LINEAR INTERPOLATION

Consider the following diagram:

Let M be the slope of the straight line PQ then

M = ^{[f(b) – f(a)]}/ _{(b – a)} ------------------- (I)

Also

M = ^{[f(c) – f(a)]}/ _{(c – a)}

Since both are the value of the slope M then [^{[f(b) – f(a)]}/ _{(b – a)}] = [^{[f(c) – f(a)]}/ _{(c – a)}] then by making subject f(c) the equation will become:

  this is called linear interpolation... Read more this article

Application of linear interpolation:

Example #1:

The function f(x) takes the following values:

Using linear interpolation find the value of 2.2.

Solution:

From the table above 2.2 is found between 2 and 3, f (2) and f (3) then from the formula

F (b) = 0.25, F (a) = 0.33, b = 3, a= 2 and c is the given number that is 2.2 so we need to find F (2.2),

By substitution from the formula F (2.2) = 0.314.

There fore the value of 2.2 is 0.314

Example #2

Given that e^1.25 = 3.4903 and e^1.26 = 3.5254 find e^1.257 by using linear interpolation.

Solution:

From the question x = 1.25 and 1.26 then in the table will be

From the above 1.257 lies between 1.25 and 1.26 then using the formula

F (b) = 3.5254, F (a) = 3.4903, a = 1.25, b= 1.26 and c is the given number that is 1.257 then we need to find F (1.257).

By substitution in the linear interpolation the answer will be 3.5149

There fore e^1.257 = 3.5149.

Example #3

If sin 35˚ = 0.5736 and sin 40˚ = 0.6428 find sin 37˚ by using linear interpolation.

Solution:

37˚ lies between 35˚ and 40˚ then from the equation F (b) = 0.6428, F (a) = 0.5736, b = 40, a = 35 and c is the given angle that is 37˚ then we need to find F (37).

After substitution to linear interpolation the value of sin 37˚ is 0.60128

Qn: Given that log (1.96) = 0.2923 and log (1.97) = 0.2945, using linear interpolation find the number for which log (x) = 0.2935.

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