Calculus

Limits

Definition

The formal definition of a limit is

<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>.

Although this won't mean much to the junior mathematician, it is important to understand since limits are used a lot more than you think.

In My Words

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

Now, if we want to confirm that it is L, we have to look at all possible values of <math>x</math>, especially small ones. Now, let's choose a number that is awfully close to <math>c</math>, and see what <math>f(c)</math> gives us. It should get us extremely close to <math>L</math>

Example

As a simple example, let's find:

<math>\lim_{x \rightarrow 0}\frac{2xy+y^2}{x}</math>

(This is what you get when you try to differentiate <math>y^2</math>.)

Given the above, the values from the definition are: <math> \begin{align} f(x) &\equiv \frac{2xy+x^2}{x} \\ c &\equiv 0 \\ \end{align} </math>

Keep in mind that we can never, ever plug in <math>x = c</math>, because <math>0 < |x - c| < \delta \,</math>. However, I know, already, that this is going to end up <math>L = 2y</math>.

<math> \begin{array}{l | l | l | l} x & \delta & f(x) & \epsilon \\ \hline 1 & > 1 & 2y+1 & > 1 \\ 0.1 & > 0.1 & 2y+0.1 & > 0.1 \\ -0.1 & > 0.1 & 2y-0.1 & > 0.1 \\ n & > n & 2y+n & > n \\ \end{array} </math>

It is clear that for any number what the limit will be as we approach 0.

Table of Limits

These are some common operations you can perform on limits:

<math> \begin{align}

\lim_{x \rightarrow n}(f(x) \pm g(x)) &= \lim_{x \rightarrow n}f(x) \pm \lim_{x \rightarrow n}g(x) \\

\lim_{x \rightarrow n}(f(x) g(x)) &= \left (\lim_{x \rightarrow n}f(x) \right ) \left ( \lim_{x \rightarrow n}g(x) \right ) \\

\lim_{x \rightarrow n}\frac{f(x)}{g(x)} &= \frac{\lim_{x \rightarrow n} f(x)}{\lim_{x \rightarrow n} g(x)} \\

\lim_{x \rightarrow n}\frac{f(x)}{g(x)} &= \lim_{x \rightarrow n}\frac{f'(x)}{g'(x)}\text{ (L'Hôpital's rule)} \\ \end{align} \,</math>

Otherwise, limits are pretty much common sense.

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.

Integrals

Integrals are a measure of the area under a curve.

Indefinite integrals do not have bounds attach. You can think of these as formulas for how to calculate definite integrals. You solve these simply by finding the anti-derivative, or the thing that gives you the derivative of what's in the integral.

Definite integrals have bounds attached. They may be infinity.

Evaluating definite integrals is really easy. Plug the high value into the indefinite integral, and subtract the low value. Don't forget that you have a constant to add at the end.

Multi-Variable, Path, etc...

I won't touch advanced integrals here. Note that all you need is a few tricks and you can do any kind of integral you can imagine. I'll cover these in advanced math.

List of Integrals

Because it's sometimes not as obvious how to find the anti-derivative, common integrals are listed here. Note that the number of integrals in books is much larger because there is no simple chain rule like there is for derivatives.

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!"