Basic Calculus for Mathematical Statistics

Introduction We live in a dynamic universe. Nothing

ever stays the same. Everything is always changing. The only thing that remains constant is change. It is because of this reality that it is difficult to give an accurate

description of events. Even as you are beginning to describe something, that thing you are describing changes.

Objectives At the end of this lesson, you will be able

to:

compute for the derivative of a given polynomial function;

find the slope of the tangent line at any point on a given polynomial curve;

determine the maximum and minimum point of a polynomial function;

find the integral or antiderivative of a given polynomial function; and

use integrals to determine areas under a given polynomial curve.

Average Rate of Change

If two quantities are functionally related, a change in one of them generally implies a corresponding change in the other.

Example: Consider the two quantities s representing

the monthly salary of an employee and t representing the number of years he has been working in a company.

We can write the relationship between s and t in mathematical notation as ( )s s t= which states that s is a function of t.

Such a function can also be represented by

a table giving the value of s corresponding to every value of t. Suppose the values of s for

different values of t are summarized in table below.

t

(years) 0 1 2 3 4 5

s (pesos) 5 000 6 000 8 000 12 000 20 000 36 000

From this table we can see that from 2t = to 4t = years, the employee's salary changed

from 8 000 to 20 000 pesos. Suppose we denote by s the increment in the employee's salary in a period of t years. Then the table above tells us that when 4 2 2t = = years,

20 000 8 000 12 000s = = pesos.

In order to get an idea of the rate at which the employee's salary charged over the two-year period, we normally divide s by t to obtain a quantity called the average rate of change of s with respect to t.

ave. rate of change of s with respect to t st

=

In this particular example, the average rate of change of the salary of the employee over the two-year period 2t = to 4t = years is

equal to 12 000 6 0002

= pesos per year.

We can get a very useful geometrical interpretation of average rate of change of one quantity y with respect to another quantity x if we plot y against x on the Cartesian plane. The average rate of change of y with respect to x is then just the ratio of the two lengths

y and x . This ratio is called the slope of the line through the points 1P and 2P of the curve. If 1P has coordinates ( )1 1,x y and 2P has coordinates ( )2 2,x y , then the slope of the line through 1P and 2P is

2 1

2 1

y y ymx x x

= =

.

If the points 1P and 2P lie on a curve described by the function ( )y f x= , then

1 1( )y f x= and 2 2( )y f x= , and the slope of the line through 1P and 2P is

2 1

2 1

( ) ( )f x f xmx x

=

.

From the definition, we'll notice that the slope of a line can be positive, negative, or zero. If a line is rising from left to right, its slope is positive. If the line is falling from left to right, its slope is negative. A horizontal line has zero slope.

Exercises:

Find the slope of the line through the following pairs of points: a) ( )0,2 and ( )3,1 b) ( )0,2 and ( )5,2 c) ( )1,3 and ( )3,6

Sequences and Limits

The values 1.0, 1.5, 1.8, 1.9, 1.999 form what mathematicians call a sequence of real numbers. As we've seen, if we continue generating numbers in this sequence, the new numbers generated will become closer and

closer to the value 2.0. The value of the real number to which the sequence of numbers approaches as more and more terms are generated is called the limit of the sequence.

An infinite sequence is usually denoted by a

succession of terms 1 2 3, , ,..., ,...na a a a where

every natural number n corresponds to a term na in the sequence. It is in this sense that we

can consider a sequence as a function of the natural numbers:

( )na f n= .

We can think of the function ( )f n as a prescription for determining the nth term of the sequence.

Example: The sequence generated by the function

21( )na f n n

= = is 1 1 11, , , ,...4 9 16

. The terms in

the sequence are obtained by successively substituting the natural numbers 1, 2, 3, in the generating function.

It frequently happens that a certain infinite sequence of numbers approaches a certain value as more and more terms of the sequence are generated. When this happens, the sequence is said to be convergent, and the value L approached by the sequence is called the limit of the sequence. Mathematicians also

say that the sequence converges to the limit L. We can write this in mathematical symbols as

lim nn a L = , which is a shortcut notation for the statement, "The limit of na as n approaches infinity is equal to L."

Two Basic Limits

1lim 0n n

=

limn

n

=

Properties of Limits:

If lim nn a A = , lim nn b B = , and C is a constant, then

1) lim

nC C

=

2) lim limn nn nC a C a C A = = 3) ( )lim lim limn n n nn n na b a b A B + = + = + 4) ( )lim lim limn n n nn n na b a b A B = =

5) lim

limlim

nn n

nn nn

aa Ab b B

= =

provided 0B .

Exercises: Given na , find lim nn a .

a) 21n

+

b) 23 nn+

c) 3 1nn+

d) 2 nn+

Limits of Functions

As in the case of sequences, the expression lim ( )x a

f x

represents the value which the function ( )f x approaches when x approaches the value of a.

Example: Evaluate ( )22

lim 5x

x

+ . Notice that for

3, 2.5, 2.4, 2.3, 2.2, 2.1, 2.01,x = 2 5x + takes the values 14, 11.25, 10.76,

10.29, 9.84, 9.41, 9.04, respectively. We can see from this sequence that the value of the function comes closer and closer to the value

9 as the value of x comes closer and closer to 2. In mathematical symbols, we write this information in the shortcut notation

( )22

lim 5 9x

x

+ = .

Notice that the limit of the function 2 5x + as x approaches 2 is the same as its value

when 2x = , i.e., in this case lim ( ) ( )x a

f x f a

= . However, this is not always true for all functions.

Limits of functions obey the same rules that

we used in connection with sequences.

Exercises: Use the limit rules to evaluate the following:

a) 2

3

8lim1x

xx+

b) 2

2

5 6lim2x

x xx +

c) 3

2

8lim2x

xx++

Derivative

If ( )f x is a function of x, we define its derivative or instantaneous rate of change at

0x , denoted by 0

00

0

( ) ( )'( ) limx x

f x f xf xx x

=

provided the limit exists.

If we let 0h x x= , we can also write this as 0 0

0 0

( ) ( )'( ) limh

f x h f xf xh

+ = .

At an arbitrary value of x, we can find the

derivative '( )f x by replacing 0x in the preceding expression by x:

0( ) ( )'( ) limh

f x h f xf xh

+ =

Notice that this last expression gives the

derivative of the function ( )f x as a function of x.

Example: Find the derivative of the following functions at an arbitrary value of x. a) ( )f x x= b) 2( )f x x= c) 3( )f x x=

We can easily generalize the results of the preceding example: The derivative of 1( )f x x x= = is

1 1 0'( ) 1 1 1f x x x= = = The derivative of 2( )f x x= is

2 1 1'( ) 2 2 2f x x x x= = =

The derivative of 3( )f x x= is 3 1 2 2'( ) 3 3 3f x x x x= = =

If we follow this pattern, we should have: the derivative of 4( )f x x= is

4 1 3'( ) 4 4f x x x= = , and so on.

We therefore see that the derivative of ( ) nf x x= must be 1'( ) nf x nx = . This gives us

the general formula for the derivative of any power of x.

Exercises:

Evaluate the derivative of the following functions at an arbitrary point x. a) 5 3 2( ) 3 2 5 6f x x x x= + + + b) 23

4( ) 2f x xx

= +

We can interpret the derivative or instantaneous rate of change of a function

0'( )f x as the slope of the tangent line to the graph of ( )f x at 0x .

It is because of this interpretation that the

derivative of a function ( )y f x= is also

represented by the symbol dydx

. (Note that this

reminds us of the expression for average rate of change, which is y

x

. Note also that the

expression dydx

does not mean dy divided by

dx, but rather the instantaneous rate of change of y with respect to x.)

The geometrical interpretation makes it easy to find the slope of the tangent line at any point on the curve. The sign of the derivative also allows us to have an idea of

whether the curve representing the function ( )f x is rising or falling as x increases. A

positive derivative means that the curve is rising (i.e., ( )f x is increasing) as x increases. A negative derivative would mean that the curve is falling (i.e., ( )f x is decreasing) as x increases. A zero value of the derivative of

( )f x would mean that the tangent line to the curve is horizontal. This can only happen if

( )f x has a maximum or a minimum point or as what mathematicians call an inflection point. The table below summarizes the behavior of ( )f x at these points.

point '( )f x behavior of ( )f x 1x negative decreasing 2x zero minimum 3x zero inflection point 4x positive increasing 5x zero maximum

Exercises:

Find the values of x for which the function 3 24 4y x x x= + has zero derivative. Sketch

the graph of the function in order to find out if each of these values of x is a minimum, maximum, or inflection point.

Antidifferentiation and Application

So far, the problems we've been considering involve finding the derivative of a given function. This is called differentiation. Now let us turn the problem inside out: Given the derivative of a function, what is the function?

(It's just like asking what the question is if you know the answer.)

The process of finding a function if its

derivative is given is called antidifferentiation or integration. If the result of differentiation is the derivative, the result of antidifferentia-

tion or integration is the antiderivative or integral.

Example:

We know that the derivative of the function 3 2( ) 5 2 3f x x x x= + + is the function

2( ) '( ) 15 4 3g x f x x x= = + + . Then we can say that an antiderivative or integral of the function 2( ) 15 4 3g x x x= + + is the function

3 2( ) 5 2 3f x x x x= + + .

If we are sensitive to the nuances of language, we'll immediately notice that we said an antiderivative or integral rather than the antiderivative or integral. This is because

3 2( ) 5 2 3f x x x x= + + is not the only possible result of the antidifferentiation of

2( ) 15 4 3g x x x= + + . In fact, we can add an

arbitrary constant c to the function ( )f x , and the resulting function will still be an antiderivative of ( )g x . Notice that the reason we can do this is that the derivative of a constant is zero. We therefore see that the antiderivative or integral of the function

2( ) 15 4 3g x x x= + + is the family of functions

3 2( ) 5 2 3f x x x x c= + + + , where c is an arbitrary constant.

In order to emphasize the inverse

relationship between differentiation and antidifferentiation, they are usually denoted by the symbol D and 1D , respectively. The

symbol ( )Df x means the derivative of ( )f x , while the symbol 1 ( )D g x means the antiderivative of ( )g x . Using this new notation, we can therefore state the derivative-integral inversion theorem:

If ( ) ( )Df x g x= , then 1 ( ) ( )D g x f x c = + .

An arbitrary constant always appears whenever we take the antiderivative of a function. The easiest functions to differentiate and integrate are the powers of x. As we learned in the preceding section, the derivative of nx is 1n nDx nx = . If n is replaced by 1n + in this formula, we'll get

( )1 1 ,n nDx n x+ = + or equivalently,

1

1

nnxD x

n

+ = +

.

If we use the derivative-integral inversion theorem, this must mean that the integral of

nx is

11

1

nn xD x c

n

+ = +

+

where c is an arbitrary constant. This formula is valid for all values of n except 1 , because this will make the denominator of the first

term on the right hand side of the equation equal to zero. Example:

5 11 5 61

5 1 6xD x c x c

+ = + = +

+.

Basic Rules for Antidifferentiation 1) The antiderivative of a constant c times a function ( )f x is equal to c times the antiderivative of the function ( )f x :

[ ]1 1( ) ( )D cf x cD f x = .

2) The antiderivative of the sum of functions is equal to the sum of their antiderivatives:

If 1 ( ) ( )D f x F x = and 1 ( ) ( )D g x G x = , then [ ]1 ( ) ( ) ( ) ( )D f x g x F x G x + = + .

In place of the symbol 1 ( )D f x for integration, a more commonly used symbol is the expression ( )f x dx . We read this symbol as "the integral of ( )f x dx" or more accurately as "the integral of ( )f x over the variable x." Using this notation, we can write the power

integration formula and the basic rules for antidifferentiation as

1

1

nn xx dx c

n

+

= ++

( ) ( )cf x dx c f x dx= [ ]( ) ( ) ( ) ( )f x g x dx f x dx g x dx+ = +

Exercise:

Evaluate the following integral using the power integral formula, and the basic integration rules stated above:

22

3 2 2x dxx

+ + .

An important application of integration is finding areas under a curve. For this

application, we need the symbol ( )b

a

f x dx , which is the definite integral of ( )f x from a to b. This is defined as follows:

If ( ) ( )f x dx F x c= + , then ( ) ( ) ( )

b

a

f x dx F b F a= provided ( )f x is continuous on the interval [ ],a b .

The definite integral ( )b

a

f x dx has a very important geometrical interpretation: It is the area in the x- y plane bounded by the curve

( )y f x= and the two vertical lines x a= and x b= .

Exercise:

Find the area under the curve 24y x x= from 1x = to 3x = .

Reference: Fundamental Concepts and Applications of Mathematics by Reuben V. Quiroga