An alternate definition for approximation is as 
If a and a+h belongs to the domain of a differentiable function ’ f ’ , then the approximate value of f (a +h) will be f (a+h) » f (a) + hf ‘ (a), f ‘(a) ¹ 0. By using this, approximate problems can be solved easily.
Example 50
Find correct to 4 decimal places, the approximate
value of
Solution : Let
f (x) =
Take a = 8 and h = 0.05 then
Now the approximation
formula is
Example 51
Find the approximate value of tan
1 (0.99)
Solution : Let f (x) = tan^{1} (x)
\ f
‘ (x) =
Take a = 1 and h = 0.01 then
f (a) = f (1) = tan
1 (1) = radians.
By the approximation formula
f ( a+h ) » f (a) + h f ‘ (a)
Example 52
Find the approximate value of sin 31^{0}
assuming that 1^{0} = 0.0175 radians and cos 30^{0}
= 0.8660.
Solution : Let f (x) = sin (x) then f ‘ (x) = cos x
Take a = 30^{0} = p/6 Radians. h = 1^{0} = 0.0175 radians.
Then f (a) = f (30^{0}) = sin 30^{0 } = 0.5
and f ‘ (a) = f ‘ (30) = cos 30^{0 } = 0.8660
using the approximation formula i.e.
f (a + h) » f (a) + h f ‘ (a) we get
\ sin 31^{0} = f (30^{0} ) + ( 0.0175 ) f ‘ (30^{0})
\ sin 31^{0} = 0.5000 + ( 0.0175 ) (0.8660)
\ sin 31^{0} = 0.5152
Errors
Let y = f (x) be a differential function of x and Dx an error in x, then
The approximate error in y is Dy = f ‘ (x) Dx
i.e. Dy = f ‘ (x) dx
Approximate relative
error in y is
Approximate percentage error in y is
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