Numerical differentiation using Stirling's formula calculator

SolutionHelp

1. From the following table of values of x and y, obtain

$(dy)/(dx)$ and

$(d^2y)/(dx^2)$ for x = 7.5

x	7.47	7.48	7.49	7.50	7.51	7.52	7.53
f	0.193	0.195	0.198	0.201	0.203	0.206	0.208

2. From the following table of values of x and y, obtain

$(dy)/(dx)$ and

$(d^2y)/(dx^2)$ for x = 900

x	0	300	600	900	1200	1500	1800
f	135	149	157	183	201	205	193

Example

1. Using Stirling's formula to find solution
x f(x)
7.47 0.193
7.48 0.195
7.49 0.198
7.50 0.201
7.51 0.203
7.52 0.206
7.53 0.208

x = 7.5

Solution:
Stirling's formula (central difference formula).
The value of table for

$x$ and

$y$

x	7.47	7.48	7.49	7.5	7.51	7.52	7.53
y	0.193	0.195	0.198	0.201	0.203	0.206	0.208

Difference table is

x	y	$Deltay$	$Delta^2y$	$Delta^3y$	$Delta^4y$	$Delta^5y$	$Delta^6y$
7.47	0.193
		0.002
7.48	0.195		0.001
		0.003		-0.001
7.49	0.198		0		0
		0.003		-0.001		0.003
7.5	0.201		-0.001		0.003		-0.01
		0.002		0.002		-0.007
7.51	0.203		0.001		-0.004
		0.003		-0.002
7.52	0.206		-0.001
		0.002
7.53	0.208

The value of

$x$ at you want to find

$f(x) : x_0 = 7.5$

$h = x_1 - x_0 = 7.48 - 7.47 = 0.01$

Stirling's Formula is

$[(dy)/(dx)]_(x=x_0) = 1/h * [1/2 * (Delta y_0 + Delta y_(-1)) - 1/12 * (Delta^3 y_(-1) + Delta^3 y_(-2)) + 1/60 * (Delta^5 y_(-2) + Delta^5 y_(-3)) + ...]$

$:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [1/2 * (0.002 +0.003) - 1/12 * (0.002 -0.001)+ 1/60 * (-0.007 +0.003)]$

$:.[(dy)/(dx)]_(x=7.5) = 1/0.01 * [0.0025-0.0000833333-0.0000666667]$

$:.[(dy)/(dx)]_(x=7.5) = 0.235$

$[(d^2y)/(dx^2)]_(x=x_0) = 1/h^2 * [Delta^2 y_(-1) - 1/12 Delta^4 y_(-2) + 1/90 * Delta^6 y_(-3) + ...]$

$:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001 - 1/12 * 0.003+ 1/90 * -0.01]$

$:.[(d^2y)/(dx^2)]_(x=7.5) = 1/0.0001 * [-0.001-0.00025-0.0001111111]$

$:.[(d^2y)/(dx^2)]_(x=7.5) = -13.61111$