Consider the function defined on the open interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Consider the function

g(x)=\tan x

defined on the open interval

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

. (1) Compute

g'(x)

and evaluate

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

. Show all steps leading to the limit value. (2) Explain what the result from part (a) indicates about the steepness of the graph of

g

x

approaches

\dfrac{\pi}{2}

from the left. (3) 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. (4) 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.

Related theory

Limits
Differentiation Rules
Rules of differentiation
Derivatives
Limits and Derivatives

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