91Ó°ÊÓ

A particle moves along the plane curve \(C\) described by \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j} .\) Solve the following problems.Describe the curvature as \(t\) increases from \(t=0\) to \(t=2\).

Given \(\mathbf{r}(t)=t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}\) and \(\mathbf{u}(t)=\frac{1}{t} \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}\), find the following:\(\frac{d}{d t}(\mathbf{r}(t) \times \mathbf{u}(t))\)

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters).[1] Use technology to graph the surface.

Sketch the curves for the following vector equations. Use a calculator if needed. $$ \text { [T] } \mathbf{r}(t)=\left\langle t^{2}, t^{3}\right\rangle $$

Find the unit tangent vector \(\mathrm{T}(\mathrm{t})\) for the following vector-valued functions.\(\mathbf{r}(t)=\left\langle t, \frac{1}{t}\right\rangle\). The graph is shown here:

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters).Find the curvature \(\kappa\) of the generating curve as a function of \(x\).

Find the unit tangent vector \(\mathrm{T}(\mathrm{t})\) for the following vector-valued functions.\(\mathbf{r}(t)=\langle t \cos t, t \sin t\rangle\)

Find a vector function that describes the following curves. Intersection of the cylinder \(x^{2}+y^{2}=4\) with the plane \(x+z=6\)

Find the unit tangent vector \(\mathrm{T}(\mathrm{t})\) for the following vector-valued functions.\(\mathbf{r}(t)=\langle t+1,2 t+1,2 t+2\rangle\)

Find a vector function that describes the following curves. Intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=y-4\)