Calculus

Limits

Definition of a limit:

<math>\lim_{x \rightarrow c} f(x) = L\,</math>

means that for all real values of <math>\epsilon > 0 \,</math>, there exists <math>\delta \,</math> for all <math>x \,</math> such that <math>0 < |x - c| < \delta \,</math>, we have <math>|f(x) - L| < \epsilon \,</math>.

In my own words:

Assume that the limit of <math>\lim_{x \rightarrow c} f(x) \,</math> is L.

Now, if we want to prove that it is L, we have to look at all possible numbers, especially small ones. Now, let's choose a number x that is awfully close to c, and see what f(c) gives us. It should get use well below this number we imagined.

Example

As a simple example, let's find:

<math>\lim_{x \rightarrow \inf}\frac{1}{x}</math>

I'm going to assume that the answer (L) is 0.

Now, let me choose a super-duper small number close to 0, 0.001, for <math>\epsilon \,</math>. Can I find another number (x) close to <math>\inf \,</math> that will give me <math>|\frac{1}{x} - 0| < 0.001 \,</math>? Of course, I can, and it would be something like 10,000, or anything bigger.

It doesn't matter how close I get to 0, I can always find another number that will get me even closer. I never have to choose infinity itself, because that's not even a real number, but I can get closer and closer if I need to.

So the limit is 0.

Derivatives

The derivative is simply the slope of a curve at a given point.

<math>\frac{d}{dx} f(x) \equiv f'(x) \equiv \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \,</math>

Forms

There are a few forms of derivatives you might see.

Liebniz: This is the oldest, and arguable, the least confusing, although it is cumbersome. Physicists don't use this much beyond introductory courses.

<math> {d \over dx} f(x) </math>

Oftentimes, the parameters to the function is dropped. You have to know that the function varies in x, otherwise the derivative is just 0.

<math> {d \over dx} f </math>

Newton: The newton form only applies to time derivatives, and involves putting dots on top. This is extraordinarily useful for basic physics.

<math> \begin{align} {d \over dt} f &= \dot f \\ {d^2 \over dt^2} f & = {d \over dt} \dot f = \ddot f \\ \end{align} </math>

Total derivative: Take Liebniz notation apart you can deal with dt, dx, dy, etc... separately. It's a bit easier to make mistake with this notation, but if you know what you are doing, it's very valuable.

Prime Notation: If you know what you are deriving against, you can use hash marks to note how many derivatives you need. For functions of one variable, the derivative is obvious.

Subscript Notation: If a function is multi-variable, you'll need to use subscripts that describe the derivative. (This is multi-variable calculus, which isn't described here.)

List

This isn't hard to solve for many formulas, but it's best to memorize as many derivatives as you can. (When we do integrals, we'll have to work backwards.)

Note that u, v, w, etc... are functions that vary in one common dimension. a, b, c, etc... are constants and do not vary in the same dimension. All derivatives are in the same dimension.

<math> \begin{align}

      a' & = 0,~a~\text{is constant.}\\

(u + v)' &= u' + v'\\

   (uv)' &= u'v + uv' \\

\left ({u \over v}\right )' &= {u'v - v'u \over v^2} \\

 (u(v))' &= u'(v) v' \text{ (chain rule)} \\

\end{align} </math>

Practice

It takes a fair bit of practice to learn how to apply the derivatives above. The ones you must be good at:

• addition rule.
• Multiplication rule.
• chain rule.
• power rule.
• logs and e.
• trigonometric rules.

The rest you can look up in a book when you get stumped. THe above should get you 99% of the way there.

Fundamental Theorem of Calculus

This is simply that the anti-derivative is the integral, as we've already said. This is one of those things that should make you go, "Wow!"