Mathematics

Problem

For

x\ge \dfrac12

, let

h(x)=\operatorname{arcsec}(2x).

A line

L

has slope

2

. (a) Show that

h'(x)=\dfrac{2}{|2x|\sqrt{4x^{2}-1}}

for

x>\dfrac12

. (b) Determine all

x>\dfrac12

at which the tangent line to the graph of

h

is parallel to

L

. Give exact values. (c) For one value obtained in part (b), write the equation of the corresponding tangent line. (d) Find the acute angle between this tangent line and the

x

-axis, expressing your answer in radians.

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