Saturday, June 28, 2014

I've put up a new post over at The Numerist that talks about Google's free online calculator that is available for everyone to use and enjoy! Please visit my site to find out more, and be sure to Like it on Facebook as well! You can read my post here: http://thenumerist.com/google-free-online-calculator/

Top 5 Most Popular Posts of May

Another month has come and gone, so it's time to look back and tally up the page views and rank my top 5 most popular posts of May! There are few surprises once again, as it seems like my most popular material is REALLY popular, and everything else is just trying to keep up. Maybe I should consider doing a "10th place to 5th place most popular posts" write up, and maybe we'd get some different stuff in there. :) All jokes aside, here are the best of the month of May.
1. Converting Point Slope Form to Standard Form - When you're learning how to write an equation of a line, there are a couple of different forms that you can use to express it. Learning how to convert from one to another is an essential skill, and this post explains how you can do it.

2. Stretching and Compressing Graphs - Coming up with a standard graph of a function isn't too hard. But what about when you are asked to stretch it or compress it, and express what you've done right in the equation of the function. Here's a primer that will show you what to do.

3. Which Measure of Central Tendency to Use - When you begin studying statistics, and learn about the various measures of central tendency, there is understandably a confusion in some cases about which measure to use with certain situations. Take a look here at some pointers to help you choose which is best for you to use.

4. Graphing Parallel and Perpendicular Lines - Graphing lines is a fairly standard and simple exercise. However, do you know what to do if you're asked to take the line you have, and come up with an equation for a line that is parallel or perpendicular to it? There are a couple of key points to keep in mind for this, and you can find out about them here!

5. Special Angles in Trigonometry - It's no secret: trigonometry is one of the most hated topics in mathematics by students. I can't explain where the fear comes from, but I will acknowledge that trig can be confusing - incredibly so, sometimes. Thankfully, there are several shortcuts that take out the heavy work and let us skip several steps. The special angles in trigonometry are one such concept. If you can remember there, then you can miss out on a lot of repetitive and potentially confusing work. Study up on these angles here!
We'll see if next month brings any further changes to these rankings! While you're here, might I suggest that you take a look at my recent series on differentiation that I've started - it's a work in progress, but so far there are posts including an introduction to differentiation and derivatives, and several differentiation rules such as finding the derivative of a constant function, how to use the power rule, finding the derivative of a constant times a function, and finding the derivative of a sum of functions or difference of functions. Take a look, and maybe we can get some of those posts up here amongst the favourites! Thanks for reading, and remember to Like my posts if you enjoy them!

Differentiation Rules - Finding the Derivative of a Difference of Functions

This post continues along in my series on calculus differentiation rules, this time talking about how to find the derivative of a difference of functions. I hope you read my last post, which applied to sums of functions, because it is nearly the same situation when you are subtracting. I'm not going to go into the same level as detail as I did there, so I highly recommend you go back and give that a read, and then come back here to see how the same kind of thing applies.

Recall the setup for these rules. Let f(x) and g(x) be two separate functions (anything you want), and let's also say that the sum of these functions, that is, f(x) + g(x), is equal to F(x). Similarly, let's say that the difference of these functions, that is, f(x) - g(x), is represented by G(x).

To find the derivative of a difference of functions, you simply determine the derivatives of the component functions and subtract them accordingly to get G'(x).

Therefore: the derivative of a difference of functions is equal to the difference of those functions' derivatives.

Compare this to the differentiation rule for adding that I covered last time. You can see how it's the same kind of thing, with no extra manipulations or reorganizations or tricks. If you can add, then you can subtract.

I'm not going to provide any examples for this rule, unless anyone leaves me a comment below to specifically request some. I think that if you can work through the example for adding, you will have a good grasp of how to perform both the addition and subtraction differentiation rules.

Thanks for checking out my latest post, even though this is one of my shorter ones. I always love getting feedback, and Facebook Likes and +1's are very much appreciated!

Differentiation Rules - Finding the Derivative of a Sum of Functions

Welcome back to my introductory calculus series on differentiation formulas.  For those who are playing along at home, I have explained several rules so far and am going to add another one today.  If you've missed those posts, then I highly encourage you to go back and take a look at them to familiarize yourself with these basic concepts.  (So far: here, here, and here.  Or just check my table of contents to find more!)  Today, I am going to talk about finding the derivative of a sum of functions.

Consider that you have two functions.  They can be whatever you want, it really doesn't matter.  While I explain this concept, let's just call these functions f(x) and g(x).  Now, suppose that you want to add these functions together, and you come up with a third function, let's call it F(x).  Sounds complicated?  It doesn't have to be.  It's actually quite simple.  But what about if we want to find the derivative F'(x) - how do we do that??  That's easy too.  You just need to understand this property of derivatives.

All you need to do is find the derivatives of the "smaller" functions, f'(x) and g'(x), and then add those together to get F'(x)!

Put simply, the derivative of a sum of functions is equal to the sum of those functions' derivatives.

In other words: F'(x) = f'(x) + g'(x).

Let's try an example, and put some numbers to this thing so that you can see it's not that crazy.

Keep in mind that when I talk about functions here, these could be anything: x2, or (x-2)5/16. Simple, or more complicated.  The same rules apply.  For now, let's keep it simple.

Suppose that f(x) = x2 and g(x) = x3.  What is the sum of these, F(x)?

Well this is as easy as it gets.  What is the sum of x2 + x3?
Your new function F(x) is just that: F(x) =x2+ x3.

So then, differentiate F(x) to find its derivative, F'(x).

By the rule that I explained above, all you have to do is find the derivative of the individual pieces of this sum, and then add those!  So, by the power rule, we have:

F'(x) = 2x + 3x2

That's it!

Obviously this can get a heck of a lot more complicated, but the principle remains the same. What if you want to find the derivative of some G(x) = x+ x2 + 4x2/3.  Think of this as a sum of three smaller functions, and you can see how this rule applies here.  You don't need to necessarily always have two distinctly different functions to apply this.  Think back to what a function is - they can have many terms to them.  (They just have to pass the vertical line test!)

My point in all of this rambling is that to find derivatives of functions that have multiple terms in them added together (like G(x) above, or F(x) before that), all you need to do is find the derivatives of the individual terms, and add them together.  It really is quite straightforward, and I hope that I have made helped to make this clear.  Once you get in the habit of applying this rule, you will do it automatically.

Please let me know if this has helped you, or should be clearer in some way.  I appreciate any feedback you would like to leave!  Also, please do me a favour and click on the Like or +1 buttons if this post helped you in any way.

Happy Fibonacci Day!

I know this may be a little late in the day, but I just realized that today's date is actually a Fibonacci sequence!  That's right, today's date is May 8, 2013, or written another way, 5/8/13!

For those who don't know, the famous Fibonacci sequence is starts off with the numbers 0, 1, and then continues by adding numbers that are equal to the sum of its preceding two numbers.  So, the classical sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.... you can keep going on and on if you wish.

Today's date, when written as I have above, forms a section of this sequence.  If you would rather write your date in the format of month/day/year, then you can expect another Fibonacci day on August 5, 2013.  Unless I'm mistaken, there won't be another one of these nice alignments of date numbers until August 13, 2021!  So, I'm sorry I didn't recognize this earlier in the day to share with everyone!