# Preparing for AP Calculus?

So you're about to take AP Calculus, eh? Here's a summary of what you should know going into it.

## What to know before taking AP Calculus

In some sense, the prerequisite for AP Calculus is to have an overall comfort with algebra, geometry, and trigonometry. After all, each new topic in math builds on previous topics, which is why mastery at each stage is so important.

However, for those of you who have taken courses in these subject, but are looking to quickly brush up on the essentials before beginning calculus, this sheet is here to highlight a few of the more crucial skills that you should have going into the course.

## Algebra

#### Manipulating expressions

- Know how to manipulate polynomial expressions.
- Adding: $(x^2 + 2x + 3) + (3x^2 - 3x) = 4x^2 - x + 3$
- Multiplying and Factoring: $(x + 2)(3x - 5) \Leftrightarrow 3x^2 + x - 10$

- Know how to solve simple linear equations.
- For example, $2x + 3 = 5x - 7$

- Know how to solve quadratic equations, such as $2x^2 + 3x - 5 = 0$
- Know the properties of exponents.
- $x^2 y^2 = (xy)^2$
- $(2^x)(2^y) = 2^{x + y}$

- Know how certain expressions are secretly exponentials in disguise
- Reciprocals: For example, $\dfrac{1}{x} = x^{-1}$
- Roots: For example, $\sqrt{x} = x^{1/2}$

- Know what logarithms are, as well as their properties.
- $y = 2^x$ says the same thing as $\log_2(y) = x$.
- $\log(x) + \log(y) = \log(xy)$.
- $\log(a^x) = x\log(a)$.

#### Functions

Calculus is all about functions, so it is helpful to be pretty fluent when it comes to thinking about functions, graphing functions, and using the appropriate terminology when talking about functions.

- Know the graphs of various elementary functions.
- Linear functions
- Quadratic functions
- Have at least a loose idea for what the graph of an $n^{\text{th}}$ degree polynomial might look like.
- Exponentials
- Logarithms

- It's also helpful to be familiar with function terminology

## Geometry

- Know how to compute the area of simple shapes.

## Trigonometry

- Be comfortable with each of the basic trigonometry functions: $\sin(x)$, $\cos(x)$ and $\tan(x)$
- Know what each one represents.
- Know the values of these functions when $x$ takes on one of the following values: $0$, $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$, $\dfrac{\pi}{3}$, $\dfrac{\pi}{2}$.
- Know what the graph of each of these functions looks like.

## Know when you need to review

The thing about calculus is that it tends to pull examples from all sorts of areas in math. While the list above does cover the main things you should be comfortable with before starting a calculus class, you will inevitably come across some example or topic which references another bit of prerequisite knowledge.

In an ideal world, you would know everything about algebra, geometry and trigonometry 100% perfectly. But more realistically, there are a few things you did not learn perfectly the first time. It's totally fine if that happens, but it can sometimes be tricky to recognize when a calculus problem is hard because you don't know the fundamentals (e.g. algebra) or if it's hard because of the new material (i.e. the calculus itself).

Just make sure you are always willing to ask yourself "Do I have a strong feel for the concepts in this problem?" If the answer is no, don't be afraid to temporarily divert your attention away from the calculus material to review the necessary algebra, geometry or trigonometry. Trust me, in the long run, it is always worth taking a step back before moving forward.

## Interview with Ben Eater

Although this is not directly about calculus preparation, we think this interview Sal did with one of our engineers, Ben Eater, serves as a telling example of why prerequisite knowledge is so important in the context of Calculus.