in approximating the value of + ∆ Note that linear approximation for f at x gives a good approximation to f x ( ) if x is close to x . Because as x approaches to x , by continutity of f at x , Error = f x L x )( ′ ... ( ) Also, if f x mx ( ) = + , then its linear approximation is L x mx m x mx for any point x a b ∈ ( , ) . That is, the linear approximation, in this case, is the original function itself (is it not surprising?).
Example . Find the linear approximation for f x x x ≥− , at x = . Use the linear approximation to estimate f ( . ) .
We know from ( ), that L x )( ′ . We have x ∆ and hence f ( ) . Also, ′ f ( ) = + x and hence ′ f ( ) . Thus, L x ( ) = gives the required linear approximation.
Now, f ( . ) = ( . ) ≈ L Actually, if we use a calculator to calculate we get