1. 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.

2. 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 $\displaystyle f(x)$ about $\displaystyle x=0$ is:

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

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

$\displaystyle f(0)=1$

and:

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

RonL