Find the derivatives of r(t) = (1 + t2)hati - tethatj + sin 2thatk and | Mathematics I 2077

Mathematics I20775 marksNumerical: Vector derivative and tangentDerivative and integrals of vector functions

Find the derivatives 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 derivatives 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: $t = 0$ . All required data present. --- Differentiate each component: $\hat{i}$ component: $\frac{d}{dt}(1+t^2) = 2t$ $\hat{j}$ component (product rule on $te^t$ ): $$\frac{d}{dt}(-te^t) = -(1\cdot e^t + t...

$$\mathbf{r}(t) = (1+t^2),\hat{i} - te^t,\hat{j} + \sin 2t,\hat{k}$$ Evaluation point: $t = 0$. All required data present. --- Differentiate each component: $\hat{i}$ component: $$\frac{d}{dt}(1+t^2) = 2t$$ $\hat{j}$ component (product rule on $te^t$): $$\frac{d}{dt}(-te^t) = -(1\cdot e^t + t...

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