Monthly Archives: August 2011

The Beauty of Openness and Real-Time

Lately I have been repeatedly impressed by the interaction and utility of Twitter. I have also been asked by numerous folks what the point of Twitter is and why I would waste my time with it. I hope to point out two areas that have struck me lately about what is really distinct and valueable about Twitter.

First I love what I call the “Openness” of it. With Facebook I can only interact with my “friends” or group-mates. With LinkedIn I can only interact with my “connections” unless I want to put someone in the awkward position of formally facilitating an introduction. Both have these barriers to meeting new people. Then you have Twitter. With Twitter I can interact with anyone, anywhere, anytime. It is such a great platform for finding or being found by new and interesting people and connecting with them.

Second, if you follow the right people and monitor your stream, you can get real-time information about what is going on anywhere. An excellent example of this is the other day I started getting updates in my feed about an accident here in town. People were warning others about it so that they could avoid the traffic congestion. They provided information about its exact location, what lanes were blocked, photographs from smartphones, highway cameras, and news coverage. Shortly AFTER I noticed this trending topic in my feed, we got a call to go and investigate the accident. Well now because of Twitter we had all this real-time data prior to even going to the site to investigate. Obviously we have to be objective and base our investigation and analysis on evidence and facts, not tweets, but the data was useful nonetheless.

I’m sure these are just some of the benefits of Twitter. Do you know of some others? Please share them in the comments. Do you agree with the two I pointed out?

Basic Derivative Rules

Previously we have discussed how a derivative is found using the limit, why the derivative is useful, and how to find it. Now to attempt to simplify or shortcut the derivative finding process, we will look at some general rules that can be used instead of going through the entire limit step-by-step every time.

The first derivative we will look at is the derivative of a linear function in slope-intercept form.

Where m is the slope and b is the y-intercept. Using the limit definition for the derivative we have:

Therefore the derivative of any linear function is, as we would expect, equal to the slope of that line. The next derivative we will look at is a special case of the linear function. If the slope is zero then the linear function is a constant function.

As above, the derivative of a linear function is its slope therefore the derivative of a constant function is also equal to the slope which is zero.

The next rule we can look at is a single independent variable raised to any power. Hence this rule is often called the Power Rule.

Applying the limit process we obtain:

Notice how we used the Binomial Theorem to expand the first term in the numerator. At this point I want us to notice a certain pattern in the coefficients that can be seen in Pascal’s Triangle. The first coefficient will always be 1 (c1=1). This can be seen by looking at the left edge of the triangle. Every first number is 1. Secondly if you look at the diagonal row parallel to, directly adjacent to, and to the right of this first row of 1’s, you notice that the numbers count sequentially as you descend the horizontal rows of the triangle: 1, 2,3,4,5,..etc. These numbers correspond to the second coefficient in the binomial expansion. If we label the horizontal row of the very first 1 at the top of Pascal’s Triangle zero and then incrementally as we go down the triangle, the second row will provide the coefficients for the binomial squared, the third row for the binomial cubed, the forth row for the binomial to the fourth power, and so on such that the nth row gives the coefficients for the binomial to the nth power and the second coefficient will be n also (c2=n). From here we have:

So you can see that the derivative of any variable to an exponent is equal to that same variable times the original exponent to the power of one less than the original exponent. For example:

In summary we can use the following rules to make our lives easier with regard to finding derivatives.

Next time we’ll look at some more helpful derivative rules.

The Binomial Theorem and Pascal’s Triangle

A binomial is a polynomial of two terms. Hence the bi- prefix. In general form it looks something like this: (a+b). If we multiply two of these together we obtain:

If we multiply three we have:

With four we have:

If we look at this examples we start to see some patterns. The number of terms in the result of 2 binomials multiplied together is 3, of 3 binomials is 4, and of 4 binomials is 5. The first term in the binomial starts in the first term of the result to the power equal to the number of multiplied binomials. Its power then successively decreases by 1 as you move through each term left to right and it is not present in the last term. In a similar manner, the second term of the binomial is found in the last term to the power equal to the number of multiplied binomials. Its power then term-by-term decreases from right to left and is not present in the first term. The coefficients at first seem to have no particular pattern besides they seem mirrored or symmetrical (1,2,1 or 1,3,3,1 or 1,4,6,4,1). In general, we can write these binomial expansions or the binomial theorem as:

where the c’s represent the coefficients.

A useful and interesting tool for determining the coefficients of a binomial expansion is Pascal’s Triangle. Blaise Pascal’s original triangle is seen in the image at the beginning of this post. We will look at and discuss the triangle in the following form.

One way of looking at how this triangle is formed is by starting with the single 1 by itself at the top of the triangle. If you add this 1 to the nothing or unseen zero on it’s left, you get 1 and this is placed under the intersection of the top 1 and this unseen zero. Likewise, the top 1 and the unseen zero to the right are added and result in the 1 to the right in the second row. The process continues all the way down the triangle adding two adjacent numbers and placing the result under those number’s intersection.

If you look carefully, you will see that the coefficients for the binomial multiplication can be found in the rows of Pascal’s Triangle. For instance the third row corresponds to the coefficients of the binomial to the power of 2, the fourth row to the power of 3, and the fifth row to the power of 4. So if we want to find the coefficients for the binomial to the power of 10, we just look at the 11th row and we have them.

Maximizing and Minimizing

In the last post about the derivative we emphasized its utility with some examples about how it can be used to find some relationships between position, velocity, and acceleration or to maximize profit and minimize cost in numerous different practical situations. Well now I want to discuss that in more depth and provide an example. Last time we used the following definition for the derivative (or slope) using the limit:

With this we can find the exact slope at any point on a curve. Now consider the following graph. Each of the red lines represents the slope at that point with a tangent line

Consider what the slope of such a tangent line would be at a minimum or maximum on a curve. That’s right, the line would be flat and horizontal like this:

Therefore the slope at these points is zero. With this geometrical understanding of a minimum or maximum we can find the derivative and then determine where the derivative (or slope) is equal to zero.

Now let’s consider an example function that relates the production level of a manufacturer to the potential revenue per unit. Obviously we want to maximize revenue so the key will be to find the maximum. Here is the function:

Now you might ask, “Well how in the world did we know this tells us about this relationship?” The short answer is that after some experimenting by the company at various different production levels and measuring the resultant revenue, the corresponding data points were plotted and the following function was mathematically obtained as a best-fit approximation to describe those data points. (More on this in a future post.) So assuming this is the correct function that we are working with, let’s first find the derivative.

Now this derivative gives us the slope at any point along the curve of the original function above. Where does this new function equal zero? This is the same as finding the roots of the equation which is a typical problem in algebra. Here is how it’s done based on our example.

Now if we put these two production level values into the original function it will tell us the revenue produced. Here are the results.

Therefore production level 2 provides the maximum revenue potential. (The root zero is an interesting point called an inflection point. But that’s another topic.) To demonstrate that this process has in fact located the maximum, here is a table of revenue values at other production levels and a graph of the function.

Next time we will look more in depth at the derivative and some basic rules or short-cuts we can use so that we don’t have to go through the entire limit process every time we want to find the derivative.

The Derivative and Why You Should Care

The first post in this series discussed the idea of the limit. And we tried to present it in a fairly simple, intuitive manner. We also discussed how a lot of the difficulty people ascribe to to calculus is due to the symbology. Above in the header you will see 4 different ways of writing the first derivative. However don’t be dismayed, just stick with me and we’ll see the derivative is not really that bad. Now with reference to the limit you might ask, what does getting really close to some point of anomaly have to do with derivatives? And what is a derivative anyway? And why should I care about all this math-ese? I suppose to motivate you to read on, I will first speak toward the usefulness of this concept.

The derivative allows us to consider the way one quantity or variable responds with regard to another quantity or variable. For instance, consider distance and speed. The more distance I cover in a period of time, the greater my speed. (For more on the relationship between distance, time, speed and acceleration see this post.) Or how about manufacturing: the less material I waste, the less my costs. Or maybe fuel economy: at a low speed I consume a lot of fuel. At a really high speed I consume a lot of fuel. But at some optimum intermediary speed, I maximize my fuel economy. Or maybe designing a structure for maximum strength with minimum weight. Or an electrical system that delivers adequate power with minimum infrastructure. All of these problems and many more utilize the concept of the derivative to arrive at their solutions.

Now with the hope that you are sufficiently convinced of its usefulness, let’s look at what exactly and derivative is. Again, I want to present it in such a way that you can intuitively grasp the idea. Take a look at the following graph.

Now the red line represents the function or the relationship between the input or independent variable, x, and the output or dependent variable, y. There are two of these key points that we want to look at. First at x=a, the function value is y=f(a). We use generic symbols so it will work for any function at any point. We will take a look at a particular function shortly to try and see this less abstractly, but for now let’s continue on. The second point is at x=a+h we get y=f(a+h). Now suppose we are curious to find the slope of the function curve (red line) at the point (a,f(a)) (the line labeled T). A rough approximation could be made by finding the slope of the line between the two points we mentioned. Recall that the slope of a line is equal to the rise divided by the run. The rise then will be equal to the difference in the two y-values, in particular f(a+h) – f(a). The run will be the difference in the x-values, in particular a+h-a which is equal to h. Therefore the approximate slope could be written as:

Now look at the graph and imagine that the point (a+h, f(a+h)) moved closer and closer and closer to the point (a,f(a)). Wouldn’t this give us a better approximation of the slope at (a, f(a))? I bet you have already noticed what concept I’m alluding to. That’s right, the limit. If we use the limit to bring the two points closer and closer together (in fact we will bring them so close together that the distance between them approaches nothing) we will have the exact slope at the first point. This is how we write that:

Now I promised an example so let’s look at a simple one. Let’s use:

Using the definition above we have:

So just like that we have found the slope or derivative of the function x squared which is 2x (“a” was just used above for a particular point, but it could really be any point “x”). In this same fashion we can find the derivative or slope of any function (provided it’s continuous – more on that later.)

But now you might say, “What does finding slope have to do with all the useful stuff you talked about at first?” We will dig into that next time.

Fluid You Can Walk On

In fluid mechanics we learn about the viscosity of a fluid. In an intuitive way we know that viscosity has to do with the rate at which a fluid will flow. For instance water will flow much more easily and quickly out of a cup than something like oil or honey. Therefore we would say that honey is more viscous. In engineering terms, we define viscosity (more precisely, absolute or dynamic viscosity) as

the constant of proportionality between the shearing stress and the rate of shearing strain.

With solids we talk about the shear modulus of elasticity, G,  as being the slope or constant of proportionality between the shear stress and shear strain. Since fluids will theoretically infinitely shear strain under a constant shear stress (which is the definition of a fluid) we must compare shear strain rate instead of shear strain. And much like with solids, the slope of the graph is often a straight line and thus a constant. This slope or constant is the absolute viscosity. Fluids with this linear type of behavior are called Newtonian Fluids.

Now this is where the interesting part comes in. Not all fluids act in this nice neat fashion. This other group of fluids are called Non-Newtonian Fluids and there are two categories. The slope of the shear stress / shear strain rate graph for Non-Newtonian Fluids is called the apparent viscosity and it changes at every point along the graph. The first group of Non-Newtonian Fluids are called shear thinning fluids or pseudoplastic fluids. With shear thinning fluids, the apparent viscosity decreases as you increase the rate of shear strain. In other words, the faster you push through it, the easier it will be. Latex paint is an example of this type of fluid. The second group is called shear thickening fluids or dilatants. These are the opposite of the other group in that the faster you push through the fluid, the harder it is to push. This results in the odd but fascinating phenomenon you can see in the following video clips.

Walking on Fluid

http://youtu.be/f2XQ97XHjVw

Make Your Own Non-Newtonian Fluid at Home

http://youtu.be/hvJikar9Vqk?t=5s

The Limit: For When You Get Really Close, But Not Quite

 

Mention calculus and many people’s eyes glaze over and they fain their inability to handle such “difficult” and/or “boring” math. In actuality calculus is no more difficult than any elementary math that most people did very well at some point during their middle or high school educational experience and it is only boring because teacher typically fail to demonstrate the useful, relevant purpose for all the mathematical mumbo-jumbo. I think a lot of the initial hesitancy is also a result of unfamiliarity with the symbols that are used. I hope to explain in relatively simple terms some of the terminology and symbology associated with the study of calculus and show how its not that complicated and why it is useful. In this post we will start with the fundamental idea of a limit.

Starting with the basic idea of a function, f(x), where f is the function and x is the input variable or what is sometimes called the independent variable (Independent because it does not depend on anything else. We can put in the function whatever we want.) We often see a function written something like this:

Here is the graph or plot of this function:

It appears that the graph is a simple parabola however if we look at the function definition closely we see that an x-value of 1 will cause us a problem. We will get a zero in the denominator and that can’t be. Therefore this function is not defined at x=1. This is where the limit comes into play. It is very useful for looking at these odd points or areas and drawing some conclusions about what is going on there. Fundamentally what the limit does is says, “Well at this point there seems to be no value, but what about really close to that point…what about even closer…and closer…and closer. Basically what happens as we get super close to the input value without actually using the value.” We write this idea with the following symbols:

where f(x) is still the function “lim” is the limit operator and L is the limit value. The subtext x->a means the limit as the input x gets really close to the value a. Let’s use the above function as an example and consider its limit as it approaches the undefined point.

To get an intuitive idea of what we are doing let’s use some values that are really close to 1 and see what we get.

As you can see here it appears that as we get really close to 1 for our input value, the output or function value seems to approach 1. Therefore in a non-rigorous way we have an idea about what the limit means and what it is and we would write the limit for this as follows:

Also note here that in the table I approached the undefined input value from both above and below. If you are looking at the graph of the function this would be the same as saying I approached the undefined point from the left and from the right. This is important to note for future discussions about the limit. I hope you have seen from this post that these ideas are not all that scary and can be understood fairly easily. In future posts we will look at the limit in more detail, see how it is fundamental to all of calculus, and see how calculus is actually practical and useful.