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

1. Assume that the limit of <math>\lim_{x \rightarrow c} f(x) \,</math> is L.
2. Choose an arbitrarily small number for <math>\epsilon</math>, something very, very close to 0 but not 0. (Hint: Choose <math>10^{-10}</math> or some other large, negative exponent.)
3. Find the value of <math>x</math> in <math>|f(x)-L|</math> that will get you something smaller than <math>\epsilon</math>.
4. Make sure that <math>|x-c|</math> gives you a real number bigger than 0. That is, you can't use <math>x=c</math>.

If you can do the above for every imaginable <math>\epsilon</math>, then you have found the limit.

In Practice

Finding the limit can be tricky, especially if you don't follow some careful rules.

1. Even if the function is not defined at <math>c</math>, the limit may still exist. Just put your finger over the graph at that point and ignore it because you can't even think about it. (If you're dealing with infinity, this is easy. You can never, ever reach infinity so you can't even consider it.)
2. Think about what happens to the function as you approach <math>c</math> from both sides.
   • If it is a different value from one side than another, the limit doesn't exist. (Unless you're doing one-sided limits, which are rare.)
   • If it switches between two values without stabilizing, the limit doesn't exist.
   • If it shoots off to infinity, the limit doesn't exist.
3. If the function smooths out before reaching <math>c</math>, and tends to a single value, that's the limit.

Notice that the definition of the limit doesn't even enter into the discussion of how to find the limit. We always work backwards, starting with the limit, and see if the definition holds up precisely, rather than the other way around.

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>, and substitute <math>\delta y = x</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.)

<math>a,\ b\ \text{are constants, and}\ f,\ g\ \text{are functions of }x.\,</math>

<math> \begin{align}

      \frac{d}{dx}a &= 0 \\
      \frac{d}{dx}(f+g) &= \frac{d}{dx}f + \frac{d}{dx}g \\
      \frac{d}{dx}(fg) &= \frac{d}{dx}(f)g + f\frac{d}{dx}(g) \\
      \frac{d}{dx}\frac{f}{g} &= \frac{\frac{d}{dx}(f)g - f\frac{d}{dx}(g)}{g^2} \\
      \frac{d}{dx}f(g) &= \frac{d}{dx}g \frac{d}{dg}f \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!"