Find the derivative of r(t) = (1 + t2)hati - te-thatj + sin 2thatk and | Mathematics I 2078

Mathematics I20785 marksNumerical: vector derivative tangentDerivative and integrals of vector functions

Find the derivative of $r(t) = (1 + t^2)\hat{i} - te^{-t}\hat{j} + \sin 2t\hat{k}$ and find the unit tangent vector at $t=0$ . [5]

Find the derivative of $r(t) = (1 + t^2)\hat{i} - te^{-t}\hat{j} + \sin 2t\hat{k}$ and find the unit tangent vector at $t=0$. [5]

$\mathbf{r}(t) = (1+t^2)\,\hat{i} - te^{-t}\,\hat{j} + \sin 2t\,\hat{k}$ Evaluation point: $t = 0$ . --- i-component: $\dfrac{d}{dt}(1+t^2) = 2t$ j-component: $\dfrac{d}{dt}(-te^{-t})$ Product rule: $\dfrac{d}{dt}(te^{-t}) = e^{-t} + t(-e^{-t}) = e^{-t}(1-t)$ So j-component:

$$\mathbf{r}(t) = (1+t^2),\hat{i} - te^{-t},\hat{j} + \sin 2t,\hat{k}$$ Evaluation point: $t = 0$. --- i-component: $\dfrac{d}{dt}(1+t^2) = 2t$ j-component: $\dfrac{d}{dt}(-te^{-t})$ Product rule: $\dfrac{d}{dt}(te^{-t}) = e^{-t} + t(-e^{-t}) = e^{-t}(1-t)$ So j-component:

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Find the derivative of r(t) = (1 + t2)hati - te-thatj + sin 2thatk and find the unit tangent vector at t=0.