Negative Exponents and Reciprocals

So, I have taught you the trick to understanding negative exponents, and how you can use that information to more easily manipulate your expressions.  (In the same post, I also described how you can simplify your exponential equations wherever you see a base raised to a zero exponent.)  But I thought that I should maybe add a little bit more information to that, so that you can understand WHY an expression raised to a negative exponent is equal to its reciprocal base with a positive exponent, and not just simply know that it does.

My friend Guillermo has recently posted a similar topic on his Mathematics and Multimedia blog.

This concept is actually amazing simple.  At first, you might think that there is some long and calculated proof involved.  In actuality, this rule for working with negative exponents can easily be found merely by considering the Exponent Laws.  (You still remember them, right?  Refer back if you need a refresher!)  ;)

To keep things simple, I will provide a brief outline that will hopefully describe this concept sufficiently, though if not, I will defer you to Guillermo's page for more details.

Briefly, let's consider the general expression x^-n.  This can be rewritten to show that the "-n" is being taken away from 0... this is always just assumed whenever we consider a negative number, but write it out this time.  So, now we have x^0-n.  Now, using the Exponent Laws, we know that when we have a subtraction function in the exponent, this is the same as dividing the exponential terms (refer back to my Exponent Laws post, linked above!).  And having rearranged things now, you can call upon the property of the zero exponent (see link above!) to show that the numerator is simply just 1, leaving you with the reciprocal base of what we started with.  Those few short steps using the Exponent rules demonstrate how x^-n is related to 1/xⁿ.

Hopefully that quick walkthrough is a good enough guide to get you through those steps.  One thing to take away from this is just how important and useful the Exponent Laws are.  Make sure you understand them!

Using Limits to Find Tangents

Having discussed in several posts the concept of limits in mathematics, in this post, I am going to describe a more practical use for limits so that you can see how useful they can be.  To do this, I would like to discuss tangents to curves (not to be confused with the tangent trigonometric function!).  You are probably already familiar with the concept of a tangent line, but here is a more descriptive definition of it:  a tangent can be defined as a line (or a three dimensional plane) that touches a curve (or a surface) at a single point, but does not intersect it or cross it.

Our problem, though, is how to express this line (let's stay away from the 3D surfaces and planes for now!) in terms of the math that we already know.  We can always draw an approximation of a tangent line to a curve, but how can we precisely define that line and express it in terms of an equation?  This is where limits come in!

Let's consider a basic example.  Consider the parabola described by the function f(x) = x².  How can we describe the line that is tangent to the curve when x = 2?

Evaluating the tangent line to the parabola at the point (2,4)

Well, think about it this way.  How would you go about finding the equation of any other line?  It doesn't have to be a tangent or any other special line.  How would we find the equation of any line?  One of the first things that we'd need to know to write the equation of the line is the slope.  Now with that thought in mind, let's connect that to the concept of limits.

To find a slope of a line, we need two points and we calculate the slope m based on the differences in x and y values.  So, if we take any other point that lies on the curve, and we connect that point to the point where we want to evaluate the tangent line (here, at x=2), we would create a very rough approximation of our tangent line and can solve its slope knowing the coordinates of the two points.  (When we connect two points on a curve like this with a straight line, this line is referred to as a secant line.)  If we take points on the curve that are progressively closer and closer to our point of interest, the created secant lines more and more approximate the actual tangent line.  I've tried to demonstrate this in the following picture:  

Approximating the tangent slope by analyzing the slopes of secant lines

As you move along the curve from point A to B to C, the slope of the secant lines AP, BP, CP get closer and closer to the actual slope of the tangent at point P.  For good practice, try solving the equations of the lines for the 3 shown secants.  I have only plotted whole numbers, but you can better see the secant slopes approaching the tangent slope if you evaluate the lines formed by points even closer to P (try when x=1.9, then 1.99, then 1.999).  Once you can see what the slope of the tangent line will be, then it is only a matter of substituting that value, and the values of the ordered pair, into our favourite formula y=mx+b to solve for b, and then you can express the equation of the line in either standard form or point-slope form.  I have already done the work, and determined that the equation of the tangent line is y=4x-4.

So, hopefully you can see here the connection to limits!  As our extra points approach our point of interest, the slope of the secant approaches the slope of the tangent.  This is essentially what we do when we consider limits; that is, we are interested in what is happening as we get infinitely closer to our point of interest.  In this case, we can say that "the limit of m (slope), as x approaches 2, is 4."

With that, you can see that limits are not just some abstract mathematical concept designed to confuse students.  Limits actually can be useful, and in this case, we have seen that they can be used to determine the equation of a line at a single point!  

Please leave a comment if you would like a little more explanation or pointers.  Also, I'd appreciate anyone's input about my graphs... I am using a new program to draw them, and while I think they look good, I am still learning how it works.  And of course, please click the +1 below if you found this post helpful!  Thanks!

Asymptotes

When working with mathematics concepts such as graphing and limits, you will soon encounter lines called asymptotes.  Despite many students' first thought being "that's a weird word!" they are really quite a simple concept to recognize.  In fact, you have probably drawn asymptotes before on your graphs and not even realized that you were using such a fancy mathematical concept.

An asymptote, quite simply, is basically any line (or maybe think of it as a border?) that your graph will approach and get infinitely close to, but never quite reach.  Now, stop and think about this for a minute, and realize how much in common this definition for an asymptotic line has with our definition of a limit which I introduced in a previous post (and then expanded on in the follow up post that discussed limit notation).  Conceptually, they both deal with your given function getting infinitely close to something, and the distance between a point on your graph and either your limit "a" value or asymptote getting infinitesimally smaller but never zero.  I should probably also emphasize this point, because it is not something to be taken for granted either.  Make sure you understand the connection between two things getting closer and closer together without ever touching, and so therefore the distance between the two things gets smaller and smaller without ever reaching zero.  Because if the distance between two points is zero, that implies that they are in the same place or touching, with no space between them to differentiate them.  This may seem like a simple concept, but I want to make sure that everyone understands it.  (For a different perspective, check out Guillermo's post on asymptotes and how they relate to curve sketching!)

So, that is how you could define an asymptote.  But, maybe you'd like to see a picture that more clearly demonstrates asymptotic behaviour of lines.  Let me try to demonstrate briefly with a few graphs:

This first graph is a common one, showing the graph of y = 1/x.

In the graph of y = 1/x, you can see that x never touches the line x = 0.  In this case, you would say that the line x = 0 is called a vertical asymptote.  Similarly for the y values, you can see that y never touches the line y = 0 either, so you would call this a horizontal asymptote.

When drawing graphs, you will often be asked to identify any vertical asymptotes or horizontal asymptotes.  Just make sure you don't mix them up!  :)

Let's look at another one, slightly different this time:

This is related to the first graph.  It is a graph of y = 1/(x-5) +2.

This second example is based on the same function as the first, that is, y = 1/x.  However, it has been horizontally translated to the right 5 units, and vertically translated up 2 units.  In this graph, you can see that the horizontal asymptote is y = 2, and the vertical asymptote is x = 5.  (Even I sometimes get horizontal and vertical asymptotes mixed up, because I instinctively want to refer to the horizontal ones in terms of x, and the vertical ones in terms of y.  However, it's easier to think of it as 'does the line go up and down,' in which case it's a vertical asymptote located somewhere 'side to side' on the x axis.  Similarly, for horizontal asymptotes, they go 'side to side' and are located somewhere along the vertical y axis.)

What we've seen with these two examples, asymptotes are lines that your graph approaches yet will never intersect or touch.  More specifically, they are lines for which your equation is not defined.  In terms of mathematics, one way an equation is undefined is when its denominator equals zero.  So, with that piece of information, it is easier to see why our asymptotes in these equations are what they are.  The equations of these two graphs are undefined, that is, have a denominator of zero, along their asymptotes.

So, that makes reading graphs to determine their asymptotes easier.  However, what about if you are merely given an equation and asked to perform asymptotic analysis on it?  How could you find asymptotes of an equation without seeing it or drawing the graph first?  Well, as I just showed, it's actually quite easy because all you have to do is determine at what points the denominator will equal zero, and you will have then found where the equation is undefined and has an asymptote.

Let's try a brief example to practice asymptotic analysis of the equation.  If you have a graphing calculator and know how to use it, you can plot these out and check.  But otherwise, let's just look at the equations.

Find asymptotes of f(x) = 200 / (x³ - 8).

For each equation you want to evaluate, it's a good idea to consider if there are both horizontal and vertical asymptotes.  To do that, you have to rearrange the expression to solve for either x or for y, and then evaluate each form to determine where the denominator equals zero.  In this example, it is already expressed in terms of x (remember that you can say y = f(x)), so you can simply examine the bottom of this form of the equation, and solve it by equating it to zero:

x³ - 8 = 0
x³ = 8
x = 2

Therefore, this shows that there is a vertical asymptote along x = 2.

To see if there are any asymptotes along the y axis, you need to rearrange the equation to isolate for the other variable.  With some simple algebra, you can rewrite the expression like so:

y = 200 / (x³ - 8)
x³ - 8 = 200 / y
x³ = (200 / y) + 8

When written this way, it is easy to see that the expression will only be undefined when the term 200/y has a zero as the denominator.  And since this is a simple expression, we can easily see that 200/y is undefined when y = 0.

With this quick asymptotic analysis, we have shown that when x = 2, the denominator equates to zero, which then means that f(2) evaluates to 200 / 0, which is undefined.  And if f(x) is undefined when x = 2, we have determined that f(x) has a vertical asymptote at x = 2.  Similarly, we have shown that there is a horizontal asymptote when y = 0, because when we rearranged the original equation to write it in terms of y, we found again that the expression is undefined with the term 200 / 0 when y = 0.

Here is what this function looks like, all graphed out.  Check it out to verify that we identified the correct values for its asymptotes!

Check the graph to see if our asymptotes make sense.  Looks like the check out ok!

So, with that, you now should be familiar enough with asymptotic lines to be able to identify them in your graphs and equations.  Remember that they are similar to limits, in that they are something that something else approaches without ever making contact.  That is a very rough description, but if you can understand that basic concept, you can build on it to be able to apply it more precisely to your mathematics as I've shown in the above graphing examples.  Another very important thought to keep in your mind is that not every equation has an asymptote.  With a little practice and work, you will be able to look at an equation and be able to say exactly where to find its asymptotes, or even declare that the equation has no asymptotes at all.  Being able to identify asymptotes is a core concept in graphing (and is usually one of the first things you want to do when converting an equation to a graph), and will allow you to work through more complicated math questions, such as ones that ask you to "find domain of rational function" or questions that involve oblique asymptotes... ones that slant on an angle!

You will find asymptotic analysis to be very useful in your math studies, so I hope that this post has at least provided a decent introduction for you to this graphing concept.  As always, please +1 the post (below) if you found it useful.  Thanks!

Limits and Continuity

In this post, I am going to explain the concept of continuity in calculus in a bit more detail than when I touched on the subject in my previous post that explained one-sided limits.  I will have even more to say about the concept of continuity when I begin my series on derivatives soon, as derivatives can quite easily provide you with an assessment of the continuity of a graph.

As we've seen, limits calculus is a branch of mathematics that describes what a graph is doing as you approach a point on the graph and get infinitely close to it without ever actually reaching it.  It may be thought of as a way of predicting what a graph will do at a point, based on what the graph is doing in the general vicinity of that point.

For instance, if you imagine an upwards facing parabola, if you begin to trace the curve with your finger, moving towards the parabola's apex, you find that you seem to be approaching a predetermined point.  This holds true, in this example, if you are moving towards the apex from either the left side or from the right side.  Based on the curvature of the parabola, you can predict what the point will be as you approach it.  Also, you can see that the point actually is exactly what the line would predict it to be.  Since this is a simple example and very easy to visualize, you can intuitively understand just by looking at it that the graph is continuous.  There are no holes or anything weird going on along the length of the curve.  It is a smooth, continuous line.

However, if you consider a graph like the one I showed you when I talked about one-sided limits in calculus, you can very easily see, without any form of analysis at all, that the graph is discontinuous. You can't draw that line without lifting your pen off the paper.  In fact, this is a very crude but simple way of checking to see if a graph is continuous or discontinuous.

In calculus, limits can explain more than just what point a graph seems to be going towards.  Limits can be used to tell us about the continuity of a graph.  In fact, limits and continuity are very important parts of graph analysis.  If you want to know if a graph is continuous at a certain point, you merely need to look at the one-sided limits on both sides of that point, as well as the point itself.  If you look at the parabola above, and want to know if it's continuous at its apex, you consider the limit as you approach from the left, and also the limit as you approach from the right, and in this case you see that they both approach the apex.  There apex is also on the curve.  If the two one-sided limits equal the same number, and that number also evaluates to the same value at that point, you can conclude that the graph is continuous at that particular point.  For similar reasons, you can see that the second graph above is a discontinuous graph at x=5.

I will leave this example for you to draw and consider: imagine a normal parabola, as above, but it's apex is undefined (a hole in the graph).  The one-sided limits from both sides approach the same value, but at the apex the graph is undefined, which obviously doesn't equal the one-sided limits result.  Therefore, the graph would be discontinuous.

It should be fairly obvious, but I'd better state it here anyways, that this doesn't say anything about the continuity of the graph at any other point.  A graph could very well be continuous in one place and discontinuous somewhere else.  When examining the one-sided limits and the value at that point, you are only evaluating the continuity of the graph AT THAT POINT.  You are not evaluating the continuity of the graph as a whole.  It's safe to say, however, that if you have a point on a graph that is discontinuous, then you have a discontinuous graph overall.  To have a continuous function, you have to have continuity at every point on the graph.  That is to say, f(x) equals both of the one-sided limits for every value of x.

Most graphs that you will come across will be continuous, though you will need to pay special attention to functions that are piecewise as they may or may not be.  By definition, all polynomials are continuous.  So are rational functions (fractional expressions) except when the denominator is 0.  This is because a rational expression with 0 in the denominator is undefined, so it has no point on the graph.  So, obviously, it isn't continuous where there is no point!

So, to quickly summarize, to evaluate the continuity of a graph of f(x) at a specific point, you need to determine the one-sided limits as you approach that point from either side, and you also have to evaluate f(x) at that specific point.  If f(x) equals both one-sided limits, you have proven that the graph is continuous at that point.

I hope that this post makes sense and explains limits and continuity calculus for you.  If you need something to be explained better, or maybe a different example, please leave me a comment and I will add some more information.  Also, please +1 me below this if you found it useful, as it really helps me!  Thanks for reading.

One Sided Limits

Continuing on from my previous post in my series about limits in calculus, in this one I would like to better explain the concept of one-sided limits.  In my last post, I introduced you to the correct notation for writing a limit, though I mentioned that the notation that I was describing did not include any description of direction.

Recall my explanation for a limit, where I basically said that a limit (L) is the value that a graph appears to have at a specific point, assuming you were to follow along the curve and approach that point.  Kind of like making a prediction of the value of a point, based on what you know the curve is doing around it.  Now, with that definition in mind, you must realize that you can obviously approach the point in question from either side on the curve.  For simple equations (or more generally, continuous ones), you will approach the same limit value as you approach the "a" value from either side.  And this should intuitively make sense.  However, what I want you to consider now is something like the following example (a piecewise function):

What you can see in this example is that the value of f(x) for all values 5 and less equals 2, and for all values greater than 5 is 4.  Also, note the open and closed circle notation of the two lines.  The closed circle indicates that it includes 5 on that line.  The open circle indicates that it does not include 5 on the line, yet it contains all values up to 5, getting infinitely closer to it but yet never quite touching it.

So, from this example, now let me put the question to you: what is the limit of f(x) as x approaches 5?

To solve this question, you must consider one-sided limits, and examine the graph from either side of x=5.  If you consider what the graph "looks like it's going to do" as it approaches 5, you see that it kind of looks like it might be doing two different things.  (In fact, I will discuss continuity in separate post, though what we will see is that if you examine the limit at a value "a" from either side, and you determine that it is the same limit value for each one sided limit, then that is proof that the curve is continuous at "a").  If you approach x=5 from the left, you can see that it looks like the line will continue on as it has been doing, and the limit of f(x) when x=5 will be 2.  This is correct.  Consider the other part of the function, and approach x=5 from the right.  What does the limit appear to be?  Well, similarly to the left part, the right part appears to keep doing what it's doing, and the limit looks like it will be 4.  This is also correct.

So then, what I've shown here is that if I ask you to find the limit of f(x) as x approaches 5, you cannot give a single answer.  There are obviously two seemingly correct answers to this.  And so to distinguish between the two, you use the concept of one sided limits.  (Again, as I will explain in my post on continuity, you can obviously see by looking at the graph that the one sided limits on each side of 5 are not the same value, and therefore you can say that the graph is discontinuous at x=5.  Looking at the graph, you can immediately see this is true.)  You would say "as x approaches 5 from the right", or "as x approaches 5 from the left."  It is not overly complicated, but it is something to be aware of.

To make things less confusing, of course there is a correct notation to indicate one sided limits.  Thankfully, it is only a very small modification to the limits notation that I have already presented to you.  Specifically, all you have to do is add a "+" or a "-" to indicate which side of the point "a" you are approaching from.  It may be more helpful to consider the notation as referring to "all values greater than a" for the +, and "all values less than a" for the -.

Here is what the one sided limit notation looks like, for when x approaches a from either the right or the left, respectively:

 

This is a commonly used notation in calculus, and you will frequently see it when you are doing your limits mathematics homework.  Hopefully, now you will be able to understand the notation when you see it and be able to instantly translate it and apply it to what you see in the graph.  As I mentioned above, I will next provide a more formal discussion about the concept of continuity of graphs, and how you can determine whether a graph is continuous or not based on its equation.  As always, please remember to +1 this post if you found it useful!  Thanks!