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>

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.) I prefer to memorize things in this form:

<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>

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!"