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Chapter 10: Problem 13
Find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=e^{-t} \mathbf{i}+4 \mathbf{j} $$
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Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{2 t}{8+t^{3}} \mathbf{i}+\frac{2 t^{2}}{8+t^{3}} \mathbf{j} $$
The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t \mathbf{i}+(2 t-5) \mathbf{j}+3 t \mathbf{k} $$
Prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. $$ D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]=\mathbf{r}(t) \times \mathbf{u}^{\prime}(t)+\mathbf{r}^{\prime}(t) \times \mathbf{u}(t) $$
Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=2 \cos ^{3} \theta \mathbf{i}+3 \sin ^{3} \theta \mathbf{j} $$
The graph of the vector-valued function \(\mathbf{r}(t)\) and a tangent vector to the graph at \(t=t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at \(t=t_{0}\) (b) Use the equations for the tangent line to approximate \(\mathbf{r}\left(t_{0}+\mathbf{0 . 1}\right)\) $$ \mathbf{r}(t)=\left\langle t,-t^{2}, \frac{1}{4} t^{3}\right\rangle, \quad t_{0}=1 $$
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