---
title: "Lagrange: f(27.5)"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:lagrange-f-27-5"
source: https://www.bctengineer.com/q/semester_4/numerical_method/lagrange-f-27-5
site: BCT Engineering
---

# Lagrange: f(27.5)

Numerical Method · Chapter 4: Interpolation

- Asked in: 2069 Poush, 6 marks
- Video: [●Lagrange Interpolation || Numerical Concept ✅️](https://www.youtube.com/watch?v=YlVOYcsyNbU) — The PrepStudio
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/lagrange-f-27-5
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Using Lagrange's interpolation formula evaluate f(27.5) from the table:
| x | 26 | 27 | 28 | 29 | 30 |
|---|---|---|---|---|---|
| f(x) | 3.846 | 3.704 | 3.571 | 3.448 | 3.333 |

## Short answer

Applying five-point Lagrange interpolation to the given data yields f(27.5) ≈ 3.6362. The data represent f(x)=100/x, and 100/27.5 = 3.6364.

## Answer

Five points are given, so Lagrange's formula gives a polynomial of degree 4:

f(x) = Σ f(xᵢ)·Lᵢ(x),   Lᵢ(x) = Π (x − xⱼ)/(xᵢ − xⱼ)  (j ≠ i)

With x = 27.5 the factors (x − xⱼ) are: 1.5, 0.5, −0.5, −1.5, −2.5 for xⱼ = 26, 27, 28, 29, 30.

```text
L₀ = (0.5)(−0.5)(−1.5)(−2.5) / [(−1)(−2)(−3)(−4)] = −0.9375/24  = -0.039062
L₁ = (1.5)(−0.5)(−1.5)(−2.5) / [(1)(−1)(−2)(−3)]  = −2.8125/−6  =  0.468750
L₂ = (1.5)(0.5)(−1.5)(−2.5) / [(2)(1)(−1)(−2)]    =  2.8125/4   =  0.703125
L₃ = (1.5)(0.5)(−0.5)(−2.5) / [(3)(2)(1)(−1)]     =  0.9375/−6  = -0.156250
L₄ = (1.5)(0.5)(−0.5)(−1.5) / [(4)(3)(2)(1)]      =  0.5625/24  =  0.023438
```

| i | xᵢ | f(xᵢ) | Lᵢ(27.5) | f(xᵢ)·Lᵢ |
|---|---|---|---|---|
| 0 | 26 | 3.846 | -0.039062 | -0.150234 |
| 1 | 27 | 3.704 | 0.468750 | 1.736250 |
| 2 | 28 | 3.571 | 0.703125 | 2.510859 |
| 3 | 29 | 3.448 | -0.156250 | -0.538750 |
| 4 | 30 | 3.333 | 0.023438 | 0.078117 |
|  |  | Σ | 1.000000 | 3.63624 |

Check: ΣLᵢ = 1.

**Answer: f(27.5) ≈ 3.6362** (the data are f = 100/x, and 100/27.5 = 3.6364).

## Check yourself

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

1. What degree polynomial do five points give?
   A. 3
   B. 2
   C. 4
   D. 5
   Answer: C. n points give degree n - 1. (Key point: Degree)

2. What should ΣLi equal?
   A. f(x)
   B. 1
   C. 0
   D. 5
   Answer: B. Coefficients sum to 1. (Key point: Sum check)

3. What is f(27.5)?
   A. About 3.704
   B. About 3.571
   C. About 2.75
   D. About 3.6362
   Answer: D. Sum of the f·L terms is 3.63624, near 100/27.5. (Key point: Result)

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