CHAPTER X
DIFFERENTIAL CALCULUS. ELEMENTARY CON
CEPTIONS AND DEFINITIONS
- In the subject of Finite Differences we were concerned with the changes in the value of a function consequent upon finite changes in the value of the independent variable. In the Differential Calculus we consider the relation of Ay to Ax when the value of Ax is made indefinitely small.
The application of the Differential Calculus is largely limited to such values of a function as are finite and continuous, and, unless otherwise stated, this limitation is to be implied in the following demonstrations. In practice these conditions are almost universally fulfilled by functions entering into actuarial calculations.
- Let y = f (x) and let x receive an increment h. Then the change in the value of y is measured by f (x+ h) — f (x) and the rate of change of y is f (x + hh —f (x) . The limit of this expression when h 0 is called the Digrerential Coefficient or First Derived Function of f (x) with respect to x.
The operation of obtaining this limit is called differentiating f (x). Using the notation of Finite Differences the differential co-efficient becomesLt AI= oz-roAx
and is variously denoted by dx , f' (x), f (x), d d(x) , D f (x).
The symbol dx or its equivalent represents an operation of the
character described ; the elements dy and dx must not be regarded as separate Y
small quantities.
- The geometrical representation of the differential coefficient is illustrated in the accompanying diagram.
Let the curve shown represent the
0 T
M N X
64 DIFFERENTIAL CALCULUS
equation y = f (x). Let 0111= x and ON = x + h, and let PM and QN be the corresponding ordinates. Let PR be the perpendicular from P on QN and let QP be produced to cut OX at T.
Then .f (x + h) — f (x) _ QN — I'M QR PM _tan PTlli
h MN =1'R=T~II —
When the point Q moves up to, and ultimately coincides with the point P, the line QPT becomes the tangent to the curve at the
point P. The limiting value off (x + h) f'x) is therefore the h
tangent of the angle which the tangent to the curve at the point (x, y) makes with the axis of x.