Mathematics

Problem

Consider the function

g(x)=\tan x

defined on the open interval

\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

. a. Compute

g'(x)

and evaluate

\displaystyle \lim_{x\to\left(\tfrac{\pi}{2}\right)^{-}}g'(x)

. Show all steps leading to the limit value. b. Explain what the result from part (a) indicates about the steepness of the graph of

g

x

approaches

\dfrac{\pi}{2}

from the left. c. Find all

x

in the interval

\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

at which the tangent line to the graph of

g

has slope

\sqrt{3}

. Justify your answer algebraically. d. For one of the

x

-values found in part (c), write the equation of the corresponding tangent line and describe its intersection with the

y

-axis.