---
title: "Stirling formula: y(35)"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:stirling-y35"
source: https://www.bctengineer.com/q/semester_4/numerical_method/stirling-y35
site: BCT Engineering
---

# Stirling formula: y(35)

Numerical Method · Chapter 4: Interpolation

- Asked in: 2068 Magh, 8 marks
- Video: [Stirling's interpolation formula](https://www.youtube.com/watch?v=UC8A9bt_-J0) — Civil learning online
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/stirling-y35
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Use Stirling's formula to compute y(35) from the following table:
| X | 20 | 30 | 40 | 50 |
|---|---|---|---|---|
| Y | 512 | 439 | 346 | 243 |

## Short answer

With x0 = 30, h = 10, p = (35-30)/10 = 0.5. Stirling's formula yields y(35) = 439 - 41.5 - 2.5 - 0.625 = 394.375.

## Answer

There are four equally spaced points (h = 10) and x = 35 lies in the middle, so Stirling's central difference formula is used with x₀ = 30 (p = (35 − 30)/10 = 0.5).

#### Difference table

| x | y | Δy | Δ²y | Δ³y |
|---|---|---|---|---|
| 20 | 512 | -73 | -20 | 10 |
| 30 | 439 | -93 | -10 |  |
| 40 | 346 | -103 |  |  |
| 50 | 243 |  |  |  |

Central differences about x₀ = 30: Δy₋₁ = −73 (from x = 20), Δy₀ = −93, Δ²y₋₁ = −20, and the only third difference available is Δ³y₋₁ = 10.

Stirling's formula:

y(x₀ + ph) = y₀ + p·(Δy₋₁ + Δy₀)/2 + p²/2!·Δ²y₋₁ + p(p²−1)/3!·(Δ³y₋₂ + Δ³y₋₁)/2 + …

Only four points exist, so Δ³y₋₂ is not available; the single third difference is used in its place.

```text
y(35) = 439 + 0.5(−73 − 93)/2 + (0.5)²/2 (−20) + 0.5(0.25 − 1)/6 (10)
      = 439 − 41.5 − 2.5 − 0.625
      = 394.375
```

Without the third-difference term the value is 395.000; with it, 394.375. The cubic through all four points gives exactly 394.375, so the correction is justified.

**Answer: y(35) ≈ 394.375**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is p for y(35) with x0 = 30, h = 10?
   A. 3.5
   B. 0.5
   C. 0.35
   D. -0.5
   Answer: B. p = 5/10 = 0.5. (Key point: p)

2. What is the first-order term?
   A. p·Δy0 = -46.5
   B. p(Δy-1 + Δy0)/2 = -41.5
   C. p·(-73)
   D. 0
   Answer: B. 0.5 × (-166/2) = -41.5. (Key point: Mean term)

3. Why is Δ³y-1 used alone?
   A. Only four points, so Δ³y-2 is unavailable
   B. It is zero
   C. Formula says so
   D. Mistake
   Answer: A. Only one third difference exists; the result matches the cubic, 394.375. (Key point: Third difference)

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