91Ó°ÊÓ

Find the slope of the line tangent to the following polar curves at the given points. \(r=2 \sin 3 \theta ;\) at the tips of the leaves

Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. $$x=-\frac{y^{2}}{16}$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y\). b. Describe the curve and indicate the positive orientation. $$x=\sqrt{t}+4, y=3 \sqrt{t} ; 0 \leq t \leq 16$$

Sketch the following sets of points. $$2 \leq r \leq 8$$

Find the slope of the line tangent to the following polar curves at the given points. $$r=1+2 \sin 2 \theta ;\left(3, \frac{\pi}{4}\right)$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y\). b. Describe the curve and indicate the positive orientation. $$x=(t+1)^{2}, y=t+2 ;-10 \leq t \leq 10$$

Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes. $$\frac{x^{2}}{4}-y^{2}=1$$

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y\). b. Describe the curve and indicate the positive orientation. $$x=3 \cos t, y=3 \sin t ; \pi \leq t \leq 2 \pi$$

Find the slope of the line tangent to the following polar curves at the given points. $$r^{2}=4 \cos 2 \theta ;\left(0, \pm \frac{\pi}{4}\right)$$

Find the slope of the line tangent to the following polar curves at the given points. $$r=2 \theta ;\left(\frac{\pi}{2}, \frac{\pi}{4}\right)$$