Two curves are given by and for .

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

Problem

Two curves are given by

y_1=\sin x + x

and

y_2 = x\tan x

for

0<x<\dfrac{\pi}{2}

. (i) Find the expressions for the slopes of the tangents to

y_1

and

y_2

at any point

x

in the stated interval. (ii) Show that the tangents to the two curves are parallel when

\sin x = x\sec^2 x -1

and hence find the common

x

–coordinate of such points, correct to two decimal places. (iii) Determine the acute angle between the two curves at

x=\dfrac{\pi}{4}

Related theory

Differentiation Rules
Angle Between Two Lines
Straight Lines
Derivatives of standard functions via chain rule
Differentiability of functions
Continuity and Differentiability

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