Rates of Change By definition, a f unction i s s imply a r elation b etween t he p oints o r m embers o f t wo s ets. I n this setting, the output of a function at a single point may be completely unrelated to the output of the function at other points. However, if a function represents a physical quantity, then it’s reasonable to believe the outputs of the function for different points in the domain are related (at least in the realm of classical mechanics). One of the basic goals of calculus is finding and understanding these relationships, in order to understand the evolution of a system. Let us suppose we know the position of an object as a function of time, at all times. We know that there must be at least some relationship between the positions of the object for closely related times. In other words, the position of the object at a given time is definitely related to the position of the object 1 second later. We may not be able to find one p osition given the other, but we know the distance traveled can only be so great, as macroscopic objects do not teleport. A very natural and interesting question that arises is as follows: given the position of an object at all times, can we find i ts v elocity? R oughly s peaking, a n o bject’s velocity ( or s peed) i s a m easure o f h ow f ast it is moving, or changing positions. Noting that velocity (or speed) is simply the rate of change of position with respect to time, we certainly should be able to find it. Definition 0 .1.1 ( Average R ate o f C hange). T he a verage r ate o f c hange o f a f unction f o ver an interval [x0 , x1 ] is the ratio of its change in output (f (x1 )−f (x0 )) over change in input (x1 −x0 = h). Mathematically, Average Rate of Change = f (x0 + h) − f (x0 ) f (x1 ) − f (x0 ) = x1 − x0 h The above definition holds in a more general context, but for the moment we’re interested in a changing position. Definition (Velocity and Speed). The velocity of an object is the rate of change of its position with respect to time. The speed of an object is the magnitude of its velocity. According to the above definition, velocity describes how fast an object is moving, and in which direction, whereas speed simply denotes how fast an object is moving. Since velocity is the rate of change of position with respect to time, average velocity is the average rate of change of position with respect to time. How should we interpret average velocity? Let x(t) denote the position of an object D, and t represent time. The average velocity of an object over an interval [a, b] in time is given by vab = x(b) − x(a) x(a + h) − x(a) = , b−a h where h = b − a. If another object E is moving with a constant velocity vab , then over the interval of time [a, b], E will travel a distance of vab · h = x(a + h) − x(a) · h = x(a + h) − x(a) = x(b) − x(a), h which is exactly the distance that D travels over the interval [a, b]. In other words, the average velocity of an object over a given interval of time is the unique number such that for any other object, if the second object travels at a constant velocity of the average velocity for that interval of time, it will travel the same distance as the original object. The key thing to note here is that object D is not necessarily traveling at a constant velocity - it could be speeding up and slowing down. In the above discussion we are implicitly referencing some mysterious nonaverage velocity, which changes throughout an interval of time. In most physical situations, we are interested in this other velocity, the one that it is changing throughout an interval of time. This velocity is called an object’s instantaneous velocity. If we think about two objects colliding, we are interested in how fast the objects are moving right when they hit, not over an interval of time before they hit. Knowing the average velocity of the objects for 5 seconds before they collide doesn’t tell us much, because we don’t know whether the objects are moving much faster or slower than average when they collide. Nevertheless, the average velocity does tell us something. If over some small period of time the objects have a given average velocity, we know that the instantaneous velocities within that interval of time are somewhat close to the average, because of the physical limits on how rapidly the velocity of an object can change. Stated more precisely, the difference between the average and instantaneous velocities is limited, because objects have finite accelerations. Thus, the average velocity over a small interval provides an approximation of the instantaneous velocity of points inside the interval. In order to find the instantaneous velocity of an object at a point, we begin by finding the average velocity of the object over an interval including that point. This provides us with an approximation of the instantaneous velocity. By decreasing the size of the interval over which we calculate the average velocity, we will get another approximation. It is possible that this approximation may be worse than the original one, but on a whole, as we decrease the length of time over which we calculate average velocity, we should increase the accuracy of our approximation. We should be able to make the approximation as accurate as we like, simply by calculating the average velocity over small enough intervals. Given that vab = x(a + h) − x(a) , h we want to make h as close to 0 as possible, in order to increase the accuracy of our approximation. The instantaneous velocity is the value approached by these successive approximations, as h approaches 0. We can think of the above process in two different but equivalent ways. As described above, the instantaneous velocity is a property of the object that already exists, that we are able to find by using average velocities. Alternatively, we can simply define it as the velocity we find by looking at the average velocity over shorter and shorter intervals. Which viewpoint you choose to hold is simply a matter of preference. Although we have outlined a process for finding instantaneous velocity, there is still one more question we need to answer. Practically speaking, how do we find instantaneous velocity? There are fundamentally two different approaches we can take here. From a theoretical perspective, if we follow this process of approximation to its logical conclusion, we should be able to find the exact value of the instantaneous velocity. However, at the present time we do not have the tools to follow this process to its logical conclusion. The tool we need to do so is called a limit, which will be the fundamental concept of study in calculus. From a practical perspective, there are physical limits to how accurate we can make our approximations. These limitations may result from computers, measuring devices, etc. The study of such limitations and how to deal with them falls primarily in the realm of numerical analysis, so they will not be our primary focus of study.