# Hard Math

Note that this is mostly concerned with college math courses, although there's a bunch of overlap with high school, and we're concerned about school math in general (all math, actually!)

The courses picked as elementary math courses are:

## Contents

Adding fractions that are functions ie $\frac{3x}{2(x-3)(x+1)}+\frac{6x+4}{(x-2)(x+1)}$

Understanding trig functions and the relationships between them. Picturing all of this on the unit circle, knowing values in degrees and radians

What is e?

Logorithms

Idea of slope

Exponential growth and decay

### First year calculus

Pictures that show convergence

Pictures that show continuous, differentiable, and smooth functions in venn diagram sort of ways.

Limits: limits that are different from different sides, limits that are infinity.

Derivatives: a picture that shows secant lines getting smaller and smaller, until you take a limit

Riemann Sums and their applications

Contrasting sequences and series

Examples of Max and Min questions

Ratio test for convergence (which asks the question, does it behave like a geometry series)

Different coordinate systems and converting between them.

Volumes of revolution, particularly volumes revolved about lines that are not the axes.

Concept of Taylor Series, what they are, how we use them, and why we use them.

Fundamental theorem of calculus

### Second year calculus (Multi)

Generalizations of fundamental theorem of calculus

Directional Derivatives

### Linear algebra

What are different basis? Eg, (1,3) can look really different in three different basis.

What does orthogonality really mean? What do orthogonal functions look like?

What do eigenvalues and eigenvectors mean? maybe something on spectral theory (Anna knows about this, and can either write a page or help someone else write it).

### Elementary statistics

Steve Wang: p-values, hypothesis testing (the logic of how tests work, Central Limit Theorem)

What do confidence intervals mean (ie that a method works 95% of the time, not that you're 95% certain of your value).

Where samples fall (ie that you don't really know if your sample was just a fluke)

Disjoint vs. Independent events (explaining the differences).

### Other directions

[E.g. Tom Hunter interview]

Functions that go from a space of one dimension to another (ie. $R\rightarrow R^2$, $R\rightarrow R^3$, $R^2\rightarrow R$, etc)