. . Linear Approximation In this section, we introduce linear approximation of a function at a point. Using the linear approximation, we shall estimate the function value near a chosen point.
Then we shall introduce differential of a real-valued function of one variable, which is also useful in applications. Let f a b :( , ) → be a differentiable function and x a b ∈ ( , ) . Since f is differentiable at x , we have lim ∆→ + ∆ ∆ ′ f ... ( ) If D x is small, then by ( ) we have + ∆ » ′ ∆ ; ...
( ) which is equivalent to + ∆ » f x ′ ∆ , ... ( ) where » means “ approximately” equal. Also, observe that as the independent variable changes from x to x + ∆ , the function value changes from f x ( ) to f x + ∆ . Hence if D x is small and the change in the output is denoted by D f or D y , then ( ) can be rewritten as change in the output = ∆= ∆= + ∆ ≈ ′ ∆ Note that ( ) helps in approximating the value of f x + ∆ using f x ( ) and ′ ∆ .
Also, for a fixed x y x , ( ) )( ), ′ ∈ , gives the tangent line for the graph of f at ( )) which gives a good approximation to the function f near x . This leads us to define Definition . (Linear Approximation) Let f a b :( , ) → be a differentiable function and x a b ∈ ( , ) . We define the linear approximation L of f at x by L x ( ) = f x a b )( ), ′ ∀∈ ...
( ) and - - Differentials and Partial Derivatives Note that by ( ) and ( ) we see that + ∆ » f x ′ ∆ , which is useful