Calculus and Probability for Actuarial Students

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CHAPTER XV
RELATION OF DIFFERENTIAL CALCULUS TO
FINITE DIFFERENCES

1.By Taylor's Theorem we have

.f(x+1)=f(x)+f'(x)+f2

^x)

+....

But

f (x +

= Ef (x),

therefore if the operation

be denoted by

D we may write symbolically

Ef(x)=(1+

_1i

D D2

+2i+...)f(x)

^=eD

)

whence

-1

andD=log

(1+A).

Therefored = A —

If the interval of differencing be

we have

f (^x+^h) = f (x)+hf'(x)+ 24 f ^""(x)+...

and therefore

^hD

-1.

Further,D' _ [log

(1 + A)]

whence

[

A— 2

³⁺

^3y

02y —

^A3y

+ iIA4y — i

^A,'y

(2).

Similarly

y —

4A'y + AA5y —

(3).

The above formula (1) gives a convenient expression for the differential coefficient in terms of advancing differences. But from the nature of the differential coefficient, it may be expected that a better result will be obtained by applying the formulas of central differences.

DIFFERENTIAL CALCULUS AND FINITE DIFFERENCES 89 Thus, taking Stirling^'s formula, we have f(x)=f(0)+x°f(0)+2Af(-1)+ ^of(-¹)
₊x(x'--1) 0 f(—1)+A3f(-2)+....
3!    2 Differentiating with regard to x,
f (x) __Of(o)+Af(—1)—A3f(—1)+O$f(—2)
2    12      +
+ terms involving x and its powers.
Putting x = 0 we obtain
₀=AJ(0)+Af(—1)—A3f(—1)+A3f(—2)+... (4).
_f, (0)    2    ₁₂Taking the first term only, we have
    f,(0)_A.f(0)+Of(—1)f(1)—f(—1)    (5),
2    2
a very useful approximation, which, by altering the origin and the unit of measurement, may be expressed more generally as
f, (x) f(^x+^h) _f(^x-^h)
If h = 1 we arrive at the approximate result
f' (x) _ A f (x — 2)     (7)
=A(1+0) ^lf(x)
=[0— ^A'+$^A3ifiA'+issA'—...]f(x). Comparing this with the value given by formula (1) we see that the error involved in taking f ' (x) = A f (x — 2) is

[ Q403+1',;04 4470A5+...]f(x),

which is approximately equal to — } 03 f (x — D.
Hence we may write
f'(x)=of(x-2)-2~4A'f(x-i) (8).
3. The introduction of the second term in equation (4) gives the useful approximation
_f, (₀₎— ⁸{.f (¹) -f(-¹)} — If (2) -f(^— 2)} (9).
12

The other formulas of central differences, when treated as above, give these and other expressions for f' (x). The above, however, are the most accurate and useful for general use.

90 DIFFERENTIAL CALCULUS AND FINITE DIFFERENCES

Using the central difference formula, we shall obtain for the differential coef^ficients of higher orders,

f"(0)=0'f(—1)—

0'f(—2)

(10)

and

,,.

(0)

off(

^—

⁰³

(—

)

= Q3 f(^— i) approx (11).

5.An example of the working of the above formulas is now given.

Example.

Find the first three differential coefficients of logx, when x =

having given the values of log

1'80, log

90, ....

Number	N log ^al	A	A^a			p5
1^.80	•587787	054067	— 1)02774	•000271	— '000038	•000007
1'90	'641854	•051293	— •002503	•000233	— '000031	•000005
2^.00	'693147	'048790	— •002270	'000202	— '000026	'000004
2^.10	•741937	•046520	— •002068	'000176	— '000022	•000004
2^.20	'788457	'044452	— 001892	•000154	— '000018
2^.30	•832909	042560	— '001738	'000136
2^.40	'875469	'040832	— '001602
2^.50	'916291	'039220
2^.60	'955511

Using the advancing difference formula (1), starting at the term 2^.00, and remembering that the interval of differencing is not unity but '1, we get

'00^.2270 '000202 '000026 '000004 f^' (x) = 10 (•048790 + ₂ + ₃ + ₄ + ₅ )

= '50000.
Similarly
f" (x) = 100 (— '002270 — •000202 — x •000026 — x '000004) — '2499,
f "' (x) = 1000 ('000202 + x '000026 +'-~ x '000004)
= '248.
The true values are, of course, given by the differential coefficients

of log x, which are respectively 1_, — 1 2 When x = 2 these x x^3'become ,}, — 4 , i; the closeness of the above approximations is apparent.
Formulas (6) and (9) give respectively for f' (x) values of '50042 and '50000.