this is calculus 2 in 5 minutes with many thanks to Dr burnhard for an enjoyable calculus 2 experience so step one terminology and pols so what exactly is a polom a polom is a function that can be represented using only arithmetic and X ra to the power of a positive integer and here's some examples of valid and invalid pols you'll see that all of them are just X raised to a positive integer and there may be certain algebraic equations that are not exactly pols but are algebraic and there are special functions called transcendental functions that are even more special because they transcend algebra and here are some examples now the question is how do computers calculate these transcendental functions chapter two McLaren series and approximation so we've probably heard of the sign function but how do we estimate a sign function using only po pooms enter the McLaren series the McLaren series works by raising x to the 0o power over 0 factorial and F to the zero derivative and then adding F to the first degree derivative * X the 1 power over 1 factorial plus F the 2 degree derivative * x^2 over 2 factorials you get the idea it'll just continue raising each degree of the derivative time x raised to that power over that same value factorial and here's that approximation in action and you're noticing that there are a few Zer that will appear such as this one because of those zeros from now on I'm going to be jumping and increasing the term and increasing the three by two every single term so the terms will increase much faster than now uh than previously because there are no zeros uh uh so with the omitting of the zeros we can now start to see the function emerge now you'll notice that with each term added X is increasing by two and it also starts off at one in terms of factorial as well as the powers so the power and the factorial is 2 n + 1 we're also noticing that -1 to the power of n will give us that alternating plus and minus now I want you to try to solve for the following equations and now pause the video memorize this and write this down because this is important chapter three special integrals so are all functions integratable so I want you to now integrate the following cosine x^2 DX and you'll quickly realize that this is impossible well sort of because you can probably see where I'm headed with this I want you to recall the tayor series for f ofx of cosine of x if you replace x with x^2 simplif if it and then add one because that's how you take an integral of something and divide by that new power and also don't forget your plus C you have the integral so this is how you do it so write down the following uh because it's important as well and now I want you to go ahead and try to solve for the following did you get this if so you're correct let's move on this is chapter 4 converging versus Divergence so on the topic of summations we're going to learn to solve some summations and these summations go to Infinity so let's rewrite the summation in this format we'll quickly see the association between some of the variables and notice how R is between 1 and 1 that will converge so now solve for this and again we're going to write it in the same format and because three is outside of the range of Nega 1 and 1 we know that this function will diverge and lastly let's go and find the sum um so let's go and bring up this problem and rewrite it in the following format now let's go ahead and try this one so this one's quite indecisive because we actually don't have an answer for it converge means to equate to a finite sum one and only one finite sum so now we're going to talk about interval of convergence the McLaren series for the approximation equates to the following and now we're going to ask can the McLaren series approximate everything the answer as you will see is sadly no you should be able to simplify this by now and notice that R is the placeholder for X so that means that when X meets the following criteria the function converges and if you graph it it shows us so what about the following 2 4 + 5x let we write in the following format and if you write the mlar approximation and also graph it you'll see that it also intersects at that point and X will only satisfy the above constraint only when the following criteria as mat and lastly ratio test so sometimes you may need to solve the following now how do we know if this converges or diverges well you take n + one over n as you can see we're going to cross out n factorial and we're going to get that value plugging in Infinity will give you one over infinity which will yield a zero and because R is zero it will converge