Durofy

The Atharva Veda, the 'fourth Veda' is a sacred Hindu text written by The Aryans. It is usually referred to as 'The Book Of Spell'. It deals with Medicine, Warfare & all other Sciences & has a magical essence to it. Its magical character can be seen in its hymns, rituals & spells which can create, heal, preserve or destroy.

The Atharva Veda, written in 2nd Millennium BC has a more efficient substitute for everything that can be done today and the things we dream of achieving tomorrow. It can cure diseases in a single prayer, it has charms for securing prosperity, to evert the evil, for obtaining a wife, a husband, & even a charm to give birth to a son, a daughter with specific features which corresponds to 'In vitro fertilization' minus its complications. It also has a charm to shrink or extend time, a concept corresponding to 'Time Travel' on which we have seen a number of movies & our physicits are still wondering if it is practically possible.

Apart from these, It also includes an amazing mathematical system. This is the system of Vedic Mathematics. It is simply a system of mathematics by the Atharvans, which can change our thoughts, beliefs and the overall scope of mathematics. These days, man has become dependant on all kinds of machines for even his basic routine works. We use the calculator for the very basic calculations we need at home, at school, in the market, at our workplace, and we kill the mathematician who lives inside each one of us.

Instead of describing the science in detail here, I will prefer demonstrating it to you with a few examples. The vedic mathematics system has certain sutras & sub-sutras which are 'methods' for various calculations. Here is an example:

A) Squaring numbers ending in 5:

Steps:

1. Add 1 to the digits that come before the digit 5.(Eg. add 1 to 2 in case of 25 squared)

2. The first part of the answer is the number(consisting of the digits that come before 5)times the number you get be adding 1 to it(here, 2x3=6)

3.The second part of the answer is simply 25.

Examples:

(35)²->3+1=4, 3x4=12 , therefore answer=>1225

(65)²->6+1=7, 6x7=42 , hence, answer=>4225

B) Multiplying a number by 11:

Steps:

1. Add the 2-digits of the number together.

2. Place the sum of the two-digits between the 2 digits to get the answer.

Examples:

21x11->2+1=3, Answer=>231

36x11->3+6=9, Answer=>396

29x11->9+2=11(a 2-digit number, hence, 1 gets carried over), Answer=>319

The Product Rule:

We use the Product Rule when we have products of two or more functions. In case of three functions, we take any two functions as as and differentiate the third as so on hence, forming three terms in the sum.

There is also a quotient rule for derivatives of functions in the form u/v but, we will stick with the product rule for the form u/v by treating (1/v) as a function of x.
But we should be familiar with the quotient rule too. According to the quotient rule:

Now, if we are dealing with composite functions, we need to use the chain rule to find their derivatives which is stated below:

Now, using the above rules and the table of derivatives in the previous post, we can find the derivative of any function and combination of functions (applying the rules as needed).

List of Derivatives of the basic functions:

All these can be derived using The first principle. However, we do need to get (very) familiar with these.
To learn the basic derivatives, Practice as many questions as you can. First do the questions referring to the table here & then without it.

So, lets see how we can apply the first principle to differentiate functions.
We'll start with the basic functions, lets take the square function first.

So the derivative of the square function is 2x which is also the slope of the tangent at any point on the curve of the function. So, for a point, x=a, the slope of the tangent would be 2a(putting x=a in 2x) And, since we know the slope of this tangent at a, we can also find its equation. Hence, we can use derivatives to find equations of tangents which is an important Application Of The Derivative.

Now, lets differentiate another function using this principle. We'll go for the rectangular hyperbola this time.


Now, lets find out a generalization for such functions(polynomial/rational & irrational).

We considered the value (phi) to collectively consist of all the higher powers of the change in x as the terms with higher powers would eventually cancel out while solving the limit.
Hence, this is the actual method for finding derivatives of functions - the first principle. However, in practice, we use properties of derivatives and the basic derivatives of the most common functions to find the derivatives of bigger and more complex functions. First, we must be aware of the derivatives of the major and most common functions.

In the previuos topic, we found out the slope of the tangent which was the derivative of the function, we had actually found something called the first principle of calculus!

So, the thing in the red box there is the first principle which we will use to find the derivatives of a function. This process, of finding derivatives(notation:dy/dx) or the differential coefficients of functions may be called Differentiation & we are said to differentiate functions while finding the derivatives.

The function y=f(x) in green, is associated with two lines in the above diagram. The line there in blue cuts the function at two points while the line in orange touches the curve at a single point. With some of our previus knowledge, we would say that the line in orange is the tangent line as it touches the curve at one single point. The statement IS ture, but does it mean that the tangent cannot cut the curve furthur at another point?
Well, the figure below answers the question.

Here, we see that the tangents at P & Q do intersect the curve of the sine function again at other points. We also notice that the curve has the same nature between point P & the point 3pi/2 & the tangent intersects the curve on the left of 3pi/2, ie when the curve has changed its nature! We find the same observation for the tangent at Q. So, for any curve, the tangent may intersect it again at a point provided the curve has changed its nature atleast once between the new point of intersection & the point of contact of the tangent.

So, that was about the orange(tangent) line. Lets talk about the blue line. Have you seen something like this before? Sure you have. Have a look at the image below.

So, this line in red on the circle, and in blue on the function, is called the secant line. Now, lets try to find ut the slope of this secant line.

So, the slope comes out to be (the change in y)/(the change in x) & here, the change in y refers to the difference between the value of the function at A and its value at B, which is:

Hence, The slope of the secant line can be written as:

Now, we need to get the orange line into the picture, keeping the blue line in mind.

Now, Lets take our observation a step furthur. You can observe in the figure below, that when the above happens, ie when A gets closer to B, the change in x goes on decreasing and finally, tends to zero when the slopes tend to be equal, or when A & B tend to coincide.

So, we basically have limiting values here which calls for the application of limits.
And that limiting value there folks, is the derivative!

Graphical Transformations help us to plot any function in calculus.
These transformations can easily be observed by changing the values of the independent variable x.

Before we study differential calculus, it is important to understand the concept of functions and their graphs. This is a major pre-requisite before any Calculus course often dealt with in a separate course called Pre-Calculus.

The Concept Of Functions, Domain & Range

Lets say we have a circle. The area of this circle depends on its radius. Hence, we have an example where a quantity depends on another. Thats exactly what the concept of a function is. Here, the area depends on the radius & is said to be ‘a function of the radius’. The area, hence, is also called the ‘dependant variable‘ while the radius being independent is called the ‘independant variable‘. Now, depending on whatever value of radius (the input) is, there is a corresponding value of the area(the output).

Every single machine in the world, works on the concept of a function. We give the machine an input & it gives us an output. Lets take a washing machine for example, we give it dirty clothes(the input) & it gives us clean clothes(the output). But, what if we leave our cell phone in our pants while giving it to the machine. It will not execute the function. You will get an output for the clean pants BUT NOT for the cellphone even though they are together.

This means that the washing machine is not programmed to take in the cell phone as an input or we say, the cellphone does not fall into the domain of the machine. Hence, the domain of a function consists of all the values that can be given to the machine as the input & the range are the values of all the ossible outputs obtained from the machine.

Mathematically, a function is described as y=f(x) where x is the input or the independant variable & y is the output or the dependant variable as explained above. Hence, all possible values of x form the Domain of the function & the corresponding output values of y form its Range.

Here is a table where functions are classified into 3 main categories based on their nature.

Algebraic Functions:

Modulus Function:

Fractional Part & Greatest Integer Function:

Signum Function:

Trigonometric Functions - Sine & Cosine Functions

Trigonometric Functions - Tangent & Cotangent Functions

Keeping these basic functions & their graphs in mind, we will move on to Graphical Transformations

On an island live 13 purple, 15 yellow and 17 maroon chameleons. When two chameleons of different colors meet, they both change into the third color. Is there a sequence of pairwise meetings after which all chameleons would have the same color?

If x = (16^³+ 18^³ + 19^³), then find the remainder if x divided by 70.

a. 0
b. 1
c. 69
d. 35