Calculus

By Jagdish Krishanlal Arora

The book "Calculus" is a foundational mathematical text that explores the principles of differentiation and integration, offering a comprehensive understanding of how functions change and accumulate quantities. It begins by introducing the concept of derivatives, which measure rates of change in functions, and provides rules and applications for finding derivatives, making it an essential tool for analyzing curves and solving real-world problems in fields like physics, engineering, and economics. The book also delves into integration, the reverse process of differentiation, which focuses on finding areas, volumes, and accumulated quantities. It illustrates how the Fundamental Theorem of Calculus connects these two fundamental concepts. Overall, "Calculus" is an integral resource for students, scientists, and mathematicians seeking to master the core principles of this critical branch of mathematics and apply them to a wide range of practical scenarios.

• Language: English
• Publisher: Jagdish Krishanlal Arora
• Release date: Oct 13, 2023
• ISBN: 9798223597377

Book preview

Introduction

Basically, we use graphs to calculate slopes, tangents, and many times we need to calculate areas of triangle, rectangle, and other 2D and 3D objects and other calculations.

Calculus helps us to solve these problems by using equations and we simply input these values and calculate the slopes, tangents, motion, velocity, physics, and areas of simple and complex objects.

The purpose of this book is to simplify mathematics which should be easy to remember as mathematics is still heavy on the mind for many people even for teachers and many teach them blindly as it is in the books.

Mathematics is very easy if remembered in the right way. First, we need to start with the basic concepts and once those are memorised then it is very easy for us to understand the advanced part.

For example, we know that all of everything has x and y components.

X is the horizontal component of a 2D object, for example a line is a 2D object.

Y is the vertical component and signifies height.

These concepts were first used in graphs and to calculate horizontal and vertical motion. In other words, motion in horizonal and vertical direction can be calculated using x and y.

X is any numerical value in the horizontal direction. It can be any number from 1-9 and can also be in decimals. (1, 2, 3, 12, 23, 23.1, 12456 anything.)

Y is any numerical value similar to the x values above.

Graphs have a limit; we cannot use graphs to calculate big values and decimal values and so Calculus helps us to solve them. We do that by making equations. Once we have the equations, we can put any x or y value in them and calculate the areas or other requirements by putting different values of x and y.

For, 3D objects, we need an additional component ‘z’ in this to calculate the area for 3D objects. That is how x, y and z come into the picture. The z component also helps us to sum the values of x and y.

Similar to addition, subtraction and multiplication, we have addition, subtraction and multiplication and even division in the use of x,y and z components making different kinds of equations in the process.

For example:

z = x + y

2x = 3y + 2z

2 = 2x + 2y + 2z

Any combination of equations is possible. We call these as quadratic equations.

We then apply the BODMAS rule to solve these equations or simplify them in addition to finding out the possible values of x, y or z so that the values on both sides of the = sign are the same and match exactly after putting the values of the components (x, y and or z).

Mathematics, or calculus in this case is all about determining the x, y and z values or using the available values. Sometimes, we make our own formulas or equations for complex objects that do not fit into the normal triangle, rectangle, parallelogram, cone, cylinder, circle, tangent etc. and make our own equations/formulas or calculate speed, motion, work, energy and in several other applications in physics and other subjects.

In area measurements, we sub divide the complex objects into simple divisions of triangles, rectangles, circles as applicable depending on the shape of the object and then totalling the sum of all the individual divisions to get the total area of the object.

For speed, time and motion we have other types of formulas as we can measure distance and time, to arrive at the speed. For external objects such as planets and stars, we cannot use distance and time as we have no data and there are other methods.

Some students are very good at reading theory and some are better off doing practical. The ones who are better off in understanding how to solve practical problems are successful, although some subjects also require to understand the theory behind the calculations.

You can either start with the practical or formulas or theory and move back and forth till you are able to understand both of them and how they are linked.

In derivatives, we use f’(x) and f(x).

A first-order ordinary differential equation of has the form:

dy/dx = f(x)

To solve this equation, you integrate both sides with respect to x:

∫ dy = ∫ f(x) dx

This integration leads to the construction of the antiderivative, allowing you to determine the general solution of the differential equation. This will become clear as you read the book chapters.

There may also be further studies where you solve complex quadratic equations using derivatives and integrals and simplify them using derivatives and integrals.

Chapter 1: The Calculus Basics

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It has two primary components: differentiation and integration.

Differentiation

Derivatives: Calculus begins with the concept of derivatives. A derivative measure how a function changes as its input (independent variable) changes. It provides information about the rate of change of a function at a specific point. The derivative of a function f(x) is denoted as f'(x) and represents