Chapter 10: Problem 62
Determine the interval(s) on which the vector-valued function is continuous. \(\mathbf{r}(t)=\langle 8, \sqrt{t}, \sqrt[3]{t}\rangle\)
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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. $$ D_{t}\left[\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)\right]=\mathbf{r}(t) \times \mathbf{r}^{\prime \prime}(t) $$
In Exercises \(53-56,\) find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=4 e^{2 t} \mathbf{i}+3 e^{t} \mathbf{j}, \quad \mathbf{r}(0)=2 \mathbf{i} $$
Find the vectors \(\mathrm{T}\) and \(\mathrm{N},\) and the unit binormal vector \(\mathbf{B}=\mathbf{T} \times \mathbf{N},\) for the vector-valued function \(\mathbf{r}(t)\) at the given value of \(t\). $$ \mathbf{r}(t)=2 e^{t} \mathbf{i}+e^{t} \cos t \mathbf{j}+e^{t} \sin t \mathbf{k}, \quad t_{0}=0 $$
Find (a) \(\quad D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)] \quad\) and (b) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) by differentiating the product, then applying the properties of Theorem 10.2. $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad \mathbf{u}(t)=\mathbf{j}+t \mathbf{k} $$
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. $$ D_{t}[\mathbf{r}(f(t))]=\mathbf{r}^{\prime}(f(t)) f^{\prime}(t) $$
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