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Problem Consider r ( t ) = ⟨ t e t 2 , t 2 ln t , tan ( t ) ⟩ \displaystyle \mathbf{r}(t)=\big\langle t e^{t^{2}},\; t^{2}\ln t,\; \tan(\sqrt{t})\big\rangle r ( t ) = ⟨ t e t 2 , t 2 ln t , tan ( t ) ⟩ t > 0 t>0 t > 0 r ′ ( t ) \mathbf{r}'(t) r ′ ( t ) ⟨ e t 2 ( 1 + 2 t ) , 2 t ln t + t , sec 2 ( t ) 2 t ⟩ \big\langle e^{t^{2}}(1+2t),\; 2t\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle ⟨ e t 2 ( 1 + 2 t ) , 2 t ln t + t , 2 t sec 2 ( t ) ⟩ ⟨ e t 2 ( 1 + 2 t 2 ) , 2 ln t + t , sec 2 ( t ) 2 t ⟩ \big\langle e^{t^{2}}(1+2t^{2}),\; 2\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle ⟨ e t 2 ( 1 + 2 t 2 ) , 2 ln t + t , 2 t sec 2 ( t ) ⟩ ⟨ e t 2 ( 1 + 2 t 2 ) , 2 t ln t + t , sec 2 ( t ) 2 t ⟩ \big\langle e^{t^{2}}(1+2t^{2}),\; 2t\ln t + t,\; \dfrac{\sec^{2}(\sqrt{t})}{2\sqrt{t}} \big\rangle ⟨ e t 2 ( 1 + 2 t 2 ) , 2 t ln t + t , 2 t sec 2 ( t ) ⟩ ⟨ e t 2 ( 1 + 2 t 2 ) , 2 t ln t + t , sec 2 ( t ) ⟩ \big\langle e^{t^{2}}(1+2t^{2}),\; 2t\ln t + t,\; \sec^{2}(\sqrt{t}) \big\rangle ⟨ e t 2 ( 1 + 2 t 2 ) , 2 t ln t + t , sec 2 ( t ) ⟩ Browse the problem library