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## Math Cracks – A Cool Approach to Integration by Parts

Introduction The idea of integration by parts sounds quite scary for many Calculus students, and I think there is a good reason for that. First of all, integration by parts is a technique that involves two steps (or more) instead of one step as most ...

## Functions: What They Are and How to Deal with Them

The concept of function is extremely important and it is absolutely omnipresent in Math. That is why we need to give it a good brush up, before attempting to understand some of the goodies that will come afterwards when go deeper into Calculus ...

## Math Cracks – What is a Derivative, Really?

It seemed important to me to go over the concept of derivative of a function. The process of differentiation (this is, calculating derivatives) is one of the most fundamental operations in Calculus and even in math. In this Math Crack tutorial I ...

## What is the Limit of a Sequence?

A sequence \(a_n\) corresponds to infinite array or list of number of the form \ where \(a_1, a_2, a_3, …\) are real numbers. For example, the sequence \ is represented by the list \ because those are the values the expression \(a_n = ...

## More About Derivatives

On the second part of this tutorial, we’ll work on some other slightly more complicated examples. Example: Given the function \(f(x) = x^3 + 2x+1\), compute the derivative \(f'(x)\) for every every point where it is defined. Solution: Notice ...

## More About Derivatives (Part 2)

Notation: The derivative \(f'(x)\) of a function \(f(x)\) is also denoted as \ This notation comes from the fact that when you compute the derivative, you compute \ The term \(f(x)-f(x_0)\) is usually referred as \(\Delta f\), and the term \(x-x_0\) ...

## The Basic Concept of Derivatives

Imagine you have a function \(f(x)\). For example you could have something like \(f(x) = x^2\) or maybe something like \(f(x) = \sin x\). We define the derivative of the function \(f(x)\) at the point \(x_0\) as \ if the limit exists. Before you ...

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