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Problem Define R ( t ) = t 2 r ( u ) R(t)=t^{2}\,r(u) R ( t ) = t 2 r ( u ) r ( u ) = ⟨ e 5 u , u sin u ⟩ r(u)=\langle e^{5u},\,u\sin u\rangle r ( u ) = ⟨ e 5 u , u sin u ⟩ u = t 3 + 2 t u=t^{3}+2t u = t 3 + 2 t R ′ ( t ) R'(t) R ′ ( t ) t t t ⟨ 2 t e 5 u + t 2 5 e 5 u , 2 t u sin u + t 2 ( sin u + u cos u ) ⟩ \big\langle 2t\,e^{5u} + t^{2}\,5e^{5u},\; 2t\,u\sin u + t^{2}(\sin u + u\cos u) \big\rangle ⟨ 2 t e 5 u + t 2 5 e 5 u , 2 t u sin u + t 2 ( sin u + u cos u ) ⟩ ⟨ 2 t e 5 t + t 2 5 e 5 t , 2 t t sin t + t 2 ( sin t + t cos t ) ⟩ \big\langle 2t\,e^{5t} + t^{2}\,5e^{5t},\; 2t\,t\sin t + t^{2}(\sin t + t\cos t) \big\rangle ⟨ 2 t e 5 t + t 2 5 e 5 t , 2 t t sin t + t 2 ( sin t + t cos t ) ⟩ ⟨ 2 t e 5 u + t 2 ( 15 t 2 + 10 ) e 5 u , 2 t u sin u + t 2 ( 3 t 2 + 2 ) ( sin u + u cos u ) ⟩ \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 ⟨ 2 t e 5 u + t 2 ( 15 t 2 + 10 ) e 5 u , 2 t u sin u + t 2 ( 3 t 2 + 2 ) ( sin u + u cos u ) ⟩ ⟨ t 2 ( 3 t 2 + 2 ) 5 e 5 u , t 2 ( 3 t 2 + 2 ) ( sin u + u cos u ) ⟩ \big\langle t^{2}(3t^{2}+2)\,5e^{5u},\; t^{2}(3t^{2}+2)(\sin u + u\cos u) \big\rangle ⟨ t 2 ( 3 t 2 + 2 ) 5 e 5 u , t 2 ( 3 t 2 + 2 ) ( sin u + u cos u ) ⟩ Browse the problem library