91Ó°ÊÓ

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Show that the particle moves on a circular cone.

Find the radius of curvature of \(6 y=x^{3}\) at the point \(\left(2, \frac{4}{3}\right)\).

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k}\). Find the following:Velocity of the particle

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Find the angle between the velocity and acceleration vectors when \(t=1.5\).

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Find the tangential and normal components of acceleration when \(t=1.5\).

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k}\). Find the following:Speed of the particle at \(t=\frac{\pi}{4}\)

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k}\). Find the following:Acceleration of the particle at \(t=\frac{\pi}{4}\)

A particle travels along the path of an ellipse with the equation \(\mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+0 \mathbf{k}\). Find the following:Velocity

Find the arc length of the curve on the given interval.Find the radius of curvature of \(y=\ln (x+1)\) at point \((2, \ln 3)\).

True or False? Justify your answer with a proof or a counterexample. $$ \frac{d}{d t}[\mathbf{u}(t) \times \mathbf{u}(t)]=2 \mathbf{u}^{\prime}(t) \times \mathbf{u}(t) $$