It is possible to calculate the derivatives of single variable functions using differentiation rules without taking limits each and every time we compute the derivatives. In this post, we introduce several rules of differentiation. However, we prove only a few of them and for the proof of the other rules, the reader should refer to…

# Month: April 2023

## One-sided derivatives

One-sided derivatives are defined by one-sided limits. In this post, we will introduce the right-hand derivative and the left-hand derivative of a function at a given point. One-sided limits Let \(f\) be a real-valued function. The reader should be able to easily prove the following result: Result. Let \(f\) be a real-valued function, \(I\) an…

## Derivative of a function

The derivative of a function at a given point is the slope of the tangent line to the function at the given point. In other words, the derivative of a function \(f\) at a point \(c\) is the slope of the tangent line to the curve \(f\) at the point \(c\). The derivative of a…

## Tangent line to a curve

Every standard calculus textbook discusses the geometric interpretation of the tangent line to a curve at a given point. However, we define the tangent line differently. The tangent line to a curve at a given point is the line which provides the best local approximation of all the lines passing through the given point. In…

## The average rate of change

The average rate of change of a function can be calculated by dividing the difference between the dependent variable and the independent variable. In this post, we discuss how to compute the average rate of change of functions in different spaces. The average rate of change in single variable calculus Let \(y = f(x)\) be…

## Regression line

A regression line is a line that fits best to a set of numerical data points in the plane. In this post, we will discuss the concept of a regression (least squares) line and a method for finding it. In statistics, regression is an analytical method to measure the association of one or more independent…

## Least squares problem

The least squares problem is to obtain a vector that the distance of the value of the given linear map at that vector from another given vector is minimized. More specifically, let \(A\) be an \(m \times n\) real matrix and \(b\) a vector in \(\mathbb{R}^m\). The least squares problem is to find a suitable…

## QR matrix factorization

In a multiplicative mathematical system, it is sometimes useful to write an object as a multiplication of smaller or simpler objects of the same kind. This process is called factorization (more generally, decomposition). Mathematicians, engineers, and data scientists use different kinds of factorization (decomposition) to simplify complex calculations, and for example, find solutions to equations….

## Extreme value theorem

The extreme value theorem states that a continuous real-valued function defined over a closed interval attains its minimum and maximum at some points in the closed interval. In other words, if \(f\) is a real-valued function continuous on a closed interval \([a,b]\), then by the extreme value theorem, there are two points \(l\) and \(u\)…

## Bolzano-Weierstrass theorem

The Bolzano-Weierstrass theorem is a fundamental result in real analysis and states that each infinite bounded real sequence has a convergent subsequence. Also, it has some applications in data science and economics. In order to prove the Bolzano-Weierstrass theorem, first we need to prove a result called the monotone subsequence theorem. Note that the monotone…