![]() ![]() ![]() ![]() A secant is a line made through two points on a curve. In any mathematical function, the derivative is found by taking a secant on the curve near the point in question. We can take multiple derivatives and find the acceleration (2nd derivative) and jerk (3rd derivative, yup its a real thing) of a function as well. time chart and want to find the velocity (tangent) at any point. The derivative where one can find the instantaneous slope or tangent of any mathematical function. Now let’s get on with the two parts of calculus! The Derivative Limits allow us to divide the function into smaller and smaller chunks of the function to evaluate things like slope and area. In calculus terms, we would express this as, “What is the limit of y, as x approaches 0.” If it is a discrete equation, you may need to evaluate the function “from the left” (meaning x values increasing) and / or “from the right” ( x values decreasing) to get the right answer. However, if we move just slightly to the right (or left) we see that the value of y is just less than 1. If we look at the equation below, we can see that the value of y when x = 0 is 0/0 or undefined. However, as the defense does get penalized, the offense gets a little closer to the intended “limit.” Image Courtesy of Jeffery Beall The main thing to keep in mind here is no matter how many times the defense is penalized, the offense will never score. If the defense keeps getting penalized every play, the ball will move closer to the goal line by half each time 6.3, 3.3, 1.6, 0.8, 0.4, 0.2 etc. If the offense is on the 25 yard line and the defense is penalized for ‘unsportsmanlike’ conduct (normally a 15 yd penalty), the ball will be placed half the distance to the goal on the 12.5 yard line and not the 10 yard line. Image Courtesy of Braindrain0000Īnother example of a limit is in (American) Football. Notice the shaded ares get smaller with as the number of sides increase. If you go to a million sided polygon, your approximation is very good. As you increase the sides to 6 (a hexagon), your approximation gets better. For example, if we want to estimate the area of a circle by inscribing an equilateral triangle in it, you will have a very bad approximation of the area. The smaller the chunks are, the better accuracy. The limit is the process of dividing a function into chunks. Before we can dive into those topics, we need to discuss the limit first. What is Calculus?Ĭalculus can be divided into two parts: the derivative and the integral. Leibniz uses the more traditional notation with the integral sign and the dx (or whatever the dependent variable is) after the integral. Another alternative to the dot is to put an apostrophe afterward for each derivative. For every derivative there were be another dot placed. Newton would puts a dot over the variable when a derivative was taken. Two this day both terminologies of calculus are still used. In their publications, both of the mathematicians use different terminology but came up with the same fundamentals. Leibniz and Newton’s developed and published their works separately and both had come up with integral and differential calculus at different times. He found that as the number of sides increased it more and more approximated the area of the circle. We call this the ‘limit’ in modern calculus.Īrchimedes was the first to pioneer integral calculus which he called ‘heuristics’ and use that to find the area of the circle by inscribing a many sided polygon. He called this the ‘method of exhaustion’. The reason for small shapes is they are easier to calculate. Whoo hoo! History of CalculusĬalculus has it’s reaches back to ancient days as far back as the fifth century BC when mathematicians like Eudoxus tried to estimate the volume of shapes by dividing them into smaller and smaller shapes. Often times, the work of calculus has already been defined in tables. (I know, I’m a nerd.) However, I have used the premises of calculus hundreds of times in design. Five of the six had to do with beam deflection and a seventh was used to determine if my water heater capacity was enough. I can honestly say that I have performed calculus only 6 times on the job. Image Courtesy of Nick Youngson CC BY-SA 3.0 ImageCreatorįor me, I have designed nearly 100 different machines or machine subsystems. At this point, everyone rolls their eyes at me. ![]()
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