91Ó°ÊÓ

Determine all values of \(t\) at which the given vector-valued function is continuous. $$\mathbf{r}(t)=\left\langle\tan t, \sin t^{2}, \cos t\right\rangle$$

Determine all values of \(t\) at which the given vector-valued function is continuous. $$\mathbf{r}(t)=\langle\cos 5 t, \tan t, 6 \sin t\rangle$$

Sketch the curve traced out by the given vector valued function by hand. $$\mathbf{r}(t)=\left\langle 3, t, t^{2}-1\right\rangle$$

Find the osculating circle at the given points. $$\mathbf{r}(t)=\left\langle t, t^{3}\right\rangle \text { at } t=0$$

Find the position function from the given velocity or acceleration function. $$\mathbf{a}(t)=\langle t, \sin t\rangle, \mathbf{v}(0)=\langle 2,-6\rangle, \mathbf{r}(0)=\langle 10,4\rangle$$

Find the unit tangent vector to the curve at the indicated points. \(\mathbf{r}(t)=\left\langle 4 t, 2 t, t^{2}\right\rangle, t=-1, t=0, t=1\)

Identify and sketch a graph of the parametric surface. $$x=\sinh v, y=\cos u \cosh v, z=\sin u \cosh v$$

Sketch the curve traced out by the given vector valued function by hand. $$\mathbf{r}(t)=\left\langle t, 1,3 t^{2}\right\rangle$$

Find the osculating circle at the given points. $$\mathbf{r}(t)=\langle\cos 2 t, \sin 2 t\rangle \text { at } t=\frac{\pi}{4}$$

Determine all values of \(t\) at which the given vector-valued function is continuous. $$\mathbf{r}(t)=\langle 4 \cos t, \sqrt{t}, 4 \sin t\rangle$$