Define , where and . Which of the following equals for all…

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Define

R(t)=t^{2}\,r(u)

, where

r(u)=\langle e^{5u},\,u\sin u\rangle

and

u=t^{3}+2t

. Which of the following equals

R'(t)

for all

t

? (a)

\big\langle 2t\,e^{5u} + t^{2}\,5e^{5u},\; 2t\,u\sin u + t^{2}(\sin u + u\cos u) \big\rangle

(b)

\big\langle 2t\,e^{5t} + t^{2}\,5e^{5t},\; 2t\,t\sin t + t^{2}(\sin t + t\cos t) \big\rangle

(c)

\big\langle 2t\,e^{5u} + t^{2}(15t^{2}+10)\,e^{5u},\; 2t\,u\sin u + t^{2}(3t^{2}+2)(\sin u + u\cos u) \big\rangle

(d)

\big\langle t^{2}(3t^{2}+2)\,5e^{5u},\; t^{2}(3t^{2}+2)(\sin u + u\cos u) \big\rangle