Exam 12: Parametric Equations and Polar Coordinates

Find the coordinates of the centroid of the curve. -Find the coordinates of the centroid of the curve $x = e ^ { 5 t } \cos 5 t , y = e ^ { 5 t } \sin 5 t , 0 \leq t \leq \frac { \pi } { 10 }$ .

Solve the problem. -The parabola $x ^ { 2 } = - 32 y$ is shifted 4 units up and 7 units to the left. Find the focus and directrix of the new parabola.

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ - $( - \sqrt { 5 } , - \sqrt { 5 } )$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 24 } { 8 + 4 \cos \theta }$

Find the length of the curve. -The spiral $r = 3 \theta ^ { 2 } , 0 \leq \theta \leq 2 \sqrt { 3 }$

Describe the graph of the polar equation. - $\mathrm { r } = 38 \sin \theta$

Find the foci of the hyperbola. - $4 x ^ { 2 } - y ^ { 2 } = 4$

Find the length of the curve. - $x = t ^ { 3 } , y = 2 t ^ { 2 } , 0 \leq t \leq 1$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=6 \cos t+2 \cos 3 t, y=6 \sin t-2 \sin 3 t, 0 \leq t \leq 2 \pi$

Determine the symmetries of the curve. - $r ^ { 2 } = 6 \cos 4 \theta$

Find the area of the specified region. -Inside the three-leaved rose $r = 8 \cos 3 \theta$

Find the Cartesian coordinates of the given point. - $\left( 16 , - \frac { \pi } { 4 } \right)$

Graph. - $x ^ { 2 } - y ^ { 2 } = 8$

Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. - $2 x - t ^ { 2 } - t = 0,2 t y + 6 t ^ { 2 } = 6 , t = 1$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 9 + 3 \sin \theta , \theta = 0$

If the equation represents a hyperbola, find the center, foci, and asymptotes. If the equation represents an ellipse, find the center, vertices, and foci. If the equation represents a circle, find the center and radius. If the equation represents a parabola, find the focus and directrix. - $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 13 = 25$

Find the area of the specified region. -Inside one loop of the lemniscate $r ^ { 2 } = 5 \cos 2 \theta$

Find the eccentricity of the ellipse. - $144 x ^ { 2 } + y ^ { 2 } = 144$

Replace the polar equation with an equivalent Cartesian equation. - $\mathrm { r } = - 7 \csc \theta$

The eccentricity is given of a conic section with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for the conic section. - $e = \frac { 1 } { 5 } , x = - 10$