Exam 12: Parametric Equations and Polar Coordinates

Provide an appropriate response. -Which of the statements below is true? A: If $\left( \mathrm { r } _ { 1 } , \theta _ { 1 } \right)$ and $\left( \mathrm { r } _ { 2 } , \theta _ { 2 } \right)$ represent the same point in polar coordinates, then $\left| \mathrm { r } _ { 1 } \right| = \left| \mathrm { r } _ { 2 } \right|$ . B: If $\left( r _ { 1 } , \theta _ { 1 } \right)$ and $\left( r _ { 2 } , \theta _ { 2 } \right)$ represent the same point in polar coordinates, then $\theta _ { 1 } = \theta _ { 2 } + 2 \pi n$ for some integer n.

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. - $x = t \sin t + t , y \cos t + y = 2 t , t = \frac { \pi } { 2 }$

Find the length of the curve. -The curve $r = 3 \sin ^ { 3 } \left( \frac { \theta } { 3 } \right) , 0 \leq \theta \leq \frac { \pi } { 2 }$

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. - $\mathrm { y } ^ { 2 } + 4 \mathrm { x } + 2 \mathrm { y } - 3 = 0$

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$ - $x = \sin t , y = 7 \sin t , t = \frac { \pi } { 3 }$

The polar equation of a circle is given. Give polar coordinates for the center of the circle and identify its radius. - $\mathrm { r } = 6 \cos \theta$

Replace the Cartesian equation with an equivalent polar equation. -2x + 3y = 6

Provide an appropriate response. -Show that if a and $b$ are not both 0 , the graph of the equation $r = a \cos \theta + b \sin \theta$ is a circle. Find the radius and the rectangular coordinates of the center of this circle.

Choose the equation that matches the graph. -

Find a parametrization for the curve. -The line segment with endpoints (-4, -7) and ( , -15)

Solve the problem. -The hyperbola $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 144 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( x - 1 ) ^ { 2 } } { 81 } - \frac { ( y + 3 ) ^ { 2 } } { 144 } = 1$ Find the foci of the new hyperbola.

Solve the problem. -Find the volume generated by revolving about the $x$ -axis the region bounded by the following graph: $y = \sqrt { 49 - x ^ { 2 } } , x = 0 , \quad x = 7$

Sketch the region which is defined by the given conditions. - $0 \leq \theta \leq \pi , r \leq 3$

Find the Cartesian coordinates of the given point. - $( - 9,2 \pi / 3 )$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 5 } { 2 + \sin \theta }$

Solve the problem. -Find the vertices and asymptotes of the following hyperbola: $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 16 } = 1$

Choose the equation that matches the graph. -

Find the foci of the hyperbola. - $16 y ^ { 2 } - x ^ { 2 } = 16$

Find the length of the curve. - $x = 6 \sin t - 6 t \cos t , y = 6 \cos t + 6 t \sin t , 0 \leq t \leq \frac { \pi } { 4 }$

Find the eccentricity of the hyperbola. - $81 x ^ { 2 } - 64 y ^ { 2 } = 5184$