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
\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
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 \(B\)?