Exam 10: Parametric Equations and Polar Coordinates

Find parametric equations for the path of a particle that moves once clockwise along the circle $x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4$ , starting at $( 2,7 )$ .

Find the polar equation for the curve represented by the given Cartesian equation. $x + y = 2$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ $x = 4 ( t + \sin t ) , y = 4 ( t - \cos t )$

$\text { The exact length of the parametric curve } x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 } \text { is } \sqrt { 2 } e ^ { x / \gamma } \text { }$ ?

The curve $x = 5 - 10 \cos ^ { 2 } t , y = \tan t \left( 1 - 2 \cos ^ { 2 } t \right)$ cross itself at some point $\left( x _ { 0 } , y _ { 0 } \right)$ . Find the equations of both tangent lines at that point.

If $a$ and $b$ are fixed numbers, find parametric equations for the set of all points $P$ determined as shown in the figure, using the angle ang as the parameter. Write the equations for $a = 12$ and $b = 4$ .

If a projectile is fired with an initial velocity of $v _ { 0 }$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after $t$ seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t , y = \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g t ^ { 2 }$ where $g$ is the acceleration of gravity $\left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ . If a gun is fired with $\alpha = 55 ^ { \circ }$ and $v _ { 0 } = 440 \mathrm {~m} / \mathrm { s }$ when will the bullet hit the ground?

If the parametric curve $x = f ( t ) , \quad y = g ( t )$ satisfies $g ^ { \prime } ( 4 ) = 0$ , then it has a horizontal tangent when $t = 4$ ?

Find parametric equations to represent the line segment from $( - 3,4 )$ to $( 12 , - 8 )$ .

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$

Find the area of the region enclosed by one loop of the curve. $r = 7 \cos 8 \theta$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = \cos \theta + \sin 2 \theta + 8 , y = \sin \theta + \cos 2 \theta + 8 , \theta = \pi$

Find an equation for the conic that satisfies the given conditions. hyperbola, foci $( 0 , \pm 6 )$ , vertices $( 0 , \pm 3 )$

Find the surface area generated by rotating the lemniscate $r ^ { 2 } = 10 \cos 2 \theta$ about the line $\theta = \pi$ .

Find a polar equation for the curve represented by the given Cartesian equation. $x ^ { 2 } = 3 y$

The planet Mercury travels in an elliptical orbit with eccentricity $0.703$ . Its minimum distance from the Sun is $8 \times 10 ^ { 7 } \mathrm {~km}$ . If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in km) from Mercury to the Sun.

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$

Find an equation of the parabola with focus $\left( \frac { 15 } { 2 } , 0 \right)$ and directrix $x = - \frac { 13 } { 2 }$ .

Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus. Its perihelion distance (minimum distance from the planet to the sun) is approximately $2.3 \times 10 ^ { 7 } \mathrm {~km}$ , and its aphelion distance (maximum distance from the planet to the sun) is approximately $2.7 \times 10 ^ { 7 } \mathrm {~km}$ . Approximate the eccentricity of the planet's orbit. Round to three decimal places.

Find an equation of the tangent to the curve at the point by first eliminating the parameter. $x = e ^ { t } , \quad y = ( t - 9 ) ^ { 2 }$