✅ Calculus (66) CBC / Practice 8: Exercise 3 (Taylor Polynomial)

Hello everyone, my name is Flor. Welcome to Practice 8 of Calculus for Exact Sciences and Engineering CBC students. In this practice, we will be looking at the Taylor polynomial. For those who have seen some of my videos from the first midterm, don't worry, we will now be following the new guide, so you won't have issues with the exercise numbering. So, let's start with the first video of Practice 8, solving Exercise 3. In Exercise 3, we are asked to calculate the Taylor polynomial of the following functions up to the indicated order at the given point. So, before starting, let's try to understand a bit what the Taylor polynomial is, because at least for me, when I took calculus, I did the Taylor polynomial exercises but didn't fully grasp what I was dealing with—a very powerful tool, a very, very useful tool that I would end up using in many subjects later on. So, this one you see here is the formal definition of the Taylor polynomial of order $n$ of $F$ centered at $x_0$. And what does this mean? I mean, translated into plain language, what does it mean that I am approximating the function with this Taylor polynomial near the point $x_0$? Well, basically, what it tells us is that if I have a function $F$, no matter how complicated my function is,