Saturday, June 1, 2013

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!

Friday, May 17, 2013

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!

Thursday, May 9, 2013

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.

Wednesday, May 8, 2013

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!

Saturday, May 4, 2013

Differentiation Rules - Finding the Derivative of a Constant Times a Function

In this post I'm going to explain another one of the differentiation rules for working with derivatives.  This time, I will show you how to find the derivative of a constant times a function.

In case you have missed them, I am creating a series of posts that explain some basic concepts in differential calculus.  So far, in my first lesson I explained how to find the derivative of a constant function, and then I followed that with a post about the power rule for derivatives.  So far, these are some of the basic rules for calculating derivatives, and it is a good idea to become very familiar with them so that you can apply them all as required on more elaborate problems later on.

This derivatives rule is a very simple one, but that doesn't make it any less important.  Consider that you have any function f(x), and it is multiplied by a constant.  Assuming that f'(x) in fact exists, f'(x) times a constant is equal to the constant times this derivative.  That may sound a little wordy, so maybe this equation will be a little clearer:

So, all you really need to be concerned with is calculating the derivative f'(x).  Then, multiplying by the constant is a simple calculation that you can add at the end.

If you're curious about what this rule means, here is one way of looking at this.  If you have an equation such as y = f(x), if you consider multiplying this function by a constant, what you are doing is essentially stretching the graph vertically.  So, comparatively, y = 2f(x) is a graph that is twice the height.  Therefore, if you then consider the slope along this function, since you have doubled the rise but not the run (the slope calculation), you have a doubled slope at every point.  And since the derivative of a function represents the slope of the line at a point, you can then see how this rule all comes together.  Basically, if you stretch the function by a constant factor, you can simply multiply the slope (the derivative) by this factor as well.

As I said, there isn't much to remember about this particular derivative rule, but it is very important to know.  It will often need to be considered in addition to other rules that I have/will outline in this series. One example would be to calculate the derivative of something like 4x3.  In this case, you'd need to draw upon this rule, as well as the power rule from my last post.  There are a few other basic differentiation rules like this one that I will cover in my next posts.  Learn all of these rules well, and you'll have no problem differentiating complicated functions!

One final comment - if you thought this post was helpful in any way, then please do me a favour and click on the Like and +1 buttons found on this page!  Thanks for your support and for visiting.  Be sure to come back if you need help with any other maths concepts.

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