Consider for

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

Problem

Consider r(t)=⟨tet2,  t2ln⁡t,  tan⁡(t)⟩\displaystyle \mathbf{r}(t)=\big\langle t e^{t^{2}},\; t^{2}\ln t,\; \tan(\sqrt{t})\big\rangle for t>0t>0. Which of the following expressions equals r′(t)\mathbf{r}'(t)? (a) ⟨et2(1+2t),  2tln⁡t+t,  sec⁡2(t)2t⟩\big\langle e^{t^{2}}(1+2t),\; 2t\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle (b) ⟨et2(1+2t2),  2ln⁡t+t,  sec⁡2(t)2t⟩\big\langle e^{t^{2}}(1+2t^{2}),\; 2\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle (c) ⟨et2(1+2t2),  2tln⁡t+t,  sec⁡2(t)2t⟩\big\langle e^{t^{2}}(1+2t^{2}),\; 2t\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle (d) ⟨et2(1+2t2),  2tln⁡t+t,  sec⁡2(t)⟩\big\langle e^{t^{2}}(1+2t^{2}),\; 2t\ln t + t,\; \sec^{2}(\sqrt{t}) \big\rangle

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