91Ó°ÊÓ

Show that the curve \(r=t \cos t i+t \sin t\) j \(+t \mathbf{k}, t \geq 0,\) lies on the cone \(z=\sqrt{x^{2}+y^{2}} .\) Describe the curve.

Describe the curve \(r=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k},\) where \(a\) \(b,\) and \(c\) are positive constants such that \(a \neq b\)

Solve the vector initial-value problem for \(y(t)\) by integrating and using the initial conditions to find the constants of integration. $$y^{\prime \prime}(t)=12 t^{2} i-2 t j, y(0)=2 i-4 j, y^{\prime}(0)=0$$

Find the radius of curvature of the parabola \(y^{2}=4 p x\) at \((0,0).\)

Use a graphing utility to generate the graph of \(\mathbf{r}(t)\) and the graph of the tangent line at \(t_{0}\) on the same screen. $$\mathbf{r}(t)=\sin \pi t \mathbf{i}+t^{2} \mathbf{j}: t_{0}=\frac{1}{2}$$

The nuclear accelerator at the Enrico Fermi Laboratory is circular with a radius of \(1 \mathrm{km}\). Find the scalar normal component of acceleration of a proton moving around the accelerator with a constant speed of \(2.9 \times 10^{5} \mathrm{km} / \mathrm{s}\)

Use a graphing utility to generate the graph of \(\mathbf{r}(t)\) and the graph of the tangent line at \(t_{0}\) on the same screen. $$r(t)=3 \sin t \mathbf{i}+4 \cos t \mathbf{j}: \quad t_{0}=\pi / 4$$

(a) Find parametric equations for the curve of intersection of the circular cylinder \(x^{2}+y^{2}=9\) and the parabolic cylinder \(z=x^{2}\) in terms of a parameter \(t\) for which \(x=3 \cos t\) (b) Use a graphing utility to generate the curve of intersection in part (a).

At what point(s) does \(4 x^{2}+9 y^{2}=36\) have minimum radius of curvature?

Let \(\theta(t)\) be the angle between \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t) .\) Use a graphing calculator to generate the graph of \(\theta\) versus \(t\), and make rough estimates of the \(t\) -values at which t-intercepts or relative extrema occur. What do these values tell you about the vectors \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t) ?\) $$\mathbf{r}(t)=4 \cos t \mathbf{i}+3 \sin t \mathbf{j}: 0 \leq t \leq 2 \pi$$