Linear Approximations

If $f (x)$ is differentiable at $x = ω$ , then

ℒ (x) = f (ω) + f^{'} (ω) (x - ω)

is the linearization of $f (x)$ at $ω$ .

The point $x = ω$ is the center of the approximation.

Find the linearization of $f (x) = \sqrt{1 + x}$ at $x = 0$

Approximate $f (x) = \sqrt{x}$ at $x = 9$ . Estimate $\sqrt{9.1}$

Approximate $sin (62^{\circ})$

Approximate ${1.01}^{3}$ using ${(1 + x)}^{3}$

Differentials

Let $f (x)$ be a differentiable function, then the differential $ⅆ x$ is an independent variable and the differential $ⅆ y$ is a dependent variable defined as

ⅆ y = f^{'} (x) ⅆ x

Find $ⅆ y$ if $y = x^{2} + 2 x$
Find $ⅆ y$ and $Δ y$ if $x = 1$ and $ⅆ x = 0.2$

Find $ⅆ y$ if $y = x^{5} + 37 x$
Find $ⅆ y$ if $x = 1$ and $ⅆ x = 0.2$