91Ó°ÊÓ

Sketch the curve and find any points of maximum or minimum curvature. \(\mathbf{r}(t)=\langle 2 \cos t, 3 \sin t\rangle\)

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

Replace \(\phi\) with \(\phi=\frac{\pi}{4}\) and determine the surface with parametric equations \(x=\rho \cos \theta \sin \frac{\pi}{4}, y=\rho \sin \theta \sin \frac{\pi}{4}\) and \(z=\rho \cos \frac{\pi}{4}.\)

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

Replace \(\phi\) with \(\phi=\frac{\pi}{6}\) and determine the surface with parametric equations \(x=\rho \cos \theta \sin \frac{\pi}{6}, y=\rho \sin \theta \sin \frac{\pi}{6}\) and \(z=\rho \cos \frac{\pi}{6}.\)

Sketch the curve and find any points of maximum or minimum curvature. \(\mathbf{r}(t)=\langle 4 \cos t, 3 \sin t\rangle\)

Evaluate the given indefinite or definite integral. $$\int_{0}^{2}\left\langle\frac{4}{t+1}, e^{t-2}, t e^{t}\right\rangle d t$$

Sketch the curve and find any points of maximum or minimum curvature. \(y=4 x^{2}-3\)

Replace \(\theta\) with \(\theta=\frac{\pi}{4}\) and determine the surface with parametric equations \(x=\rho \cos \frac{\pi}{4} \sin \phi, y=\rho \sin \frac{\pi}{4} \sin \phi\) and \(z=\rho \cos \phi.\)

Beginning with Newton's second law of motion, derive the equations of motion for a projectile fired from altitude \(h\) above the ground at an angle \(\theta\) to the horizontal and with initial speed \(v_{0}.\)