Chapter 10: Problem 58
Determine the interval(s) on which the vector-valued function is continuous. \(\mathbf{r}(t)=\sqrt{t} \mathbf{i}+\sqrt{t-1} \mathbf{j}\)
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Find \((a) r^{\prime \prime}(t)\) and \((b) r^{\prime}(t) \cdot r^{\prime \prime}(t)\). $$ \mathbf{r}(t)=8 \cos t \mathbf{i}+3 \sin t \mathbf{j} $$
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 \(\mathbf{r}\left(t_{0}+\mathbf{0 . 1}\right)\) $$ \mathbf{r}(t)=\left\langle t, \sqrt{25-t^{2}}, \sqrt{25-t^{2}}\right\rangle, \quad t_{0}=3 $$
The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j},(3,0) $$
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.$$ D_{t}[\mathbf{r}(t) \pm \mathbf{u}(t)]=\mathbf{r}^{\prime}(t) \pm \mathbf{u}^{\prime}(t) $$
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. $$ D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]=\mathbf{r}(t) \times \mathbf{u}^{\prime}(t)+\mathbf{r}^{\prime}(t) \times \mathbf{u}(t) $$
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