Let on the open interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

f(x)=\tan x

on the open interval

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

, and let

g=f^{-1}

. (a) Explain why

f

is invertible on the given interval and state the domain and range of

g

. (b) Using analytic methods, derive a formula for

g'(x)

expressed entirely in terms of

x

. (c) Evaluate

g'(1)

, giving your answer correct to three decimal places. (d) The point

(1,g(1))

lies on the graph of

g

. Write the equation of the tangent line to the graph of

g

at that point.