Quadratic approximation

Quadratic approximation

Basic Concepts: Linear approximation

Lessons

The formula for quadratic approximation is:

Q(x)=f(a)+f(a)(xa)+f(a)2(xa)2Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^{2}

Where:

f(a)+f(a)(xa)f(a) + f'(a)(x - a) is the linear part

f(a)2(xa)2\frac{f''(a)}{2}(x - a)^{2} is the quadratic part.

  • Introduction
    Quadratic Approximation Overview

    What is quadratic approximation?


  • 1.
    Approximating values using Quadratic Approximations

    Find the Quadratic approximation to the function at the given point:

    a)
    f(x)=2cosxf(x) = 2\cos{x} at a = π2\frac{\pi}{2}

    b)
    g(x)=x3+2x2+5x+4g(x) = x^{3} + 2x^{2} + 5x + 4 at a = 1


  • 2.
    Consider the function f(x)=xf(x) = \sqrt{x}
    a)
    Find the quadratic approximation of the function at a=4a = 4

    b)
    Approximate 5\sqrt{5} and 6\sqrt{6}

    c)
    Compare the exact values of 5\sqrt{5} and 6\sqrt{6} with your approximated values in part b). How close were we?


  • 3.
    Approximate ln2\ln{2}