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.
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.