91Ó°ÊÓ

Use a graphing utility to generate the intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and the plane \(z=y+2 .\) Identify the curve and explain your reasoning.

Find the value of \(x, x>0,\) where \(y=x^{3}\) has maximum 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 \(\mathbf{r}^{\prime}(t) ?\) $$\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j} ; 0 \leq t \leq 1$$

Use the given information to find the normal scalar component of acceleration at time \(t=1\). $$\mathrm{a}(1)=\mathrm{i}+2 \mathrm{j}-2 \mathrm{k} ; \quad a_{T}(1)=3$$

(a) Find the points where the curve $$ r=t i+t^{2} j-3 t k $$ intersects the plane \(2 x-y+z=-2\) (b) For the curve and plane in part (a), find, to the nearest degree, the acute angle that the tangent line to the curve makes with a line normal to the plane at each point of intersection.

Find the maximum and minimum values of the radius of curvature for the curve \(x=\cos t, y=\sin t, z=\cos t.\)

Find where the tangent line to the curve $$ r=e^{-2 t} i+\cos t \mathbf{j}+3 \sin t \mathbf{k} $$ at the point (1,1,0) intersects the \(y\) -plane.

Find the minimum value of the radius of curvature for the curve \(x=e^{t}, y=e^{-t}, z=\sqrt{2} t.\)

Show that the graphs of \(\mathrm{r}_{1}(t)\) and \(\mathbf{r}_{2}(t)\) intersect at the point \(P .\) Find, to the nearest degree, the acute angle between the tangent lines to the graphs of \(\mathbf{r}_{1}(t)\) and \(r_{2}(t)\) at the point \(P\) $$\begin{array}{l} \mathbf{r}_{1}(t)=t^{2} \mathbf{i}+t \mathbf{j}+3 t^{3} \mathbf{k} \\ \mathbf{r}_{2}(t)=(t-1) \mathbf{i}+\frac{1}{4} t^{2} \mathbf{j}+(5-t) \mathbf{k}: \quad P(1,1,3) \end{array}$$

An automobile travels at a constant speed around a curve whose radius of curvature is \(1000 \mathrm{m}\). What is the maximum allowable speed if the maximum acceptable value for the normal scalar component of acceleration is \(1.5 \mathrm{m} / \mathrm{s}^{2} ?\)