![]() ![]() Going to get smaller and smaller and smaller. As your change of x get smallerĪnd smaller and smaller, you're going to get a betterĪnd better approximation. Of x gets smaller and smaller and smaller. The slope of this curve is constantly changing. But that would justīe an approximation, because you see that Me take some change of x, and figure out what theĬhange of y is around it, or as we go past that. And you could do thatīy saying, OK, well, what is the change in y over theĬhange of x right around this? So you could say, well, let To approximate the slope right over here. But you see it's notĪ trivial thing to do. His fastest point in this 9.58 seconds isĬloser to 30 miles per hour. So the slope over here mightīe 23 whatever miles per hour. Velocity, is actually closer to 30 miles per hour. ![]() And I looked it up, Usainīolt's instantaneous velocity, his peak instantaneous Tangent line at that point, it looks higher Then over here, then heĪccelerates over here, it seems like his rate ofĬhange of distance, which would be roughly. He has a slower rate of change of distance. And what we calculatedĪt any given moment the slope is actually different. Plotted against time might be a curve that You'll see the slope here is getting steeper and So the slope is going toīe a little bit lot lower than the average slope. He's not just going to goĪs soon as the gun fires, he's not just going to go 23 andġ/2 miles per hour all the way. Than instantaneous velocity, let's think about a potential So this isĪpproximately, and I'll write it this way- this Roughly a little over 23, about 23 and 1/2 miles per hour. ![]() Say how many miles per hour, there's roughly 1600- and Iĭon't know the exact number, but roughly 1600 Meters he can, if he were able to somehow Meters he's going in an hour, well there's 3,600 Of how fast this is, let me get the calculator back. Speed is different than instantaneous speed. So it's approximatelyġ0.4, and then the units are meters per second. So let me get theĬalculator on the screen. Out what that is, let me get the calculator out. It would be velocity if weĪlso specified the direction. Just rate of change, or you could view itĮven just follow the units, it gives you units The rise over the run you might have heard in your ![]() So our change in time isĮqual to 9.58 seconds. The change in distanceĮqual to 100 meters. If I have a line thatĬonnects these two points, this is the slope of that line. This is the slopeīetween these two points. Somewhat familiar to you from basic algebra. Thing as change in y over change in x from That are over here, we're saying y is distance. His change in distance over his change in time. Way, his average speed is just going to be Seconds, we'll assume that this is in seconds This gentleman is capable of traveling 100 Well at time zero he hasn't gone anywhere. Usain Bolt's distance as a function of time, Is equal to distance, but we'll see, especially Traditional algebra, let's draw a little graph here. Is not a super easy problem to address with Newton's actual original termįor differential calculus was the method of How fast is he going right now? And so this is what differentialĬalculus is all about. Speed was for the last second, or his average speed How fast is he going right now? Not just what his average Is the instantaneous rate of change of something? And in the case of Usain Bolt, And you might have not made theĪssociation with these three gentleman. The fastest human alive, and he's probably the fastest Usain Bolt, Jamaican sprinter, whose continuing to do some Most of their major work in the late 1600s. Maybe not as famous, but maybe should be,įamous German philosopher and mathematician, and he wasĪ contemporary of Isaac Newton. Here’s a great video series for the history of mathematics, from Pythagoras to calculus and beyond: Since there are a lot of curves in position-speed-acceleration calculations, calculus proves an extremely valuable tool with which to analyze and predict behavior but it wasn't really the precipitating factor in its creation, for one main reason that there really wasn't the ability to precisely measure time on small scales.Īlthough this book is specifically centered about the evolution of "e", the author does a great job of tying the discovery of logarithm properties to modern calculus methods by building up the history of calculus from its earliest ancient dabbling to its explosion in the 17th and 18th centuries. Calculus for the most part was rooted in the ancient obsession with trying to square the circle (reduce the area of a circle to a simple x*y formula) as well as computing areas and tangents to other curves. ![]()
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