91Ó°ÊÓ

Use graphing technology to sketch the curve traced out by the given vector- valued function. $$\mathbf{r}(t)=\langle 2 \cos 3 t+\sin 5 t, 2 \sin 3 t+\cos 5 t\rangle$$

Find a parametric representation of the surface. The portion of \(z=x^{2}+y^{2}\) below \(z=4\)

For the linear path traced out by \(\mathbf{r}(t)=\langle a+b t, c+d t, e+f t\rangle\) find the tangential and normal components of acceleration.

Sketch the curve traced out by the endpoint of the given vector-valued function and plot position and tangent vectors at the indicated points. $$\mathbf{r}(t)=\left\langle t, t^{2}-1\right\rangle, t=0, t=1, t=2$$

Use graphing technology to sketch the curve traced out by the given vector- valued function. $$\mathbf{r}(t)=\langle 4 \cos 4 t-6 \cos t, 4 \sin 4 t-6 \sin t\rangle$$

Find the curvature at the given point. \(f(x)=\sin x, x=\frac{\pi}{2}\)

Find the binormal vector \(\mathbf{B}(t)=\mathbf{T}(t) \times \mathbf{N}(t)\) at \(t=0\) and \(t=1 .\) Also, sketch the curve traced out by \(r(t)\) and the vectors \(\mathbf{T}, \mathbf{N}\) and \(\mathbf{B}\) at these points. $$\mathbf{r}(t)=\left\langle t, 2 t, t^{2}\right\rangle$$

Sketch the curve traced out by the endpoint of the given vector-valued function and plot position and tangent vectors at the indicated points. $$\mathbf{r}(t)=\langle\cos t, t, \sin t\rangle, t=0, t=\frac{\pi}{2}, t=\pi$$

Find the curvature at the given point. \(f(x)=e^{-3 x}, x=0\)

Use graphing technology to sketch the curve traced out by the given vector- valued function. $$\mathbf{r}(t)=\langle 8 \cos t+2 \cos 7 t, 8 \sin t+2 \sin 7 t\rangle$$