8.2.1 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 )( ), ′ ∀∈ ... ( )