CATCULUS

Tavishi If cats were academics, they'd all be mathematicians. This is my cat-hematician, Billy. He wears his math hat.

As you can tell, he is clearly very pleased by the math hat. Cat-culus is very fun for my kitty curmudgeon. Select topics that fascinate the Cat include, but are not limited to: MeowcLaurin series, meowptimization, and Lapawnge multipliers. The MeowcLaurin series is just the Taylor series where c (variable for Cat) is equal to zero. The Taylor series is an infinite sum representation of a function, using its derivatives to approximate the value of said function. As n increases, the approximation becomes better and better. If a Taylor paw-lynomial with three terms was used to approximate something, versus a Taylor polynomial with twelve terms, the polynomial with twelve terms would have significantly less error than the polynomial with three terms. Take the function e^x (suddenly wishing I knew how to use LaTeX....). The Taylor series representation here is:

This here is a Meowclawrin series, as rather than (x-a) being raised to the nth power, it is just x. Thus, that means the Taylor series approximation is about x = 0. However, if the function was e^(x-1), the value for c would be 1 as well.

The Taylor/Meowclawrin series finds its way into basically everything, including: the kinematic equations, trigonometric function calculation in calculators, bell curves, and more. Take, for example: Felix the cat is stranded on Mount Doom at (1.25 , 1) on the x-y plane with Frodo Baggins and Samwise Gamgee. When the eagles come to rescue Felix, they have to dodge flying seals being shot at them, and thus, follow the path of sin(x^2). Felix sees a bird in the air and jumps off of an eagle at x = 2.96. A Taylor polynomial could be used to figure out the approximate y-value of where Felix flies freely in the air without tedious trigonometry. Take the Taylor approximation of the function to a certain degree, plug and chug, and you'll find a pretty good approximation of how far off the ground Felix leaped.

In the end, everything boils down to a Taylor approximation. Next, meowptimization (optimization in non-catculus terms). Meowptimization uses the critical points of certain functions (where first derivative is 0), or places where the function is at a minimum or maximum (not all critical points are mins or maxes) to find the optimal conditions for a certain thing. Meowptimization is one of the select catculus topics where you can very clearly see a "practical" application of the idea. Take, for example:

Felix the cat is sitting at (0,0) on the x-y plane. He wants to get to his favorite cat toy, which is at (4,6). He can walk in any direction at a speed of 1 unit per second, but he notices, to his horror, that above the line y=1, the solid ground turns to mud, so that he can only walk at a speed of sqrt(3)/3 units per second. What is the minimum possible time it could take for Felix to reach his toy? This problem would involve Furmat's Principle of least time, leading into Snell's law and the like- essentially, take this problem but instead of a kitty cat, light! Leave a comment if you want a Lapawnge multipliers catculus post!