The Triangle Inequality Theorem

This is a post based on a question.

I am looking to write a proof (without breaking it into cases) for the following:

Prove that |x+y| ≤ |x| + |y| using the following assumptions:

-|x| ≤ x ≤ |x|

if c ≤ r, then c ≥ -r

if r ≥ -c, and r ≤ c, then -c ≤ r ≤ c

Before I go on, I have to apologize. I was unable to come up with a proof of my own (I kept getting stuck), so I searched the internet (this property is famously known as the "Triangle Inequality", and has applications in number theory, calculus, physics, and linear algebra) and found two different proofs that appeared side-by-side on numerous sites. I have copied and pasted these proofs into this post BUT I must warn you that I myself do not believe either of them. Though these proofs seem unanimously accepted by the people of the internet, they do not seem correct to me (I will explain why later). I am doing my best to answer your question, but just know that.

The red question marks indicate areas where the proof fails for me. If you can find an explanation for these areas, let me know.

Find a (Math) Tutor Near You

When I first created this blog, I wanted to keep it simple because I personally don't like it when I'm browsing other blogs and they take forever to load.

However, I've added another useful tool to the sidebar (in addition to the widget I added yesterday) which I hope you will use. I just discovered a revolutionary website called WyzAnt. This site gives tutors and students a place to find each other without going through the hassle and creepiness-factor of websites like cragslist. All of the tutors must complete an extensive application process and YOU, the student, can have THEM, the tutor(s), background-checked. How neat is that?

WyzAnt not only for math, either. There are tutors there that specialize in everything from SAT-prep to violin lessons to Mandarin Chinese.

If you think that you could benefit from meeting regularly face-to-face with a tutor in your area, type your zip code (or a nearby zip code, if you're internet-scared like I am) into the widget I added to the sidebar. Otherwise, just browse on over to the site anyway and check it out.

Sorry, international friends, WyzAnt is restricted to only the U.S. at the moment (I think, though I didn't see anything that explicitly said that).

Basic Integration By Parts

This is a post based on a question.

If it isn't a problem, can I ask you how to integrate by parts?

The properties of integrals let us split up addition, but not multiplication.

So, when we have two functions MULTIPLIED together, we must use the technique of Integration By Parts.

(Note: In this post I will show the formula and explain how to use it, but I will not show why it works. If anyone would like to see where the formula comes from, please let me know in a comment.)

Notice that there is an integral in the answer! This means that to evaluate the problem completely you must also do that integral at the end. This also means that there is potential for the process repeating over and over (you get the answer, but then you have to use Integration By Parts again), but I titled this post Basic Integration By Parts for a reason; repetition would be Advanced Integration By Parts.

Anyway, this formula is critical. After extensive practice, you will know it by heart just because you will use it so much. Before I was entirely familiar with it, however, I had a chant that helped me remember it. All I said was "u v minus v du," but it was enough for me remember it forever.

Now, let's do an example.

Memorization

This is a post based on a question.

I'm going to sixth grade and I still haven't memorized my multiplication tables. Help me?

This is more of a series of helpful hints than an actual lesson.
While the vast majority (and most exciting parts) of math involves critical thinking, problem solving, and multi-step processes, there are certain parts that just require some simple memorization. For example, once you have the concept of what multiplication actually is, it is immensely helpful (perhaps even necessary) to simply memorize multiplication tables for all combinations of numbers 0-12. Likewise, in Calculus it can greatly improve your test-taking speed by just memorizing certain basic derivatives and antiderivatives.

Before I go any further, I must make an extremely important point: the goal of memorization is not to substitute for understanding, but to increase mental speed. That is, you should know what I mean when I ask what 4x6 is, and not just know the answer. You must first deeply understand the concepts before memorizing the numbers.

I have a few tried-and-true methods for memorizing simple facts, such as times tables and derivatives. It is best to try all of them and see which works best for you. Everyone's mind works differently and there are numerous learning styles to take into account when selecting a method.

Wolfram Alpha

This website is an invaluable resource for calculus (or any math or any other class, really) students: http://wolframalpha.com/

Find answers to questions in an immense variety of areas: Mathematics, Statistics, Physics, Chemistry, Engineering, Finance, Weather, Music, Nutrition; it can do almost anything!
Take a look at examples of many things it can do: http://www.wolframalpha.com/examples/

The Power Rule

This is a calculus lesson based on a question:

 Would you be able to explain how to derive a square root? 

The power rule is used to find the derivative of (or "differentiate" or "find the instantaneous rate of change of") an algebraic expression of the form x to a numerical power, or xⁿ.

The derivative of a function ƒ(x) = xⁿ is ƒ'(x) = n * x^n-1. If it is easier to remember think of it as dragging the exponent down (to multiply the x by) and then subtracting one in the exponent.

Here are some examples to clarify: