differential Equations video:
      This is why you're learning differential equations
  • A fantastic video series about the Laplace Transform:
    This 3-video series is a fantastic explanation of how the Laplace Transform works, what the s-plane represents, and how this applies intuitively to mass-spring system applications:
    •   The Physics' of Euler's Formula | Laplace Transform Prelude
    •   But what is it a Laplace Transform?
    •   Why Laplace Transforms are so useful
  • A great video about how to visually interpret what a Fourier Transform represents (which is a part of what a Laplace Transform represents):
      But what is the Fourier Transform? A visual introduction
  • Another great video about the intuitive meaning of Laplace Transforms:
      What does the Laplace Transform really tell us? A visual explanation (plus applications)
  • A great way to visualize the Fourier Series using added curves as linked 'rods':
      What is a Fourier Series?
  • This video explains the concept behind the Runge-Kutta method of numerical approximation, a more accurate and stable algorithm than Euler's Method for approximating solutions to differential equations:
      Why Runge-Kutta is SO Much Better Than Euler's Method

...for AP Calculus

• Secant/Tangent Definition of Derivative
  
• Sketch the Derivative Curve
  
• Riemann Sums
  
• Geogebra 3D graphing
  
• Ti-83/84 calculator emulator
  

<-- You should look at the videos in the Calculus III/Differential Equations section, because everything you are learning in AP Calculus is needed to eventually explore these fascinating topics!

...for AP Statistics

• Normal Model Parameters
  
• LSRL as minimum residuals
  
• LSRL and outliers
  
• Rossman Chance: Great Statistics related online simulation applets
  
• Ti-83/84 calculator emulator
  
• Programs for Ti Calculators:
  Chi-Squared Goodness-of-fit Test
  Linear Regression T Interval
  Inverse T function
  Pivot Program File for Ti-83/84

• Good short video about spotting misleading graphs:
    How to spot a misleading graph
• Good examples which help in understanding Simpson's Paradox in Statistics:
    Simpson's Paradox
• A great 1-hour video which introduces much of what we cover in AP Statistics and talks about why it is important in the world today:
    Nova - Prediction by the Numbers
• Interesting discussion of 'cognitive ease' and how it explains the way we can be mislead to believe things that are not true:
    The Illusion of Truth
• A discussion of problems with the way statistics is used in scienc:
    Is Most Published Research Wrong?
• Another great video on p-values and how they can cause issues in scientific research. This video also suggests some alternatives which attempt to address the issue:
    P-values Broke Scientific Statistics - Can We Fix Them?

Other interesting math resources (in no particular order)...

• A great overview of all areas of mathematics:
  The Map of Mathematics
• Same as above, but for engineering:
  The Map of Engineering
• This website describes many careers which use math and includes specific examples and quotes from people working in these fields, along with job data (salaries, number of positions in the U.S., etc.):
  We Use Math website
• The Art of Problem Solving: a great site focused on innovative approaches to learning math and problem solving, and a great resource for math competitions such as the American Math Competition:
  The Art of Problem Solving
• Modulation Types: learn how information is encoded onto telecommunication signals
  All Modulation Types Explained
• QAM modulation: the method of encoding used in cellphones and wifi
  QAM (Quadrature Amplitude Modulation)
• A fantastic presentation on the underlying mathematics that are used in neural networks, one of the underlying structures used in artifical intelligence and machine learning systems:
  Neural Networks
• Learn how .jpg images are encoded (quite involved and interesting!):
  The Unreasonable Effectiveness of JPEG: A Signal Processing Approach
• Mathematics involved in various ways to conduct elections:
  Why Democracy Is Mathematically Impossible
• Very interesting, intuitive explanation of what the mathematical object, a Tensor, represents:
  I never understood why this math object is found everywhere...until now!
• Great introduction to how binary counting is used within a computer:
  Computers Are Just Rocks Doing Math
• (Not a video) This webpage has great 3D illustrations which explain a concept in signal processing which is really important - how to use In-Phase and Quadrature (IQ) Sampling of a signal for analysis:
  IQ Data for Dummies
• A team competition with scholarship awards for winning teams:
  Modeling the Future Challenge Website
• Pixar collaborated with Khan Academy to produce a great set of videos and online activities which show how Pixar developers and artists use mathematics to create amazing movies:
  Pixar in a Box
• The Code Book - An excellent introduction to cryptography with history as well as great medium-technical explanations of how code making and code breaking works:
  The Code Book
• Fascinating video about how we perceive color and some surprising facts about light and color:
  There's No Purple Light
• Copper's Surprising Reaction to Strong Magnets - Force Field Motion Dampening:
  Interesting physical effects of Lenz's Law
• Fascinating, and surprising explanation of magnetism via special relativity:
  How Special Relativity Makes Magnets Work
• A three-part, 3-hour video about the structure of the universe and string theory:
  PBS - The Elegant Universe
• A little known, but important topic about cars:
  Understanding Automobile Wheel Alignment.
• Intuitive explanation for the Lorentz Transformation which explains Special Relativity effects such as length contraction and time dilation:
  Lorentz Transformations
• A 1-hour video about fractals and how they can be used to model the imperfect real world:
  Nova - Hunting the Hidden Dimension
• An older video, but provides a really excellent explanation of Chaos Theory and how it arises from non-linear systems:
  Chaos Theory