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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)