Basic Concepts: Linear approximation

#### Lessons

The formula for quadratic approximation is:

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

Where:

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

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

• Introduction

• 1.

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

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

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

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

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

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

• 3.
Approximate $\ln{2}$