Rates of ChangeThe heart of the sciences resides in change - upon observing the past and present behaviorof a system, we wish to make predictions about what will happen in the future. Once weunderstand the physics of a situation, there is great interest in exploiting this understandingto build our own systems, for whatever purpose we may desire; this is the task of appliedscientists, or engineers. This leads us to the question of how relate changing quantities.There are essentially two questions of this typ e:1. If we know the value of a quantity at all times, can we determine the rate at whichit changes at all times? For example, if we know the position of a moving body atall times, can we determine its instantaneous velocity (change in position) at each ofthese times? If so, how? Similarly, if we know the volume of water in a reservoir at alltimes, can we describe the flow of water in and out?2. If we know the way in which a quantity is changing at all times, can we determine thevalue of the quantity itself? For instance, if we know the velocity of a moving body atall times, can we find its position at all times?Let us try and make some headway on the first of these questions. Suppose it took you45 minutes to drive to class today, and you know that you drove 30 miles to class. Withthis information it is not difficult to c alculate your average speed was30 miles.75 hours= 40mileshour. Ingeneral, we calculate the average rate of change of a function f (x) over an interval [x1, x2]by dividing the change in f (∆f = f (x2) − f(x1)) by the length of the interval over whichit occured (∆x = x2− x1= h). Thus,∆f∆x=f(x2) − f(x1)x2− x1=f(x1+ h) − f (x1)h(geometrically, the line connecting these two points is called a secant line, and the averagerate of change between the two points is the slope of the secant line).This however, does not answer our first question. You likely reached some stoplights on theway to class, so we know at least at some times you were not moving; ie. your instanta-neous veloc ity (or speed) was 0. Thus, we know that at least at some points in time yourinstantaneous velocity differed from your average velocity (in fact, it like ly differed the vastmajority of the time). Suppose on your way to class, after 40 minutes into your journey yourcar hit a tree. In order to analyze this collision, we need to know exactly how fast your carwas traveling at that instant. If you were traveling at 5 mph the outcome would be verydifferent from if you were traveling at 60 mph.Let us shift our attention back to those times in your journey when your car was not mov-ing. Since your car is not moving, but time is passing, it is clear that time spent sittingat stoplight is slowing down your travel; ie. time spent at a stoplight reduces your averagespeed. Other than the fact that it takes some time for your car to accelerate, it’s not clearthat time spent at a stoplight should influence your current velocity. In fact, it makes verylittle sense. Thus, if enough time has passed between a stoplight and a given instant of timeto allow you to accelerate to the desired velocity, the fact that the car was stopped shouldhave no influence on the car’s current velocity. The key to making headway in the problemlies in the previous sentence: “if enough time has passed between a stoplight and a giveninstant of time.” Since there are limits to the rate at which your car can accelerate, if twoinstants in time are sufficiently close, there is a limit to the difference between the velocityof your car at those two times; if your car is stopped at a stoplight at one point in time, itis infeasible that your car is moving at 100 mph after 1 second has passed. Over a shorterperiod of time, it follows that your car’s velocity will have to be even closer to 0, as your carwill have had less time to accelerate.Using the above idea, we should be able to make a reasonably accurate estimate of our in-stantaneous velocity at a given time by considering the average velocity over a small intervalof time containing the point in time we are interested in. By decreasing the length of theinterval of time in question, we should be able to get increasingly accurate estimates of theinstantaneous velocity. However, this leads to a new problem. No matter how small aninterval of finite length is, it is possible to create a yet smaller interval. Thus, how can wetruly be sure we have found the exact value of the instantaneous velocity, rather than just anestimate? We would like to look at an interval of 0 length, except for the fact that we cannotdivide by 0. Using a computer we can calculate the value for increasingly small intervals, upto the point where we can feel reasonably comfortable with our estimate. Nevertheless, weare still considering an estimate, and would like to find an analytical solution.Unfortunately, we cannot solve this problem using just the tools currently at our disposal; wewill need to introduce a new concept, the limit. The concept is one of the most powerful inmathematics, and lies at the heart of calculus. Because all of calculus is built upon this con-cept, it is of the utmost important you invest the time required to have a cle ar understandingof the limit. The solution to the problem at hand is that the instantaneous velocity at apoint in time is given by the average rate of change over an interval of time around that time,in the limit that the length of that interval approaches 0. We will have to digress from thisproblem for now, and return to it when we have sufficiently de veloped this new tool, the