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differentibily

We present a new proof of the differentiability of exponential functions. It is based
entirely on methods of differential calculus. No current or recent calculus text gives or
cites a proof of the differentiability that depends only on such elementary tools. Our
proof makes it possible to give a comprehensive treatment of the derivative properties
of exponential and logarithmic functions in that order in differential calculus, building
on the standard introduction to these topics in precalculus courses. This is the logical
order and has considerable pedagogical merit.
Most calculus books defer the treatment of exponential and logarithmic functions to
integral calculus in order to prove differentiability. A few texts introduce these topics in
differential calculus under the heading of “early transcendentals” but defer the proof of
differentiability to integral calculus. Both approaches have serious pedagogical faults,
which are discussed later in this paper.
Our proof that exponential functions are differentiable provides the missing link
that legitimizes the “early transcendentals” presentation.
Preliminaries
We assume that ar has been defined for a > 0 and r rational in a precalculus course
and that the familiar rules of exponents are known to hold for rational exponents. It is
natural to define ax for a > 0 and x irrational as the limit of ar as r → x through the
rationals. In this way, ax is defined for all real x.

Basic properties of ax for real x are inherited by limit passages from corresponding
properties of ar for r rational. These properties include the rules of exponents with
real exponents and
ax is positive and continuous,
ax is increasing if a > 1,
ax is decreasing if a < 1.
It is not especially difficult to justify the definition of ax for x irrational and to
derive the foregoing properties of ax for x real, but there are a lot of small steps. A
program along these lines is carried out by Courant in [2, pp. 69–70]. The general idea
of each step is well within the grasp of students in typical calculus classes. However,
just as properties of ar with r rational are routinely stated without proof, it is better to
give just an overview of the basic properties of ax with x real, illustrated with graphs,
and move on to the question of differentiability, which is more central to differential
calculus.
A more complete development, beginning with the derivation of properties of ar
with r rational, might be given in an honors class. The properties can be extended to
ax with x real with the aid of the density of the rationals in the reals and the squeeze
laws for limits. The conclusion that ax with a > 1 is increasing also relies on the
following proposition which should seem evident from graphical considerations:
If f is a continuous function on a real interval I
and f is increasing on the rational numbers in I,
then f is increasing on I.
The same proposition will provide a key step in the proof that ax is differentiable.
Henceforth, we restrict our attention to properties of ax with a > 1. Corresponding
properties of ax with 0 < a < 1 follow from ax = (1/a)
−x .
The differentiability of ax
Consider an exponential function ax with any a > 1. In order to prove that ax is differentiable
for all x, the main task is to prove that it is differentiable at x = 0. Our proof
of this depends only on methods of differential calculus. It is motivated by the fact
that the graph of ax (see Figure 1) is concave up, even though this fact is not assumed
a priori

figure 1

Graph of ax with B = (x, ax ) and C = −x, a−x for x > 0

In Figure 1, imagine that x → 0 with x > 0 and x decreasing. Then B and C slide
along the curve toward A. The upward bending of the curve seems to imply that
slope AB decreases, slope AC increases,
and slope AB − slope AC → 0.
It follows that the slopes of AB and AC approach a common limit, which is the slope of
the tangent line T in Figure 1 and the derivative of f (x) = ax at x = 0. This geometric
argument will be made rigorous.
The curve in Figure 1 is actually the graph of f (x) = 2x . The following table gives
values of the slopes of AB and AC rounded off to two decimal places. It appears that
the slopes of AB and AC approach a common limit, which is f
(0) = slope T ≈ 0.7.


x                    1               1/2         1/4          1/8         1/16        1/32
slope AB        1              .83         .76          .72 .       71           .70
slope AC       .50            .59        .64 .         66         .67            .69


With this preparation, we are ready to prove that f (x) = ax is differentiable at
x = 0. The foregoing geometric description of the proof and the numerical evidence
should be informative and persuasive to students, even if they do not follow all the
details of the argument.

Theorem 1. Let f (x) = ax with any a > 1. Then f is differentiable at x = 0 and
f
(0) > 0.
Proof. To express our geometric observations in analytic terms, let
m(x) = f (x) − f (0)/x − 0
= ax − 1/x
.
In Figure 1, x > 0 and
slope AB = m(x),
slope AC = m(−x).
We shall prove that, as x → 0 with x > 0 and x decreasing, m(x) and m(−x) approach
a common limit, which is f
(0).
To begin with, m(x) is continuous because ax is continuous. The crux of the proof,
and the only tricky part, is to show that
m(x) is increasing on (0,∞) and (−∞, 0).
We give the proof only for (0,∞) since the proof for (−∞, 0) is essentially the same.
We show first that m is increasing on the rationals in (0,∞). Fix rational numbers r
and s with 0 < r < s and let a vary with a ≥ 1. Define
g(a) = m(s) − m(r ) =a^s -1/s = a^r-1/r

.
Then g(a) is continuous for a ≥ 1 and
g
(a) = a^s -1-a^r-1>0 for a>0

Thus, g(a) increases as a increases and g(a) > g(1) = 0 for a > 1, so
m(r) < m(s) for 0 < r < s.
Thus, m(x) is continuous on (0,∞) and m(x) increases on the rational numbers in
(0,∞). As noted earlier, this implies that m(x) increases on (0,∞). The argument for
the interval (−∞, 0) is similar.
For x > 0,
                         m(−x) = m(x)a−x ,
                        0 < m(−x) < m(x),
                    0 < m(x) − m(−x) = m(x) 1 − a−x .

Let x → 0 with x decreasing. Then
m(x) decreases, m(−x) increases, m(x) − m(−x) → 0.
It follows that m(x) and m(−x) approach a common limit as x → 0, which is f
(0).
Furthermore, 0 < m(−x) < f '(0) < m(x) for x > 0, which implies that f ' (0) > 0.
We believe that this proof is new. We have been unable to find any other proof
that depends only on methods of differential calculus. Theorem 1 and familiar reasoning give the principal result on the differentiability


special refrence to www.maa.org

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What is function

What Are Functions?

Functions are what we use to describe things we want to talk about mathematically. I find, though, that I get a bit tongue tied when I try to define them.

The simplest definition is: a function is a bunch of ordered pairs of things (in our case the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one another.
Thus, here is an example of a function:

[{1, 1}, {2, 1}, {3, 2}]

This function consists of three pairs, whose first members are 1, 2 and 3.
It is customary to give functions names, like f, g or h, and if we call this function f, we generally use the following notation to describe it:

f(1) = 1, f(2) = 1, f(3) = 2

The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of f here are 1, 2 and 3, and the set consisting of these three numbers is its domain.

The second members of the pairs are called the values of the functions, and the set of these is called the range of the function.

The standard terminology for describing this function f is:

The value of f at argument 1 is 1, its value at argument 2 is 1, and its value at argument 3 is 2, which we write as f(1) = 1, f(2) = 1, f(3) = 2.

We generally think of a function as a set of assignments of values (second members of our pairs) to arguments (their first members).

The condition that the first members of the pairs are all different is the condition that each argument in the domain of f is assigned aunique value in its range by any function.

Above Consider the function g, defined by the pairs (1, 1), (2, 5), (3, 1) and (4, 2). What is its domain? What is the value of g at argument 3? What is g(4)?

If you stick a thermometer in your mouth, you can measure your temperature, at some particular time. You can define a function T or temperature, which assigns the temperature you measure to the time at which you remove the thermometer from your mouth. This is a typical function. Its arguments are times of measurement and its values are temperatures.

Of course your mouth has a temperature even when you don't measure it, and it has one at every instant of time and there are an infinite number of such instants.

This means that if you want to describe a function T whose value at any time t is the temperatures in your mouth at that time, you cannot really list all its pairs. There are an infinite number of possible arguments t and it would take you forever to list them.

Instead, we employ a trick to describe a function f: we generally provide a rule which allows you, the reader, to choose any argument you like in f's domain, and, by using the rule, to compute the value of your function at that argument. This rule is often called a formula for the function. The symbol x is often used to denote the argument you will select, and the formula tells you how to compute the function at that argument.

The simplest function of all, sometimes called the identity function, is the one that assigns as value the argument itself. If we denote this function as f, it obeys

f(x) = x

for x in whatever domain we choose for it. In other words, both members of its pairs are the same wherever you choose to define it.

We can get more complicated functions by giving more complicated rules, (These rules are often called formulae as we have noted already). Thus we can define functions by giving any of the following formulae among an infinity of possibilities:

These represent, respectively, 3 times x, x squared, 3 divided by x, x divided by the sum of the square of x and 1, and so on.

We can construct functions by applying the operations of addition, subtraction, multiplication and division to copies of x and numbers in any way we see fit to do so.

There are two very nice features of functions that we construct in this way, and the first applies to all functions.

We can draw a picture of a function, called its graph on a piece of graph paper, or on a spreadsheet chart or with a graphing calculator. We can do it by taking argument-value pairs of the function and describing each by a point in the plane, with x coordinate given by the argument and y coordinate given by the value for that pair.

Of course it is impossible to plot all the pairs of a function, but we can get a pretty good idea of what its graph looks like by taking perhaps a hundred evenly spaced points in any interval of interest to us. This sounds like an impossibly tedious thing to do and it used to be so, but now it is not. On a spreadsheet, the main job is to enter the function once (with its argument given by the address of some other location). That and some copying is all you have to do, and with practice it can be done in 30 seconds for a very wide variety of functions.

The second nice feature is that we can enter any function formed by adding, subtracting, multiplying, dividing and performing still another operation, on the contents of some address very easily on a spreadsheet or graphing calculator. Not only that, these devices have some other built in functions that we can use as well.

The two of these facts mean that we can actually look at any function formed by adding subtracting multiplying or dividing copies of the identity function x and other built in functions, and any number we want, and see how they behave, with very limited effort.

We will soon see that we can use the same procedure used for constructing functions to construct their derivatives as well, but that is getting ahead of the story. We can compute derivatives for most functions numerically with only a small amount of effort as well.

Ref From math.MIT.edu

Bionomial expansion

Binomial Theorem
For some basic values:

Binomial Expansion

For any power of n, the binomial (a + x) can be expanded


This is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression. There will always be n+1 terms and the general form is:

Some basic properties
Funda: There is one more term than the power of the
exponent, n. That is, there are terms in the expansion
of (a + b)n
.
Funda: In each term, the sum of the exponents is n,
the power to which the binomial is raised.
Funda: The exponents of a start with n, the power of
the binomial, and decrease to 0. The last term has no
factor of a. The first term has no factor of b, so
powers of b start with 0 and increase to n.
Funda: The coefficients start at 1 and increase
through certain values about “half”-way and then
decrease through these same values back to 1.
Funda: To find the remainder when (x + y)n
is divided
by x, find the remainder when yn
is divided by x.
Funda: (1+x)n ≅ 1 + nx,

When binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly?  Consider the following expansions: What observations can we make in general about the expansion of (a + b)n ? 1.  The expansion is a series (an adding of terms). 2. The number of terms in each expansion is one more than n.  (terms = n + 1) 3. The power of a starts with an and decreases by one in each successive term ending with a0.  The power of bstarts with b0 and  increases by one in each successive term ending with bn. 4. The power of b is always one less than the "number" of the term.  The power of a is always n minus the power of b. 5.  The sum of the exponents in each term adds up to n.  6. The coefficients of the first and last terms are each one. 7.  The coefficients of the middle terms form an interesting (but perhaps not easily recognized) pattern where each coefficient can be determined from the previous term.  The coefficient is the product of the previous term's coefficient and a's index, divided by the number of that previous term.          Check it out:   The second term's coefficient is determined by a4:  The third term's coefficient is determined by 4a3b:  To Get Coefficient From the Previous Term: (This pattern will eventually be expressed as a combination of the form n C k..) 8. Another famous pattern is also developing regarding the coefficients.  If the coefficients are "pulled off" of the terms and arranged, they form a triangle known as Pascal's triangle.  (The use of Pascal's triangle to determine coefficients can become tedious when the expansion is to a large power.) 1 1    1 1    2    1 1    3    3    1 1     4    6     4    1(notice the symmetry of the triangle)   (The two outside edges of the triangle are comprised of ones.  The other terms are each the sum of the two terms immediately above them in the triangle.) By pulling these observations together with some mathematical syntax, a theorem is formed relating to the expansion of binomial terms: Binomial Theorem(or Binomial Expansion Theorem) Most of the syntax used in this theorem should look familiar. The notation is just another way of writing a combination such as n C k  (read "n choose k"). (Ref from wild math)  Our pattern to obtain the coefficient using the previous term (in observation #6), , actually leads to the n C k used in the binomial theorem. Here is the connection.  Using our coefficient pattern in a general setting, we get: Let's examine the coefficient of the fourth term, the one in the box. If we write a combination n C k  using k = 3, (for the previous term), we see the connection: The Binomial Theorem can also be written in its expanded form as: Remember that  and that  Examples using the Binomial Theorem: 1.  Expand  . Let a = x, b = 2, n = 5 and substitute.  (Do not substitute a value for k.)      Ref from (purplemaths,regentspre.org)

Foundation of maths

Introduction to the foundations of mathematics

Mathematics and theories

Mathematics is the study of systems of elementary objects, whose only nature is to be exact, unambiguous (two objects are equal or different, related or not; an operation gives an exact result...). Such systems are conceived independently of our usual world, even if many of them can resemble (thus be used to describe) diverse aspects of it. Mathematics as a whole can be seen as «the science of all possible worlds» of this kind (of exact objects). 
Mathematics is split into diverse branches, implicit or explicit frameworks of any mathematical work, that may be formalized as (axiomatic) theories. Each theory is the study of a supposedly fixed system (world) of objects, called its model. But each model of a theory may be just one of its possible interpretations, among other equally legitimate models. For example, roughly speaking, all sheets of paper are systems of material points, models of the same theory of Euclidean plane geometry, but independent of each other.

Foundations and developments

Each theory starts with a foundation, that is the data of a list of pieces of description specifying what it knows or assumes of its model(s) (its kind or shape). This includes a list of formulas (statements) called axioms, expressing the required properties of models, i.e. selecting its accepted models as the systems where the axioms are true, from the whole range of possible systems where they can be interpreted. 
Then, the study of a theory progresses by choosing some of its possible developments : new concepts and information about its models, resulting from its given foundation, and that we can add to it to form its next foundation.
In particular, a theorem of a theory, is a formula deduced from its axioms, so that it is known as true in all its models. Theorems can be added to the list of axioms of a theory without modifying its meaning.

Other possible developments (not yet chosen) can still be operated later, as the part of the foundation that could generate them is preserved. Thus, the totality of possible developments of a theory, independent of the order chosen to process them, already forms a kind of «reality» that these developments explore (before the Completeness theorem will finally show how the range of possible theorems precisely reflects the more interesting reality of the diversity of possible models).

There are possible hierarchies between theories, where some can play a foundational role for others. For instance, the foundations of several theories may have a common part forming a simpler theory, whose developments are applicable to all.
A fundamental work is to develop, from a simple initial basis, a more complete foundation endowed with efficient tools opening more direct ways to further interesting developments.

The cycle of foundations

Despite the simplicity of nature of mathematical objects, the general foundation of all mathematics turns out to be quite complex (though not as bad as a physics theory of everything). Indeed, it is itself a mathematical study, thus a branch of mathematics, calledmathematical logic. Like any other branch, it is made of definitions and theorems about systems of objects. But as its object is the general form of theories and systems they may describe, it provides the general framework of all branches of mathematics... including itself.

And to provide the framework or foundation of each considered foundation (unlike ordinary mathematical works that go forward from an assumed foundation), it does not look like a precise starting point, but a sort of wide cycle composed of easier and harder steps. Still this cycle of foundations truly plays a foundational role for mathematics, providing rigorous frameworks and many useful concepts to diverse branches of mathematics (tools, inspirations and answers to diverse philosophical questions).

(This is similar to dictionaries defining each word by other words, or to another science of finite systems: computer programming. Indeed computers can be simply used, knowing what you do but not why it works; their working is based on software that was written in some language, then compiled by other software, and on the hardware and processor whose design and production were computer assisted. And this is much better than at the birth of this field.) 
It is dominated by two theories:

• Set theory describes the universe of «all mathematical objects», from the simplest to the most complex such as infinite systems (in a finite language). It can roughly be seen as one theory, but in details it will have an unlimited diversity of possible variants (not always equivalent to each other).
• Model theory is the general theory of theories (describing their formalisms as systems of symbols), and their possible models.

Each one is the natural framework to formalize the other: each set theory is formalized as a theory described by model theory; the latter better comes as a development from set theory (defining theories and systems as complex objects) than directly as a theory. Both connections must be considered separately: both roles of set theory, as a basis and an object of study for model theory, must be distinguished. But these formalizations will take a long work to complete, especially for this following last piece:

• Proof theory completes model theory by describing a possible formal system of rules of proofs giving the theorems of any theory. A theory is consistent if its theorems will never contradict each other. Inconsistent theories cannot have any model, as the same statement cannot be true and false on the same system.

Model theory and proof theory are essentially unique, giving a clear natural meaning to the concepts of theory, theorems and consistency of each theory. 
From settheory.net
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Set theory

Set theory

The set theory was developed by George Cantor in 1845-1918. Today, it is used in almost every branch of mathematics and serves as a fundamental part of present-day mathematics.

In set theory we will learn about representation in roster form and set builder form , types of sets (Empty set, singleton set, finite and infinite sets, equal and equivalent sets), cardinal number of a set, subsets (Proper subset, super set, power set), number of proper subsets, universal set, operation on sets (Union, intersection, difference and complement of sets).

In everyday life, we often talk of the collection of objects such as a bunch of keys, flock of birds, pack of cards, etc. In mathematics, we come across collections like natural numbers, whole numbers, prime and composite numbers. 

(Ref Britannia. Math)

Let us examine the following collections:

1 Even natural numbers less than 20, i.e., 2, 4, 6, 8, 10, 12, 14, 16, 18. 

2Vowels in the English alphabet, i.e., a, e, i, o, u. 

3 Prime factors of 30 i.e. 2, 3, 5. 

4Triangles on the basis of sides, i.e., equilateral, isosceles and scalene. 

We observe that these examples are well-defined collections of objects.

Let us examine some more collections. 

1 five most renowned scientists of the world. 

2 Seven most beautiful girls in a society. 

3 Three best surgeons in America. 

These examples are not well-defined collections of objects because the criterion for determining as most renowned, most beautiful, best, varies from person to person. 

Sets:

A set is a well-defined collection of distinct objects.

We assume that,

● The word set is synonymous with the word collection, aggregate, class and comprises of elements.

● Objects, elements and members of a set are synonymous terms.

● Sets are usually denoted by capital letters A, B, C, ....., etc.

● Elements of the set are represented by small letters a, b, c, ....., etc.

If ‘a’ is an element of set A, then we say that ‘a’ belongs to A. We denote the phrase ‘belongs to’ by the Greek symbol ‘∈‘ (epsilon). Thus, we say that a ∈ A.

If ‘b’ is an element which does not belong to A, we represent this as b ∉ A.

Some important sets used in mathematics are

N: the set of all natural numbers = {1, 2, 3, 4, .....}

Z: the set of all integers = {....., -3, -2, -1, 0, 1, 2, 3, .....}

Q: the set of all rational numbers

R: the set of all real numbers

Z+: the set of all positive integers

W: the set of all whole numbers

In other words

1. the branch of mathematics which deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets.

Concept of set theory:- 

set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the defination of complex and sophisticated mathematical concepts.

Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements  or members of the set.

The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinite of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.

At just that time, however, several contradictions in so-called naive set theory were discovered. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Nicolas baurbakis (begun 1939; “Elements of Mathematics”): “Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets.”

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Numbers history

A decimal place system has been traced back to ca. 500 in India. Before that epoch, the Brahmi numeral system was in use; that system did not encompass the concept of the place-value of numbers. Instead, Brahmi numerals included additional symbols for the tens, as well as separate symbols for hundredand thousand.

The Indian place-system numerals spread to neighboring Persia, where they were picked up by the conquering Arabs. In 662, Severus Sebokht - a Nestorian bishop living in Syria wrote:

I will omit all discussion of the science of the Indians ... of their subtle discoveries in astronomy — discoveries that are more ingenious than those of the Greeks and the Babylonians - and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe that because they speak Greek they have arrived at the limits of science would read the Indian texts they would be convinced even if a little late in the day that there are others who know something of value.

The addition of zero as a tenth positional digit is documented from the 7th century byBrahmagupta, though the earlier Bakhshali Manuscript, written sometime before the 5th century, also included zero. But it is in Khmer numerals of modern Cambodia where the first extant material evidence of zero as a numerical figure, dating its use back to the seventh century, is found.

As it was from the Arabs that the Europeans learned this system, the Europeans called them Arabic numerals; the Arabs refer to their numerals as Indian numerals. In academic circles they are called the Hindu–Arabic orIndo–Arabic numerals.

The significance of the development of the positional number system is probably best described by the French mathematician Pierre Simon Laplace (1749–1827) who wrote:

It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.

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