Composition of functions

Ramanujan

Mathematician Ramanujam didn’t have any close friends and someone asked him the reason. He replied that although he wanted to have close friends but nobody was up to his expectations.

When pressed how he expected his friend to be,  he replied,  "like numbers 220 and 284!"

The person got confused and asked what is the connection between friendship and these numbers!

Ramanujam asked him to find the divisors of each number!

With much difficulty, the person derived and listed them as,

220 → 1,2,4,5,10,11,20,22,44,55,110,220

284 →1,2,4,71,142,284

Ramanujam then asked the person to exclude the numbers 220 and 284 and asked the sum of the remaining divisors.

The person was astonished to find,

220 → 1+2+4+5+10+11+20+22+44+55+110=284

284 →1+2+4+71+142=220

Ramanujam explained that an ideal friendship should be like these numbers, to complement each other. Even when one is absent, the other should represent the friend!

To all my 220 & 284's 😎
Happy friendship day...!!! 😊
From
Collection

MATHEMATICS IS CONSISTENT WITH AXIOMS

Consider the situation we know that the sum of the angles of a triangle is 180 degree. This theorem is from Eucledean plane geometry in which sides of triangle are straight line segments.

But our earth being spherical we can not draw straight line segment any where on the surface of the earth. So there exist other geometries known as Non Euclidean Geometries to deal with such situations.
 Important among these geometries are 1)Reimannian geometry

There were two friends. One was a mathematician and the other a politician. They were fast friends from their childhood and they had maintained their friendship throughout, even though they belonged to different professions.
Once the politician told his mathematical frined: "we are birds of the same feather. We both talk nonsencse. The day before yesterday I came to meet you in your school. you were teaching in a class. I did not want to disturb you; therefore I did not call you outside. But your voice being loud, I could hear from outside what you were teaching. You were telling your students that the 'sum of the angles of a traingle is 180 degree. We did study some thing like this, but I don not remember exactly what it was. So I took it that what you were teaching in the class was correct. Yesterday also I came to meet you and that time also you were teaching. But his time I heard you telling your students that the ' sum of the angles of a traingle is less than 180 degree, and in the other class you tell that it is less than 180 degree. . Mathematics being an exact subject, only one of these two statements can and must be true. So your case is like only.
We politicians also tell one thing one platform and exactly opposite thing on the other platform. So I say that we both are birds of the same feather; we both talk nonsense".

To this mathematician replied: " Yes, my dear friend, we both talk nonsense. But there is a difference. You talk inconsistent nonsense, while we talk consistent nonsense. Our statements, though they may look contradictory, have to consistent with the axioms with which we start our subject.
YES THAT WAS POWER OF MATHEMATICS

Is all differentiable functions are integrable?

While attending teleconference as a resource person in bangalore. I and panel members was asked to answer the following questions by my fellow lecturers.
Is integrable functions are continous?, Is every continous functions are integrable?, Is all functions are integrable etc.
Really I find these questions are good and we should have to clarify this questions. Here I tried to answer this : students are competent can give your suggestion

-->1. -->Are all functions that can be differentiated, integratable?
All differentiable functions are integrable.
True because all differentiable functions are continuous(Because Differentiability implies continuity but continuity need not implies differentiability) and by FTC, fundamental theorem of integral calculus all continuous functions are integrable.

2.Is every continuous functions are integrable?
True by fundamental theorem of integral calculus

3. Is all integrable functions are continuous?
-->This doesn't follow from the FTC, but I'm having trouble thinking of a counter-example. I looked around on the web and saw a couple people say that this is false, but never explain why. Can you integrate piecewise functions? If so then I can think of an easy counter-example. We've never talked about doing so in class. but think!


Some of the following integrals are not integrable:
LIST OF SUCH INTEGRALS --> --> --> 1. int sinx / x dx

2. int cosx/ x dx

3. int root (sinx) dx

4. int root (cosx) dx

5.int sin (x ² ) dx

6. int cos (x ² ) dx

7. int e ^{-x 2} dx

8. int e ^{x 2} dx

10. int root(1+x ³ ) dx

11. int x tan x dx

12. int 1/ log x dx . and many more ok dear. try to remember it.

WHAT IS INTEGRALS?

Calculus is the study of how things change.

In differentiation we are finding a derivative i.e. slope of tangent to the curve i.e how the curve changes at different points for a given function

In integration we are finding a original function whose derivative or slope of the tangent or change at different points is known

Slopes of curves is also can be considered as rate of change at different cases. This is called instantaneous change. Derivative of a curves tell us the  instantaneous rate of change of a curve.

A curvy function changes at different rates throughout its domain—sometimes it’s increasing quickly and the tangent line is steep (causing a high-valued derivative). At other places the curve may be increasing shallowly or even decreasing, causing the derivative to be small or negative, respectively.

Look at the graph of f(x) in Figure 

If instantaneous change is given to find the original state we need to integrate.

To find the change at different points we need to differentiate.

Calculus analyses things that change, and physics is much concerned with changes. For physics, you'll need at least some of the simplest and most important concepts from calculus.

A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the "derivative") of various functions. (The process of doing so is called "differentiation".)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called "integration".)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.

We know how to find the derivative of a function by different methods, Now we are going back in integration.

If we know that

 dy/dx=2x

and we need to know the function this derivative came from, then we "undo" the differentiation process. (Think: "What would I have to differentiate to get this result?")

y=x²  is ONE antiderivative of   dy/dx =2x

There are infinitely many other antiderivatives which would also work, for example:

y=x²+4

y=x²+π

y=x²+27.3

In general, we say y=x²+K is the indefinite integral of 2x. The number K is called the constant of integration.

Note: Most math text books use C for the constant of integration, but for questions involving electrical engineering, we prefer to write "+K", since C is normally used for capacitance and it can get confusing.

Notation for the Indefinite Integral

We write: ∫2x  dx=x²+K and say in words:

"The integral of 2x with respect to x equals x² + K."

The Integral Sign

The ∫ sign is an elongated "S", standing for "sum". (In old German, and English, "s" was often written using this elongated shape.) Later we will see that the integral is the sum of the areas of infinitesimally thin rectangles.
∑ is the symbol for "sum". It can be used for finite or infinite sums.
∫ is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
This "long s" notation was introduced by Leibniz when he developed the concepts of integration.
GEOMETRICAL INTERPRETATION OF INDEFINITE INTEGRAL:
Let f (x) = 2x. Then ∫ f (x) dx = x² + C. For different values of C, we get different integrals. But these integrals are very similar geometrically.
Thus, y = x² + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis.

Clearly, for C = 0, we obtain y = x², a parabola with its vertex on the origin.
The curve y = x² + 1 for C = 1 is obtained by shifting the parabola y = x² one unit along y-axis in positive direction.
For C = – 1, y = x² – 1 is obtained by shifting the parabola y = x² one unit along y-axis in the negative direction.
Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig
 


Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x²,
y = x² + 1, y = x² + 2, y = x² – 1, y = x² – 2 at P₀, P₁, P₂, P_-1, P_-2 etc., respectively, then dy / dx at these points equals 2a.
This indicates that the tangents to the curves at these points are parallel. Thus, ∫2x² dx = x + C = F_C (x) (say), implies that the tangents to all the curves y = F_C (x), C ∈ R, at the points of intersection of the curves by the line x = a, (a ∈ R), are parallel.

Further, the following equation (statement) ∫ f (x) dx = F (x) + C = y (say) , represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral.
∫ f (x) dx = F (x) + C = Family of all curves which are geometrically similar and the tangents drawn to all curves by the line x=a (at any particular point) are parallel.

NOW THE QUESTIONS?

1.      Is All differentiable functions are integrable.

Ans: True because all differentiable functions are continuous and by FTC all continuous functions are integrable.

2.      Is All integrable functions are continuous.?

Ans:  This doesn't follow from the FTC, Now for this it is difficult to answer at this level. We have to learn advance mathematics

Can you integrate piecewise functions? If so then I can think of an easy counter-example. We've never talked about doing so in class.

3.      Is All integrable functions are differentiable.

Ans: Even though 1 is true this doesn't follow from it. Same difficulty as 2.

{Differentiable functions} ⊂ {Continuous functions} ⊂ {Integrable functions}

In order for some function f(x) to be continuous at x = c, then the following two conditions must be true:

i)                    f(c) is defined and the limit of f(x) as x approaches c is equal to f(c).

In order for some function f(x) to be differentiable at x = c, then it must be continuous at x = c and it must not be a corner point (i.e., it's right-side and left-side derivatives must be equal).

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists.

While all continuous functions are integrable, not all integrable functions are continuous. To understand this idea we need to study advanced mathematics i.e Reimann integral.

Hope this will satisfy your needs    : Happy learning   : by KHV, SAGAR