Another Linear Approximation Question

• Sep 10th 2007, 11:35 PM
Wyau
Another Linear Approximation Question

Question:
A useful linear approximation to
1/(1+tan x)
can be obtained by combining the approximations
1/(1+x) is about equal to 1 - x and tan x is about equal to x
to get 1/(1+tan x) is about equal to 1-x

Show that this is the standard linear approximation of 1/(1+ tan x)

I'm really slow on this linear approximation stuff. Thanks again for helping me out.
• Sep 10th 2007, 11:44 PM
CaptainBlack
Quote:

Originally Posted by Wyau

Question:
A useful linear approximation to
1/(1+tan x)
can be obtained by combining the approximations
1/(1+x) is about equal to 1 - x and tan x is about equal to x
to get 1/(1+tan x) is about equal to 1-x

Show that this is the standard linear approximation of 1/(1+ tan x)

I'm really slow on this linear approximation stuff. Thanks again for helping me out.

The standard linear approximation of $f(x)$ about $x=0$ is:

$
f(x) = f(0) + x f'(0)
$

So put $f(x) = \frac{1}{1+\tan(x)}$ then you need to show that:

$
f(0)=1
$

and:

$
\left[ \frac{df}{dx}\right]_{x=0} = -1
$

RonL