### A Different Way to Memorize the Taylor Series and a Cry For Mathematical Help

A Different Way to Memorize the Taylor Series and a Cry For Mathematical Help

As an undergrad, and throughout my graduate career until now, I've always had a hard time remembering how to apply a Taylor series.  I knew there were coefficients in front of powers of x and that those coefficients somehow involved derivatives of the function I was trying to aproximate with the Taylor series, but that was aobut it.  Even when I looked up the formula, there was still always some initial confusion.  I'd arrive at an equation that looked like the following

followed by a little bit of text blithely informing that if I'd only take the nth derivative of both sides of the equation, and evaluate the derivative at x = b, then of course I'd see that the coefficient a sub n could be written as

This inevitably led to having to remember that the first n-1 terms in the sum would become zero after they were differentiated n times, and that the terms of n+1 and above would become zero when x was equal to b so that only the term containing a sub n was left.  Perhaps I'm a bit slow, but that was my process every single time.

Looking at the formulas last night for the umpteenth time, it occurred to me that the first two terms in the Taylor series are how we approximate functions when we only want to add a small change in x.  We get the value for the function at x equals b, then approximate the function as a straight line and multiply the slope of that line times a small change in x to get the approximate value of the function at the location b plus our small change in x

In other words, we as the question, how does our function change for a small change in x, (what's it's slope with respect to x?), and then multiply that slope by our small change in x to get the associated small change in the function.

Now, look at a Taylor series with three terms:

The second term is the slope with respect to x I just mentioned.  If we write the second derivative in a different notation, we have

which, when I squint at it in just the right light looks for all the world like the amount my function will change for a small change in x squared.  An then with a second look, I see that the derivative is in fact multiplied by x squared.  It's slightly more visible if we take b equal to zero, and rewrite the series as:

This looks kind of great!  A Taylor series is just a series of corrections that ask what's the slope of my function with respect to a power of x and then multiplies that slope by the particular power of x.  That's far easier for me to remember, but there's still one issue left.  What about the annoying factors of one over n factorial out in front of each term?

And, that brings me to my second mathematical sin for the day, or more politely worded, my second opportunity to be mathematically clarified. I'm calling it the inverse chain rule.  The chain rule provides a way to take the derivatives of composite functions like

by first taking the derivative of the outer function, (cosine in this case), with respect to its argument, (x squared in this case), then determining how the argument changes with respect to x and finally determining the entire change with respect to x by multiplying the two changes together. So, using the chain rule, I can ask the question: how does my function of x squared change with respect to x?

A Little Formal Math Help Please?
In my above reasoning, I've asked the inverse of the chain rule question.  I'm asking, (for the third term), how my function of x varies with respect to x squared.  The terms of n factorial are just one over the nth derivative of the power of x I'm interested in

Since I've asked the inverse of the chain rule question question, I'm guessing that instead of multiplying [the derivative of the my function with respect to x to the n] by [the derivative of x to the n with respect to x] that multiplying by the inverse of [the derivative of x to the n with respect to x].  I know this is correct because it gives the correct terms for a Taylor series, but I don't know how to say it formally or explain it completely.  Would anyone like to clarify... please?

Summary of the Method
So, at the end of the day, the way I will now always remember the Taylor series is as a set of patches in powers of x to an approximtaion.  I multiply each power of x by the slope of the function with respect to that power of x.  To take care of the inverse chain rule, if the power is n, I then divide by the nth derivative of the nth power of x, (also known as n factorial).

Picture of the Day:
 From 1/27/13

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…