009A Sample Final A, Problem 8

8. (a) Find the linear approximation $L(x)$ to the function $f(x)=\sec x$ at the point $x=\pi /3$.
    (b) Use $L(x)$ to estimate the value of $\sec \,(3\pi /7)$.

Foundations:
Recall that the linear approximation $L(x)$ is the equation of the tangent line to a function at a given point. If we are given the point $x_{0}$, then we will have the approximation $L(x)=f'(x_{0})\cdot (x-x_{0})+f(x_{0})$. Note that such an approximation is usually only good "fairly close" to your original point $x_{0}$.

 Solution:

Part (a):
Note that $f'(x)=\sec x\tan x$. Since $\sin(\pi /3)={\sqrt {3}}/2$ and $\cos(\pi /3)=1/2$, we have
$f'(\pi /3)\,\,=\,\,\sec(\pi /3)\tan(\pi /3)\,\,=\,\,{\frac {1}{1/2}}\cdot {\frac {{\sqrt {3}}/2}{\,\,1/2}}\,\,=\,\,2{\sqrt {3}}.$
Similarly, $f(\pi /3)=\sec(\pi /3)=2.$ Together, this means that
$L(x)=f'(x_{0})\cdot (x-x_{0})+f(x_{0})$
$=2{\sqrt {3}}(x-\pi /3)+2.$

Part (b):
This is simply an exercise in plugging in values. We have
$L\left({\frac {3\pi }{7}}\right)=2{\sqrt {3}}\left({\frac {3\pi }{7}}-{\frac {\pi }{3}}\right)+2$
$=2{\sqrt {3}}\left({\frac {9\pi -7\pi }{21}}\right)+2$
$=2{\sqrt {3}}\left({\frac {2\pi }{21}}\right)+2$
$={\frac {4{\sqrt {3}}\pi }{21}}+2.$

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