Interpolation Examples

Practical examples demonstrating linear interpolation and polynomial interpolation in real-world scenarios. Learn through hands-on examples from science, engineering, and finance.

Temperature vs. Altitude (Linear)

Scientific Data

A meteorologist measures temperature at different altitudes and needs to estimate the temperature at 1500 meters.

Given Data Points:

Altitude (m)	Temperature (°C)
1000	15.2
2000	8.7

Find: Temperature at 1500m

Solution using Linear Interpolation:

Formula:

T = T₁ + (T₂ - T₁) × (h - h₁) / (h₂ - h₁)

Substitution:

T = 15.2 + (8.7 - 15.2) × (1500 - 1000) / (2000 - 1000) T = 15.2 + (-6.5) × (500) / (1000) T = 15.2 - 3.25 = 11.95°C

Result: Temperature at 1500m = 11.95°C

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Population Growth (Polynomial)

Statistical Analysis

A city planner has population data for several years and needs to estimate the population for 2018 using polynomial interpolation.

Given Data Points:

Year	Population
2015	45000
2017	48500
2019	52800
2021	57200

Find: Population in 2018

Solution using Polynomial Interpolation:

Using Newton's divided differences method, we construct a polynomial that passes through all four data points.

Divided Difference Table:

x	f[x]	f[x₀,x₁]	f[x₀,x₁,x₂]	f[x₀,x₁,x₂,x₃]
2015	45000	1750	-87.5	10.42
2017	48500	2150	25
2019	52800	2200
2021	57200

Result: Population in 2018 ≈ 50,425 people

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Material Stress Testing (Linear)

Engineering

An engineer tests material strength at different stress levels and needs to estimate the strain at an intermediate stress value.

Given Data Points:

Stress (MPa)	Strain (mm/mm)
100	0.0012
200	0.0025

Find: Strain at 150 MPa

Solution:

Strain = 0.0012 + (0.0025 - 0.0012) × (150 - 100) / (200 - 100) Strain = 0.0012 + 0.0013 × 0.5 = 0.00185 mm/mm

Result: Strain at 150 MPa = 0.00185 mm/mm

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Investment Returns (Polynomial)

Financial Analysis

A financial analyst has quarterly return data and needs to estimate returns for a specific month using polynomial interpolation.

Given Data Points:

Quarter	Return (%)
1	5.2
2	7.8
3	6.1
4	8.5

Find: Estimated return for Quarter 2.5

Solution using Polynomial Interpolation:

The polynomial method captures the non-linear relationship in the quarterly returns data.

Result: Estimated return at Q2.5 ≈ 6.95%

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Practice Problems

Try these problems to test your understanding of interpolation methods:

Problem 1: Linear Interpolation

Given points (2, 10) and (8, 34), find the value at x = 5.

Show Solution

Using linear interpolation: y = 10 + (34-10) × (5-2)/(8-2) = 10 + 24 × 0.5 = 22

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Problem 2: Polynomial Interpolation

Given points (0, 1), (1, 4), (2, 9), find the value at x = 1.5.

Show Solution

Using polynomial interpolation with Newton's method, the result is approximately 6.25.

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Tips for Choosing Methods

Use Linear When:

Data shows approximately linear trends
You have only two data points
Quick estimates are sufficient
Working with monotonic data

Use Polynomial When:

Data exhibits curved relationships
You have multiple data points (3+)
High accuracy is required
Data represents smooth physical phenomena