The Riemann Integral: Part 1

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Introduction

The theory of integration forms an important part of mathematical analysis. Historically integration was used to find areas of plane figures. Archimedes used the very same process to find areas of parabola but he called it the method of exhaustion. The idea used by Archimedes was to divide the desired area in terms of smaller and smaller areas so that the sum of the areas of these smaller parts tended to a finite limit. It was the genius of Newton (and Leibniz too) to recognize that the process of integration could be viewed as the inverse process of differentiation. This greatly helped in finding areas of curves for which summing the areas of smaller parts was difficult. After Newton people started thinking of integration as the inverse of differentiation and the older approach based on summation was put at the back front.

Functions of Bounded Variation: Part 2

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Continuity and Bounded Variation

In the last post we saw that continuity is not essential to the property of being a function of bounded variation. However monotonicity is absolutely essential in the sense that every function of bounded variation can be expressed as a sum or difference of monotone functions. But does that mean that continuity is not at all required? Can we have a function which is discontinuous everywhere and still be of bounded variation? The answer is NO! As an example the function $ f(x) = 0$ when $ x$ is irrational and $ f(x) = 1$ when $ x$ is rational is not of bounded variation. We can choose a partition to consists of equal number of rational and irrational points lying alternately and then the variation can be seen as a linear function of the number of points of subdivision so that the variation is not bounded.

Functions of Bounded Variation: Part 1

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Introduction

In the last two posts we studied monotone functions which vary in the same direction in a given interval. Here we will study functions which do not vary too much. In a sense continuous functions also don't vary too much (for example they are bounded on closed intervals). But here we need to discuss variation in a different sense. More technically we try to measure variation in smaller parts of an interval and then add up these variations to form total variation. We formalize these concepts now.

Monotone Functions: Part 2

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In the last post we established certain conditions for the monotonicity of a function in an interval. In this post we will establish the same results via a different approach. This is based on the standard theorems of differential calculus, namely the Rolle's Theorem and the Langrange's Mean Value Theorem. We first need to establish these theorems.

Monotone Functions: Part 1

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Introduction

Few posts ago we discussed continuous functions and their properties. In this series of posts we discuss another class of functions namely the monotone functions and their extensions. The word monotone crudely suggests that these functions should have a single tone which translates properly to "variation in a single direction". In other words such functions either increase all the time or decrease all the time.

Logarithms using Square Roots

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Introduction

In the last post we discussed that a proper theory of logarithmic function can not be developed without using the analytical approach (or methods of calculus in simpler language). Here we discuss more about the common logarithms (i.e. logarithms to the base 10) which are normally introduced to students in secondary classes without using any calculus. The basic idea is simple enough: if $ y = 10^{x}$ then we write $ x = \log_{10}y$ and say that $ x$ is the common logarithm of $ y$. The difficulty with this approach is that the meaning of $ 10^{x}$ cannot be explained properly without using calculus if $ x$ is irrational. However in the examples and exercises given the exponent is either a symbol (like $ a, b, c, x$) or is a rational number so there is no confusion when the concept of logarithm is presented via this approach. And the student is able to learn the fundamental properties of logarithms and their practical usage in computation with relative ease.