Question 1

Graph the polar equation.
-$$\mathrm { r } = 2 - \cos \theta$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 2

Solve the problem.
-The parabola $x ^ { 2 } = - 12 y$ is shifted down 6 units and right 3 units. Find an equation for the new parabola and find the new vertex.

Accepted Answer

A)  $( x - 3 ) ^ { 2 } = - 12 ( y + 6 )$; vertex; $( 3 , - 6 )$ 
B)  $( y + 6 ) ^ { 2 } = - 12 ( x - 3 )$; vertex; $( - 6,3 )$ 
C)  $( x + 6 ) ^ { 2 } = - 12 ( y - 3 )$; vertex; $( - 6,3 )$ 
D)  $( x + 3 ) ^ { 2 } = - 12 ( y - 6 )$; vertex; $( 3 , - 6 )$ 
A)  $( x - 3 ) ^ { 2 } = - 12 ( y + 6 )$; vertex; $( 3 , - 6 )$ 
B)  $( y + 6 ) ^ { 2 } = - 12 ( x - 3 )$; vertex; $( - 6,3 )$ 
C)  $( x + 6 ) ^ { 2 } = - 12 ( y - 3 )$; vertex; $( - 6,3 )$ 
D)  $( x + 3 ) ^ { 2 } = - 12 ( y - 6 )$; vertex; $( 3 , - 6 )$

Question 3

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$
-$\left( \frac { 1 } { 6 } , \frac { \sqrt { 3 } } { 6 } \right)$

Accepted Answer

A)  $\left( \frac { 1 } { 3 } , \frac { \pi } { 3 } \right)$ 
B)  $\left( \frac { 1 } { 3 } , \frac { 2 \pi } { 3 } \right)$ 
C)  $\left( \frac { 1 } { 3 } , \frac { \pi } { 6 } \right)$ 
D)  $\left( \frac { 1 } { 6 } , \frac { \pi } { 6 } \right)$ 
A)  $\left( \frac { 1 } { 3 } , \frac { \pi } { 3 } \right)$ 
B)  $\left( \frac { 1 } { 3 } , \frac { 2 \pi } { 3 } \right)$ 
C)  $\left( \frac { 1 } { 3 } , \frac { \pi } { 6 } \right)$ 
D)  $\left( \frac { 1 } { 6 } , \frac { \pi } { 6 } \right)$

Question 4

Find the area of the specified region.
-Inside the lemniscate $r ^ { 2 } = a ^ { 2 } \sin 2 \theta , a > 0$

Accepted Answer

A)  $a ^ { 2 }$ 
B)  0 
C)  $\frac { a ^ { 2 } } { 2 }$ 
D)  $\pi a ^ { 2 }$ 
A)  $a ^ { 2 }$ 
B)  0 
C)  $\frac { a ^ { 2 } } { 2 }$ 
D)  $\pi a ^ { 2 }$

Question 5

$$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$$
-$$x = 3 \sin t , y = 3 \cos t , t = \frac { 3 \pi } { 4 }$$

Accepted Answer

A)  $2 \sqrt { 2 }$ 
B)  $- 2$ 
C)  $- \frac { 2 \sqrt { 2 } } { 3 }$ 
D)  $\frac { \sqrt { 2 } } { 3 }$ 
A)  $2 \sqrt { 2 }$ 
B)  $- 2$ 
C)  $- \frac { 2 \sqrt { 2 } } { 3 }$ 
D)  $\frac { \sqrt { 2 } } { 3 }$

Question 6

Provide an appropriate response.
-True or false? If $\mathrm { n } > 0$ is an even integer, then the area of the region enclosed by $\mathrm { r } = \sin \mathrm { n } \theta$ is twice the area of the region enclosed by $r = \sin [ ( n + 1 ) \theta ]$.

Accepted Answer

A) True 
 B)False

Question 7

Find an equation for the line tangent to the curve at the point defined by the given value of t.
-$$x = \csc t , y = 18 \cot t , t = \frac { \pi } { 3 }$$

Accepted Answer

A)  $y = - 36 x + 18 \sqrt { 3 }$ 
B)  $y = 6 \sqrt { 3 } x - 36$ 
C)  $y = 36 x + 6 \sqrt { 3 }$ 
D)  $y = 36 x - 18 \sqrt { 3 }$ 
A)  $y = - 36 x + 18 \sqrt { 3 }$ 
B)  $y = 6 \sqrt { 3 } x - 36$ 
C)  $y = 36 x + 6 \sqrt { 3 }$ 
D)  $y = 36 x - 18 \sqrt { 3 }$

Question 8

Parametric equations and and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
-$x=5 t+5, y=15 t+9,-\leq t \leq \infty$

Accepted Answer

A)  $y = 3 x - 6$; Entire line, from right to left
  
B)  $y = 3 x + 6$; Entire line, from right to left
  
C)  $y = 3 x - 6$; Entire line, from left to right
  
D)  $y = 3 x + 6$; Entire line, from left to right
  
A)  $y = 3 x - 6$; Entire line, from right to left
  
B)  $y = 3 x + 6$; Entire line, from right to left
  
C)  $y = 3 x - 6$; Entire line, from left to right
  
D)  $y = 3 x + 6$; Entire line, from left to right

Question 9

Find the vertices and foci of the ellipse.
-$196 x ^ { 2 } + 64 y ^ { 2 } = 12,544$

Accepted Answer

A)  Vertices: $( \pm 14,0 )$; Foci: $( \pm 2 \sqrt { 33 } , 0 )$ 
B)  Vertices: $( \pm 8,0 )$; Foci: $( \pm 2 \sqrt { 33 } , 0 )$ 
C)  Vertices: $( 0 , \pm 14 )$; Foci: $( 0 , \pm 2 \sqrt { 33 } )$ 
D)  Vertices: $( 0 , \pm 8 )$; Foci: $( 0 , \pm 2 \sqrt { 33 } )$ 
A)  Vertices: $( \pm 14,0 )$; Foci: $( \pm 2 \sqrt { 33 } , 0 )$ 
B)  Vertices: $( \pm 8,0 )$; Foci: $( \pm 2 \sqrt { 33 } , 0 )$ 
C)  Vertices: $( 0 , \pm 14 )$; Foci: $( 0 , \pm 2 \sqrt { 33 } )$ 
D)  Vertices: $( 0 , \pm 8 )$; Foci: $( 0 , \pm 2 \sqrt { 33 } )$

Question 10

Find the length of the curve.
-$$x = 5 \sin t + 5 t , y = 5 \cos t , 0 \leq t \leq \pi$$

Accepted Answer

A)  30 
B)  10 
C)  20 
D)  $5 \pi$ 
A)  30 
B)  10 
C)  20 
D)  $5 \pi$

Question 11

Graph.
-$36 y^{2}-4 x^{2}=144$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 12

Describe the graph of the polar equation.
-$$\mathrm { r } ^ { 2 } = 38 \mathrm { rcos } \theta - 6 r \sin \theta - 9$$

Accepted Answer

A)  Parabola opening upward with vertex (38, -6) 
B)  Circle with radius 19 and center (19, -3) 
C)  Circle with radius 3 and center (38, -6) 
D)  Parabola opening to the right with vertex (38, -6) 
A)  Parabola opening upward with vertex (38, -6) 
B)  Circle with radius 19 and center (19, -3) 
C)  Circle with radius 3 and center (38, -6) 
D)  Parabola opening to the right with vertex (38, -6)

Question 13

Describe the graph of the polar equation.
-$$5 r \cos \theta + r \sin \theta = 7$$

Accepted Answer

A)  Vertical line passing through (5, 0) 
B)  Parabola with vertex (5, 7) opening upward 
C)  Line with slope -5 and y-intercept (0, 7) 
D)  Line with slope 7 and y-intercept (0, 5) 
A)  Vertical line passing through (5, 0) 
B)  Parabola with vertex (5, 7) opening upward 
C)  Line with slope -5 and y-intercept (0, 7) 
D)  Line with slope 7 and y-intercept (0, 5)

Question 14

Find the Cartesian coordinates of the given point.
-$\left( 8 , \frac { 11 \pi } { 3 } \right)$

Accepted Answer

A)  $( - 4 , - 4 \sqrt { 3 } )$ 
B)  $( - 4 \sqrt { 3 } , - 4 )$ 
C)  $( 4 \sqrt { 3 } , - 4 )$ 
D)  $( 4 , - 4 \sqrt { 3 } )$ 
A)  $( - 4 , - 4 \sqrt { 3 } )$ 
B)  $( - 4 \sqrt { 3 } , - 4 )$ 
C)  $( 4 \sqrt { 3 } , - 4 )$ 
D)  $( 4 , - 4 \sqrt { 3 } )$

Question 15

Find the area of the specified region.
-Shared by the circles $r = 3 \cos \theta$ and $r = 3 \sin \theta$

Accepted Answer

A)  $\frac { 3 } { 2 } ( 2 - \sqrt { 2 } )$ 
B)  $\frac { 9 } { 4 } \pi$ 
C)  $\frac { 3 } { 8 } ( 4 \pi - 3 \sqrt { 3 } )$ 
D)  $\frac { 9 } { 8 } ( \pi - 2 )$ 
A)  $\frac { 3 } { 2 } ( 2 - \sqrt { 2 } )$ 
B)  $\frac { 9 } { 4 } \pi$ 
C)  $\frac { 3 } { 8 } ( 4 \pi - 3 \sqrt { 3 } )$ 
D)  $\frac { 9 } { 8 } ( \pi - 2 )$

Question 16

Find the focus and directrix of the parabola.
-$x ^ { 2 } = 36 y$

Accepted Answer

A)  (9, 0); y = 9 
B)  (0, -9); x = - 9 
C)  (9, 0); x = 9 
D)  (0, 9); y = -9 
A)  (9, 0); y = 9 
B)  (0, -9); x = - 9 
C)  (9, 0); x = 9 
D)  (0, 9); y = -9

Graph the polar equation. - $\mathrm { r } = 2 - \cos \theta$ $Graph the polar equation. - \mathrm { r } = 2 - \cos \theta$

Solve the problem. -The parabola $x ^ { 2 } = - 12 y$ is shifted down 6 units and right 3 units. Find an equation for the new parabola and find the new vertex.

$\text { Find the polar coordinates, } 0 \leq \theta < 2 \pi \text { and } r \geq 0 \text {, of the point given in Cartesian coordinates. }$ - $\left( \frac { 1 } { 6 } , \frac { \sqrt { 3 } } { 6 } \right)$

Find the area of the specified region. -Inside the lemniscate $r ^ { 2 } = a ^ { 2 } \sin 2 \theta , a > 0$

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$ - $x = 3 \sin t , y = 3 \cos t , t = \frac { 3 \pi } { 4 }$

Provide an appropriate response. -True or false? If $\mathrm { n } > 0$ is an even integer, then the area of the region enclosed by $\mathrm { r } = \sin \mathrm { n } \theta$ is twice the area of the region enclosed by $r = \sin [ ( n + 1 ) \theta ]$ .

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = \csc t , y = 18 \cot t , t = \frac { \pi } { 3 }$

Find the vertices and foci of the ellipse. - $196 x ^ { 2 } + 64 y ^ { 2 } = 12,544$

Find the length of the curve. - $x = 5 \sin t + 5 t , y = 5 \cos t , 0 \leq t \leq \pi$

Graph. - $36 y^{2}-4 x^{2}=144$ $Graph. - 36 y^{2}-4 x^{2}=144$

Describe the graph of the polar equation. - $\mathrm { r } ^ { 2 } = 38 \mathrm { rcos } \theta - 6 r \sin \theta - 9$

Describe the graph of the polar equation. - $5 r \cos \theta + r \sin \theta = 7$

Find the Cartesian coordinates of the given point. - $\left( 8 , \frac { 11 \pi } { 3 } \right)$

Find the area of the specified region. -Shared by the circles $r = 3 \cos \theta$ and $r = 3 \sin \theta$

Find the focus and directrix of the parabola. - $x ^ { 2 } = 36 y$

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Exam 12: Parametric Equations and Polar Coordinates

Graph the polar equation. - r=2−cos⁡θ\mathrm { r } = 2 - \cos \thetar=2−cosθ

Solve the problem. -The parabola x2=−12yx ^ { 2 } = - 12 yx2=−12y is shifted down 6 units and right 3 units. Find an equation for the new parabola and find the new vertex.

Find the area of the specified region. -Inside the lemniscate r2=a2sin⁡2θ,a>0r ^ { 2 } = a ^ { 2 } \sin 2 \theta , a > 0r2=a2sin2θ,a>0

Find an equation for the line tangent to the curve at the point defined by the given value of t. - x=csc⁡t,y=18cot⁡t,t=π3x = \csc t , y = 18 \cot t , t = \frac { \pi } { 3 }x=csct,y=18cott,t=3π​

Find the vertices and foci of the ellipse. - 196x2+64y2=12,544196 x ^ { 2 } + 64 y ^ { 2 } = 12,544196x2+64y2=12,544

Find the length of the curve. - x=5sin⁡t+5t,y=5cos⁡t,0≤t≤πx = 5 \sin t + 5 t , y = 5 \cos t , 0 \leq t \leq \pix=5sint+5t,y=5cost,0≤t≤π

Graph. - 36y2−4x2=14436 y^{2}-4 x^{2}=14436y2−4x2=144

Describe the graph of the polar equation. - r2=38rcosθ−6rsin⁡θ−9\mathrm { r } ^ { 2 } = 38 \mathrm { rcos } \theta - 6 r \sin \theta - 9r2=38rcosθ−6rsinθ−9

Describe the graph of the polar equation. - 5rcos⁡θ+rsin⁡θ=75 r \cos \theta + r \sin \theta = 75rcosθ+rsinθ=7

Find the Cartesian coordinates of the given point. - (8,11π3)\left( 8 , \frac { 11 \pi } { 3 } \right)(8,311π​)

Find the area of the specified region. -Shared by the circles r=3cos⁡θr = 3 \cos \thetar=3cosθ and r=3sin⁡θr = 3 \sin \thetar=3sinθ

Find the focus and directrix of the parabola. - x2=36yx ^ { 2 } = 36 yx2=36y