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Chapter 10: Problem 39
Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=2 x^{2}+3, \quad x=-1 $$
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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. $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad \mathbf{u}(t)=\mathbf{j}+t \mathbf{k} $$
The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle $$
The \(z\) -component of the derivative of the vector-valued function \(\mathbf{u}\) is 0 for \(t\) in the domain of the function. What does this information imply about the graph of \(\mathbf{u}\) ?
In Exercises \(\mathbf{3 7}\) and \(\mathbf{3 8 ,}\) 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. $$ \mathbf{r}(t)=t \mathbf{i}+2 t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad \mathbf{u}(t)=t^{4} \mathbf{k} $$
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. $$ \mathbf{r}(t)=(6-t) \mathbf{i}+t \mathbf{j},(3,3) $$
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