2. Using Stirling's formula to find solution
| x | f(x) |
| 0 | 135 |
| 300 | 149 |
| 600 | 157 |
| 900 | 183 |
| 1200 | 201 |
| 1500 | 205 |
| 1800 | 193 |
x = 900
Solution:
Stirling's formula (central difference formula).
The value of table for `x` and `y`
| x | 0 | 300 | 600 | 900 | 1200 | 1500 | 1800 |
|---|
| y | 135 | 149 | 157 | 183 | 201 | 205 | 193 |
|---|
Difference table is
| x | y | `Deltay` | `Delta^2y` | `Delta^3y` | `Delta^4y` | `Delta^5y` | `Delta^6y` |
| 0 | 135 | | | | | | |
| | 14 | | | | | |
| 300 | 149 | | -6 | | | | |
| | 8 | | 24 | | | |
| 600 | 157 | | 18 | | -50 | | |
| | 26 | | -26 | | 70 | |
| 900 | 183 | | -8 | | 20 | | -86 |
| | 18 | | -6 | | -16 | |
| 1200 | 201 | | -14 | | 4 | | |
| | 4 | | -2 | | | |
| 1500 | 205 | | -16 | | | | |
| | -12 | | | | | |
| 1800 | 193 | | | | | | |
The value of `x` at you want to find `f(x) : x_0 = 900`
`h = x_1 - x_0 = 300 - 0 = 300`
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=900) = 1/300 * [1/2 * (18 +26) - 1/12 * (-6 -26)+ 1/60 * (-16 +70)]`
`:.[(dy)/(dx)]_(x=900) = 1/300 * [22+2.6666666667+0.9]`
`:.[(dy)/(dx)]_(x=900) = 0.08522`
`[(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=900) = 1/90000 * [-8 - 1/12 * 20+ 1/90 * -86]`
`:.[(d^2y)/(dx^2)]_(x=900) = 1/90000 * [-8-1.6666666667-0.9555555556]`
`:.[(d^2y)/(dx^2)]_(x=900) = -0.00012`
This material is intended as a summary. Use your textbook for detail explanation.
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