Let and let for

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Let

w(u)=\ln(u^4+1)

and let

u(t)=\sqrt{\tan(3t)}

for

0<t<\dfrac{\pi}{6}

. Consider the vector function

\mathbf R(w)=\langle w^2,\,\sqrt{w},\,e^{w}\rangle.

Throughout, treat all expressions as defined on their natural domains. (a) Express

\mathbf R

as a composite function of

t

and write the resulting position vector

\mathbf r(t)

explicitly. (b) Derive a simplified formula for

\dfrac{d\mathbf R}{dt}

in terms of

t

alone. Present each component in its simplest algebraic form. (c) Verify that the magnitude of

\dfrac{d\mathbf R}{dt}

is increasing at

t=\dfrac{\pi}{18}

. Justify your conclusion analytically.