Approximations by Differentials

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Let y = f (x) be a differentiable function of x, errors in x and y are denoted by dx and dy, we have

\ Error in y = f ' (x) dx.

Note 1:

Note 2:

Note 3:

dy and dy are not usually the same and dy is the approximate value of dy.

Note 4:

dx and dy are called the differentials of x and y respectively.

Example:

Find the approximate value of fourth root of 82 upto 3 decimal places.

Solution:

Let y = f (x) = x^1/4

Let x = 81, dx =1. Taking these values we have