Tangent Lines – Slope of a Graph at a Poin
We begin our tangent lines series.
The key idea here, for the moment, is to imagine that we have a graph of a function y = f(x), and it’s shown in green. Now suppose we take a particular point x = a. So right here is x = a. We want to ask how fast f(x) changes at x = a. It’s a challenging question that we often take for granted. It’s like saying that, at this point in the road, I’m moving 55 miles per hour. That doesn’t mean I will have traveled 55 miles in the next hour or that it will take one hour to travel the next 55 miles. It says I am currently moving at 55 miles per hour. It’s a very tricky concept. Anyway, we often use calculus to describe an instantaneous rate of change. The graph of a function f(x) has a tangent line at x = a if the graph’s slope is equal to the slope of the line tangent to the graph at that point. It is this red line I’ve drawn right here. That’s a key geometric concept. If you think back, way back to a video several videos ago, we learned that the only thing we know how to take slopes of are lines, and we understand how to take the slope of a line. This is a line, and its slope is the instantaneous rate of change. That’s what we’ve got here. This is the tangent line to the graph of the function at the point and its slope gives us the rate at which the function changes there. If you understand this idea, it is the only thing that matters. Even though the graph does not present a straight line, it is possible to draw lines tangent at each point on the graph and their slopes are what we call instantaneous rates of change. This is also called the derivative of the function at that point, and at this symbol here represents it: f(a). Now how do you actually compute the derivative of f(a)? That’s tricky because if you want to compute a slope of a line, you need two points – on the line – and all you have is this one point here. This tricky formula is how you solve it. Don’t be intimidated by the limit we’ll unpack that idea as we work through the screens. Here are the key concepts to keep in mind. In order to work up to a more complex example, let me start with a simpler one. Suppose we have the equation y = 3x. For the moment, let’s consider a fanciful example. Any business majors in the audience should not be offended by this simplification. Let’s pretend that this model represents the revenue generated by selling a particular item. I want to sell it for $x-that’s the price- and then I want to see how much revenue comes in. Okay, so if I increase the price, then the revenue goes up forevermore. Although y = 3x is an unrealistic function, notice that it’s simply the graph of y = 3x. Therefore, the price should be set at $1,000.000,000.000.000. One million bajillion trillion dollars. Of course, that cannot be realistic. Now suppose that my business has set the price of the item at dollars. I want to ask whether increasing the price will increase revenue. Obviously it will, so let me ask how much. What is the rate of change at this point? It is a simple question with a line. Let us see why that is true. Let us go to the point on the graph for which a=3a. Now suppose I increase the price of my unit by $1, so that we have a+ 1. Where is that point located on the graph? That was our a+1 and 3(a+1). Now let’s calculate the slope of this line. To compute the slope of that line segment, I first need to calculate the difference in y values by subtracting 3a from 3(a+1). Then I will divide that result by the difference in x values by subtracting at 1 – a. And if you solve the algebra problem – not really the point here – you will get three. The key idea here is that there is nothing special about increasing a dollar; if I increase two dollars, the rise-over-run calculation will be exactly the same – that is what makes lines special. The slope of the line is not important. If we increase $1, the revenue will go up three times $1. If we increase $2, the revenue will go up three times $2. The slope of the line merely represents this relationship. If you remember back to the slope-of-line videos, you’ll recall that the slope of a line is its steepness. That’s a key point about lines. Let’s return to a model that is more realistic. So if you look at the same picture as in the very first, notice that this time I’ve added y = f(x), and I said that I’m graphing price versus revenue. On this green curve, I pointed out to you that this is a more realistic price-revenue curve. It says that normally when I raise my prices, my revenue goes up as I get more money for each product sold. However, eventually I reach a point where I’m raising prices so much that people are mad at me and say, “I’m not really going to buy your product. It should not be that expensive” The revenue stream shows a gradual increase until it starts to decline. That is what graph represents. Let us now ask the same question again: if my price is $1 and I want to raise it, then does my revenue increase? And if so, by how much? Does it decrease? And if so, by how much? To put it another way, what is the revenue’s instantaneous rate of change at this price point? The following is an important consideration. What makes this different from a line? The answer depends on my price point. Suppose my price point was right here – at B – and let’s look right up there. If I increase the price from B, my revenue will increase but not as much as if I were to increase it from a. And it seesm even worse over here at c. If my price is already at the maximum point of my target market’s willingness to pay and I raise my price, I will lose revenue. So it is a bad idea for me to put it at that price. Business majors would likely argue that it is a bad idea to have a price at C in the first place–that’s another matter. The key mathematical takeaway, however, is that the slope of the tangent line changes depending on where you are on the curve. Notice that’s not the case when my green curve was a line, but it is true here. The answer to the question, ‘By how much will my revenue increase if I increase my price a little bit from the current price point?’ is the slope of this red line. That’s what the derivative of the function at the price value x=a means. The derivative of a function at a given point is denoted f(a), where f(x) is the original function and x=a. Great. So it has a name, but now we need to figure out how to calculate it. How to calculate the slope of the line segment between slopes of lines? That is a fair question: I think you are all pretty good at that by now. But here is unfair one. Calculate the slope of that red line using only the coordinates of that red dot. I will now explain to you what a is, explain what f(a) is, and then ask you to give me the slope of the line. I understand that you are upset with me because I asked that question. So let me ask you another question. Although the answer won’t be the answer you want, what I am going to do is draw a little line segment between those two points. I’m gonna take another point of the line and I’m gonna draw a little line segment. Therefore, we can think about what all those points are. Let us say that here is a point ath. h stands for a little bit. In other words, right here, this is h Let us consider the coordinates of this point. We know that the coordinates for this point are (a, f(a). We also know that the coordinates for this point are (ath, f(ath)). The slope of the line segment is equal to the difference in y values divided by the difference in x values. The difference in y values is (f(ath) – f(a)), and the difference in x values is (a+h – a), so we can set those equal to each other and solve for h, which will give us the slope of the line segment. Let’s pause here for a moment to consider the situation. I don’t know the slope of the tangent line, but I would like to find it. I know the slope of this piece of the curve if you give me values for the two endpoints, but if you give me values for the function itself, I can find the slope at any point along it. Here is where the concept of limits comes into play. A mathematician can write an equation, a = sign, and say that h approaches zero. Later we’ll calculate exactly what this means. But for now, the really conceptual point is that it says I don’t really want this point to be here. I want h to be zero because if I move h toward zero, this little line segment snaps toward pointing in the same direction as the tangent line. As I move h towards zero, the slope of that little line segment snaps to the slope I care about. The limit as h goes to zero here is not h equal zero but rather that h gets closer and closer to zero. [Graph][Graph][Graph] [Graph]
