Page 28 - Essentials of physical chemistry

xxvi Introduction: Mathematics and Physics Review

beginning chapter since at a later time we may use it for an approximation when x is small. The
general formula for the Taylor expansion is
4
3
2
x d x x x x
1 n n
X
f (x) ,
n! dx 2 3! 4!
f (x) ¼ f (0) þ so e ffi 1 þ x þ þ þ þ
n¼1 x¼0
n
x
Note for future reference, in any limiting case involving e compared to any single function of x , the
x
x
behavior of e will dominate since e contains every power of x and more higher powers besides; e x
will always ‘‘beat’’ any single power of x in a limiting case. That is an example of a tedious proof of
something you can safely memorize in that when x is small it can be seen from the series expansion
x
that only the first two terms are needed so that we can often use: e ffi 1 þ x, when (x 1).
Along the way we used n! which we will also find very useful later on; it is called ‘‘n-factorial’’ and
is defined as n! 1 2 3 4 5 6, . . . n, that is, the product of successive integers up to n. The use of
factorial notation will be common later in the text and it should be clear that it is a convenient form to
represent what may be a very large product string of integers. Many of the inexpensive calculators
have a special key for n! so try to find out the limit of your calculator and probably you will get an
overflow message at about 69! Be prepared to stretch your imagination later on to use the factorial of a
number as large as Avogadro’s number! We will see later there is a good approximation for factorials
of large numbers and Stirling’s approximation will be discussed in later chapters.
The next thing to review is that we will often have to take a derivative of a function, which is the
product of several functions so we need to review the derivative of a product.

dy d dV dU
Given: y ¼ U(x) V(x), then ¼ (UV) ¼ U þ V .
dx dx dx dx
This principle can be extended to multiple products as well.
Next, we need to consider the case of a quotient. I prefer to use the ‘‘fall off the roof’’ idea as
applied to a negative power.

U(x) d U(x) d 1 1 dU 2 dV
, then (UV ) ¼ (V ) þ U( 1)(V ) .
V(x) dx V(x) dx dx dx
Given: y ¼ ¼
This is equivalent to the form usually given in calculus books if the last term is multiplied by

d U V(dU=dx) U(dV=dx)
=
VVÞ to obtain the form as .
dx V V
ð ¼ 2
This exercise is shown to illustrate another example:

d 1 d n (nþ1) d 4 4
3
[x ] ¼ ( n) x so that 1 x ¼ ( 3)(x ) ¼ 3 x :
¼
dx x n dx dx
Now we need to venture into what may be new to you in the form of integration. Basically,
integration is the reverse of taking a derivative so that when you do the step mentally, you say to
yourself, what is the function whose first derivative is the function in the integrand, the function
under the integral sign? There are two basic types of integrals, a definite integral, which is evaluated
between two certain limit values of the argument and results in a definite numerical value. A more
general type of integral is an indefinite integral, which produces a function whose derivative is the
integrand function in the integration formula, but (!) since the derivative of a constant is zero, there
might have been some constant number associated with the integrated function, which might need
some sort of additional information such as a boundary condition to evaluate.
b dg(x)
ð
Definite integral: f (x)dx ¼ g(b) g(a), where ¼ f (x) and we note the order of the limits
a dx
implies g(upper) g(lower). Also note that in one variable, the integral is an area which is the
product of a sliding value of the function with a tiny slice dx. In some cases, a definite integral can
be approximated by plotting the integrand function versus x, cutting out and weighing the paper area