Calculus : Overwiew and contents

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$$\mathrm{The}~\varepsilon-\delta~\mathrm{blog}$$


Calculus : Overview


These series of posts on calculus are going to be my first "lecture" and I will happily accept any suggestions that you offer in the comments at the end. I was initially planning to start with trigonometry but I still have plenty of conceptual doubts regarding it and I thought it would be better to master it completely before posting it. So, I will post about trigonometry once I finish it, which might take up to a month and a half but I think the wait would be worth it. Anyway, let's begin.


An important note

The calculus that I will discuss in these series of posts is merely to be treated as a "crash course" for the calculus required for physics. So, I will not go much in depth about the topics of calculus but I will share some additional, interesting stuff in order to boost your interest in calculus, for example, the arc length formula, finding volumes of 3 dimensional shapes using calculus (cone, sphere, frustum etc). I will also discuss about something called Gabriel's horn, a 3d shape said to have a finite volume but an infinite surface area.


Contents :

First, I will post about calculus in general, why it is necessary, how it all began etc., what ways there are to approach it (infinitesimals and limits, in general) and other basic stuff so that you know why we're doing all of this.


This is what all I'll discuss :

  • General discussion, Infinitesimals and limits, what are they, which one's better and why?
  • Limits : definition, meaning, visualization and methods of evaluating them.
  • Derivatives : What exactly are they? How do we define and evaluate them?
  • Higher order derivatives : What are they? What are their applications in calculus?
  • Maxima and minima : Their applications in physics and using them to find the extremum of polynomial functions, mostly quadratic functions.
  • Integration : Uses and the two general approaches : limit of Riemann sum and the fundamental theorem of calculus.
  • Rules related to integrating using the fundamental theorem of calculus
  • Arc length formula and some miscellaneous applications of integration (like finding the average value of a function over an interval).
  • Finding volumes of some 3d shapes using calculus.

I will also go over approximations at the end and discuss things like "what makes a good approximation?", the staircase paradox etc.

By the way, the staircase paradox leads to the "deduction" that $\pi = 4$ and succeeds in confusing many students. Getting around it is a good way to know what makes a good approximation.


Regarding approximations, I will be discussing sine approximations separately at the end of the posts regarding trigonometry as it doesn't seem to have anything to do with calculus per se.


Enjoy :)


$\text{~}\rho\alpha\xi\delta\epsilon\epsilon\pi~\sigma\iota\nu\delta\eta\theta$

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