Problem source

Explain briefly, by means of a diagram, or otherwise, why 
\[
\mathrm{f}(\theta+\delta\theta)\approx\mathrm{f}(\theta)+\mathrm{f}'(\theta)\delta\theta,
\]
when $\delta\theta$ is small. 

Two powerful telescopes are placed at points $A$ and $B$ which are
a distance $a$ apart. A very distant point $C$ is such that $AC$
makes an angle $\theta$ with $AB$ and $BC$ makes an angle $\theta+\phi$
with $AB$ produced. (A sketch of the arrangement is given in the
diagram.) 

\noindent \begin{center}
\psset{xunit=0.8cm,yunit=0.8cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=3pt 0,linewidth=0.5pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-4.18,-0.94)(4.4,5.22) \psline(-4,0)(4,0) \psline(-2,0)(2,5) \psline(2,5)(1,0) \rput[tl](-2.3,-0.14){$A$} \rput[tl](1.08,-0.14){$B$} \rput[tl](-1.6,0.46){$\theta$} \rput[tl](1.24,0.52){$\theta+\phi$} \rput[tl](2.14,5.1){$C$} \end{pspicture*}
\par\end{center}

If the perpendicular distance $h$ of $C$ from $AB$ is very large
compared with $a$ show that $h$ is approximately $(a\sin^{2}\theta)/\phi$
and find the approximate value of $AC$ in terms of $a,\theta$ and
$\phi.$ 

It is easy to show (but you are not asked to show it) that errors
in measuring $\phi$ are much more important than errors in measuring
$\theta.$ If we make an error of $\delta\phi$ in measuring $\phi$
(but measure $\theta$ correctly) what is the approximate error in
our estimate of $AC$ and, roughly, in what proportion is it reduced
by doubling the distance between $A$ and $B$?