In this article, we will be introduced to the concept of the Mean Value Theorem. We will see both its definition and how it can be used in practice.
Before we dive into the Mean Value Theorem, we should first consider another theorem first. This is due to the fact that it will be used later, and this is Rolle’s Theorem. It is defined like so:
Let us now take a look at what this theorem can be used for, by using an example. Imagine that we have the following function:
In this article, we will find out how to find the minimum and maximum values of a function. We will also learn how to find the absolute extrema on a function, which simply means that the largest value is the absolute maximum, and the smallest value is the absolute minimum. To be able to do this, it is important to have a basic knowledge of how to take a derivative of a function.
Let us first look at the concept of critical points and define what they are. Imagine that you have a function, which looks like so:
In this article, we will consider the concept of continuity. We will also consider continuity in relation to limits, and we will also be introduced to the concept of the Intermediate Value Theorem.
Let us first get a definition of continuity. It can be defined as:
From this definition, we can conclude the following fact:
In a former article, we talked about the limits of functions. We have used what we learned in that article to calculate the limit of both sequences and infinite series. In this article, we will be given the tools to prove whether a given point of convergence is accurate or not. This will only be done for a limit at a finite point — in the next article, we will look at the left-hand-side and right-hand-side limits. In the articles after that, we will look at cases where we find the limits at infinity — or the limit is infinity.
In the former articles, we looked at infinite series and specific cases of these: geometric and harmonic series. We were told that harmonic series are always divergent, but we did not show why. In this article, we will see how we can prove it through the so-called Integral Test.
Let us first remember what the notation of the harmonic series is. It looks like the following:
For us to understand why it is divergent, let us first look at the function defined as:
In a former article, we were introduced to infinite series. We were also briefly introduced to the concept of absolute and conditional convergence, but we lacked some tools before we could look at examples. In this article, we will look at two of the concepts we needed: Geometric and Harmonic Series.
We will start by looking at Geometric Series. Let us first introduce its general notation, which can be denoted in two manners:
These two series are identical, they are just written differently. Therefore, they will also have the same value in case of convergence.
As we saw in our…
In this article, we will cover the topic of infinite series. We will start out by showing the notation for an infinite series and after we will handle the subjects of divergence, convergence, and absolute convergence. If you are not already familiar with sequences and limits, then I recommend you to read these two articles first: ‘Finding Your Limits’ and ‘An Introduction to Sequences’.
Let us first try to understand what is meant by an infinite series. To do so, we will first need to consider a sequence. Let us imagine that we have the following sequence:
This means that…
In a former article, we looked at how we could find the limits of functions. In this article, we will look at sequences, both in defining them and then relating them to our knowledge about limits. We will start out by defining what a sequence is and to notation, we will use.
The primary question is: What is a sequence? Simply put, a sequence is a list of numbers written in a specific order. This list of numbers can either have a finite or infinite number of numbers, but in this article, we will only consider sequences of infinite size.
In this article, we will get an easy-to-understand walkthrough of various subjects concerning limits. We will get the informal definition and use it to solve different limit calculations. In a future article, we will also look at the formal definition and use it to solve with. All of this is done to have a basic understanding of limits, such that we can draw upon this knowledge in the next two articles. These two articles will be looking at sequences and series and how we can find their point of convergence, i.e., their limit. …
Said in simple terms, probability theory is the mathematical study of the uncertain — it allows us (and thereby also the computer) to reason and make decisions in situations where complete certainty is impossible. Probability theory plays a center-stage role in machine learning theory, as many learning algorithms rely on probabilistic assumptions about the given data. In this article, we will consider a specific probability bound — Chebyshev’s Bound.
This article is meant to understand the inequality behind the bound, the so-called Chebyshev’s Inequality. It will try to give a good mathematical and intuitive understanding of it. In two other…