Finite Differences: Solved Example Problems

Example 5.1

Construct a forward difference table for the following data

Solution:

The Forward difference table is given below

Example 5.2

Construct a forward difference table for y = f(x) = x ³ + 2x + 1 for x = 1,2,3,4,5

Solution:

y = f(x) = x³ + 2 x + 1 for x =1,2,3,4,5

Example 5.3

By constructing a difference table and using the second order differences as constant, find the sixth term of the series 8,12,19,29,42…

Solution:

Let k be the sixth term of the series in the difference table

First we find the forward differences.

Given that the second differences are constant

∴ k – 55 = 3

 k = 58

∴ the sixth term of the series is 58

Example 5.4

Find (i) ∆e^ax (ii) ∆²e^x  (iii) ∆logx

Solution:

Example 5.5

Evaluate  by taking ‘1’ as the interval of differencing.

Solution:

Example 5.5

Evaluate  by taking ‘1’ as the interval of differencing.

Solution:

Example 5.7

Prove that f(4) = f(3) + Δf(2) + Δ²f(1) + Δ³f(1) taking ‘1’ as the interval of differencing.

Solution:

We know that f(4) - f(3) = Δ f(4)

f(4) - f(3) = Δ f(3)

= Δ [ f(2) + Δf(2) ]    i.e.[ f(3) - f(2) = Δf(2) ]

= Δ f(2) + Δ²f(2)

= Δ f(2) + Δ²[f(1)+ Δf(1)]

f(4) = f(3) + Δf(2) + Δ² f(1) + Δ3 f(1)

Example 5.8

Given U₀ =1, U₁ =11, U₂ =21, U₃ =28 and U₄ =29 find Δ⁴U₀

Solution:

Δ⁴U₀ = (E-1) ⁴U₀