As you'll see, calculus is really about developing as fully and completely as you possibly can, an understanding of how things change. We work with functions which as you know provide a link or connection between inputs and outputs. In this video, we explore the way outputs of functions change on average as we vary the inputs. We restrict inputs to some given finite interval of the real line and capture this idea of averaging by finding the slope of the line joining two points on the graph of the function. This leads to the notion of the average rate of change of a function. Graphs of functions visualize relationships between quantities represented by variables say x and y. Here's the graph of some fairly wild fluctuating function y equals f of x. Let's just focus on an interval of inputs x between a and b, where a is less than b. There are two points on the curve with inputs x equal to a and b with outputs f of a and f of b respectively. We can tidy things up by throwing away extraneous points restricting the domain to the interval from a to b. We'd like to know or try to gain insight about how one quantity affects or causes changes in another quantity. This is especially important if you want to make predictions. The reserve bank wants to know, what effect changing the base interest rate might have on certain measures associated with the economy. It could do some kind of mathematical modeling to make reasonable estimates. Almost certainly sophisticated functional relationships will come into play and the functions could involve many variables. In our case we simplify things and have just two variables. An independent variable x moving between a and b and a dependent variable y whose values are constrained by the graph pictured here. Now, that is produced by functions like this one can fluctuate a lot. We can ask on average about the overall fluctuation. Here f of b is bigger than f of a but how much bigger? Well, the overall change in y values is the difference f of b minus f of a. But we want to interpret this in context. The rate at which this change has occurred is relative to the length of the interval which is the difference b minus a. The shorter the interval, the faster we think of this change as occurring, the longer the interval the slower we think of this as occurring. To capture the average rate of change we should divide the change or difference in the outputs by the change or difference in the inputs. This becomes a formal definition. The average rate of change y equals f of x of the interval from a to b is f of b minus f of a divided by b minus a. In a moment you'll realize this is an instance of something familiar. If we joined the endpoints color pink in the diagram by straight line color blue, then this quotient is just the slope of that blue line segment. The vertical rise divided by the horizontal run. We can think of the line between the endpoints of the curve as smoothing out or ignoring wriggles or fluctuations. So, what? How does this relate to reality? Well, road traffic authorities are very interested in the concept of average rate of change. Here's a graph of the function in the x y plane. It describes one of my car trips from Sydney to Melbourne. The curve emanates from the origin. Vertical axis denotes distance in kilometers from the starting point in Sydney corresponding to zero. The horizontal axis denotes time since the journey began. Observed that 873 kilometers is the full length of the journey finishing in Melbourne. On the horizontal axis zero hours, minutes and seconds matches the usual zero. The end point of the domain on the x axis denotes the total time spent on a journey which happens to be just over 11 and a half hours. Notation for counting time in hours, minutes and seconds originated with Babylonian mathematicians who invented the base 60 counting system. Based on dividing units into 60 parts an hour is 60 minutes and a minute is 60 seconds. At this highlighted part of the journey where the graph is flat, the car isn't moving and I'm at Holbreak, just over halfway to Melbourne, having a rest. Here's another flat part, Benalla having an even longer rest. Even though there's a lot of fluctuation including periods of not moving, we can still ask about the overall average rate of change associated with this journey. We join the endpoints using a straight line and find its slope, which is 873 divided by just overall 11 and a half hours, which you can calculate to be approximately 75.4 kilometers per hour. This says that of the entire journey, my average speed was about 75 kilometers per hour. Let's zoom in on this part of the graph highlighted here which looks roughly like a piece of straight line but with a steeper slope than the average. Here is the highlighted piece enlarged and tidied up a little. The endpoints in fact correspond to speed cameras located at Coolac and North Gundagai. These are average speed cameras which means that they are used to work out exactly the average rate of change. That is the average speed for that part of the journey. If it turns out to be more than the speed limits, 110 kilometers per hour, then the driver is breaking the law and receives a fine in the mail a few days later. Let's say what happened in my case. I drove fairly smoothly but not exactly uniformly over that stretch. To get the average rate of change, we join the endpoints and find the slope. Which is the vertical rise over the horizontal run, which becomes 16 kilometers divided by nine minutes and nine seconds. The denominator converts to a decimal and the fraction evaluates to about 104.9 kilometers per hour. So my average speed for this segment appears to be just under 105 kilometers per hour, safely within the speed limit. But we have to be cautious about accuracy. Noticed that though the time has been measured very accurately, and indeed the Babylonians had a reputation for accuracy using their base 60 system, the distance on the y axis is only quoted to the nearest kilometer. Suppose for example that the true distance to the camera at Coolac was slightly less than the whole number on the y axis and the true distance to the camera at North Gundagai was slightly more, then we have to make some adjustments to the previous calculation. We get an approximation of 110.2 kilometers per hour, which in fact is over the speed limit. You can see how sensitive these calculations can be with regard to possibilities of rounding errors. So, one must exercise caution. Estimates of about 110 and 105 agree fairly well to two significant figures. So, the mathematics is working but the outcomes in terms of staying within the speed limit could be drastically different. There are a couple of important issues. The first is the sensitivity of conclusions to rounding errors which we've just discussed. The second issue is that even if the estimate of 105 kilometers per hours correct, it's only an average over that leg of the journey, and it's possible to exceed the speed limit at some moment. The graph isn't a straight line. If you are looking at the speedometer of the car you would notice that my instantaneous speed is in fact fluctuating from moment to moment. It's possible even though the average is about 105, that the speedometer was reading over 110 at some moment. We want to work towards understanding what we might mean by instantaneous speed. This will be formalized soon using the notion of a derivative. Here's a sneak preview. If you take a line segment with the same slope as the average and move it across the curve, there will be at least one moment in the journey when it becomes a tangent line to the graph. On my journey between Coolac and North Gundagai you can see this occurred five times. At those moments the speedometer of my car will be showing an instantaneous speed that agreed with the overall average of about 105 kilometers per hour. But if you take a series of snapshots at a cascade of moments, the tangent lines can have varying slopes. Sometimes steeper and sometimes shallower than the average. It's the slopes of these tiny miniature tangent lines that become the speeds that are showing instant by instant on speedometer. If your average speed exceeded 110 then you can be absolutely sure that at some moment your speedometer was showing this speed in excess of 110, which is the basis for recieving a fine. But if the average speed was less, there's no way of knowing for sure that the instantaneous speed was always less than 110. If you're used to glancing at speedometers either as a driver or as a passenger, then you already have first hand practical experience with derivatives. We'll work towards formalizing this later in the module. Today we discussed the average rate of change of a function defined on an interval, and saw how to interpret this in terms of the slope of the line joining the endpoints of the curve. This gives us information about how quantities change on average. We looked in detail at an example involving a car trip, interpreting the average rate of change in terms of average speeds over certain legs of the journey. When we start to think about instantaneous speeds or rates of change such as what you see on the speedometer in a car, then we're well on the way to formalizing the notion of a derivative.. Please read the notes accompanying this video and when you're ready please attend the exercises. Thank you very much for watching and I look forward to seeing you again soon.
