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 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.