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Chapter 10: Problem 10
Find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\frac{1}{t} \mathbf{i}+16 t \mathbf{j}+\frac{t^{2}}{2} \mathbf{k} $$
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The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=\langle 4 t, 3 \cos t, 3 \sin t\rangle $$
True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a particle moves along a sphere centered at the origin, then its derivative vector is always tangent to the sphere.
Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ $$ \text { Find the velocity vector and show that it is orthogonal to } \mathbf{r}(t) $$
The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=3 t \mathbf{i}+t \mathbf{j}+\frac{1}{4} t^{2} \mathbf{k} $$
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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