L(x)= f(a) + f' (a)(x-a) A quadratic approximation is an extension of linear approximation. It is created by adding one more term related to the second derivative to the aboves. The formula for the quadratic approximation Q(x) of a function f(x) near a is given by:
Q(x)=f(a)+f'(a)(x-a)+ f''(a)/2(x-a)2
Linear Part Quadratic Part
This approximation is more complicated than the linear approximation and is only used when higher accuracy is needed
In particular, consider the function f(x) =√ x+1.
(a) Find √ 1.1 with a calculator.
(b) Determine the linearization of f(x)= √ 1+x near x=0
(c) Using the linear approximation from (b), approximate √ 1.1.
(d) Find the quadratic approximation to f(x)= √ 1+1 at x=0
(e) Using the quadratic approximation, approximate √ 1.1
(f) Circle one: Which is closer to the actual value? Linear or Quadratic
(g) Find the formula for the quadratic approximation to f(x)=) (1+x)α at x=0.