Calculus and Probability for Actuarial Students

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FUNCTIONS

Rectilinear Co-ordinates.

A function of two variables may be represented graphically according to the scheme of the figure shown.

Let two straight lines of indefinite length

OX, 0Y

be drawn at right angles to each other. The point 0 is

called the

origin, OX is

called the

axis of

and

the

axis of y.

Then if

is measured along the axis of

equal in value to

and at that point a

—

perpendicular line PH is drawn equal in

_ -

N X

value to

y = f (x), ON

is called the

abscissa_

and

the

ordinate

of the point

The

convention is taken that measurements of x in the direction OX are considered to be positive and those in the contrary direction negaSimilarly measurements of

in the direction

0 Y

are treated as positive and those in the contrary direction as negative. The point

written for convenience as

(x, y),

is thus completely deter-mined from given values of

and

If for every value of

the corresponding value of

were plotted on a diagram such as the above a continuous curve would be obtained which would be the graphical representation of the equa

y = f (x).

It is, of course, impossible in practice to plot every value of the function, but generally a few values can be filled in so as to enable the curve to be drawn by sight.It should be noted that the lines

and

OY,

and consequently the origin 0, can be chosen quite arbitrarily and that the position of the point P can be fixed, in the manner indicated, with reference to any suitable axes of co-ordinates.

The point P can be fixed with reference to its distance from two straight lines not at right angles to each other, the distance PN = y being measured along a line parallel to the axis of y.

The values

and

corresponding to a given point

are called the

rectilinear co-ordinates

(or simply the

co-ordinates)

of the point P. Where the axes are at right angles to each other the system can be distinguished, if necessary, by referring to

and

as the

rectangular co-ordinates of P.

As a rule rectangular co-ordinates are the more convenient to use in practice and lead to simpler results. Unless otherwise expressed it is to be understood that rectangular co-ordinates are implied.

GRAPHICAL REPRESENTATION 3

5. The following examples give simple cases of the graphical representation of explicit functions.

(i)
The equation x = a clearly represents a straight line parallel to the axis of y and at distance a from it; for the value of x at any point is constant and equal to a.

represents a straight line passing

through the origin and making an angle

with the axis of

where

tan

since at any point the ratio

y : x

is constant and equal to tan O.

Let

be any straight line cutting the axes of

and

respecat the points

and

so that

and

OB = b.

Let P be any point on the line AB, of which the co-ordinates are (x, y). Then, if perpendiculars PN and PM be dropped upon the axes of x and y, MP = x and NP = y.

Alsox _

_ PB

a OA

and

yNPAP

b OB AB'

yPB+AP

+ _

bAB

=1.

Hence

= 1 is the equation of the straight line

AB.

1—2

4 FUNCTIONS

An implicit function can be similarly represented. For ex-ample, it is obvious from the ordinary properties of the circle that the implicit relationship x' + y' = a' represents a circle of radius a with its centre at the origin.

Note.

The function

bx + cx' + de + ...

is sometimes called

parabolic

function, since the equation

bx + cx'

is represented graphically by a curve which is known as a parabola.

It does not follow that for every value of x there will always be a real value of y.

Thus, consider the function

(x — a) (x —

b) (x — c),

where

c > b >

is negative, the right-hand side of the equation is negative and

can have no real value. If

is positive and <

the position is the same. If, however, x >

and

< b,

then the right-hand side is positive and y has a real value; but when x

> b

and < c, y is again unreal and remains so until x > c when a revalue of

results for each value of

The form of the curve is shown below, where

OB = b

and OC = c.

In circumstances such as these,Y

where one or more parts of acurve are isolated from theothers, the function and the

curve representing it are said to

0 4B C

discontinuous.

8.It is convenient here to

introduce the conception of the

limiting value

of a function, or simply a

limit.

If y = f (x) and y continuously tends towards a certain value, 1, and can be made to differ by as little as we please from that value by making x approach some fixed value a, then 1 is said to be the limiting value of f (x) when x tends to the value a.

convenient notation is as follows :

when

Also

would be expressed as Lt

f (x).