Question 1

Graph.
-$48 x^{2}+27 y^{2}=432$

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Find an equation for the line tangent to the curve at the point defined by the given value of t.
-$$x = 10 t ^ { 2 } - 7 , y = t ^ { 3 } , t = 1$$

Accepted Answer

A)  $y = \frac { 3 } { 20 } x - \frac { 11 } { 20 }$ 
B)  $y = \frac { 3 } { 10 } x + 1$ 
C)  $y = \frac { 20 } { 3 } x - \frac { 11 } { 20 }$ 
D)  $y = \frac { 3 } { 20 } x + \frac { 11 } { 20 }$ 
A)  $y = \frac { 3 } { 20 } x - \frac { 11 } { 20 }$ 
B)  $y = \frac { 3 } { 10 } x + 1$ 
C)  $y = \frac { 20 } { 3 } x - \frac { 11 } { 20 }$ 
D)  $y = \frac { 3 } { 20 } x + \frac { 11 } { 20 }$

Question 4

Find the area of the specified region.
-Inside the smaller loop of the limacon $r = 4 + 8 \sin \theta$

Accepted Answer

A)  $16 \pi$ 
B)  $4 ( 3 \sqrt { 2 } - \pi )$ 
C)  $\frac { \pi ^ { 2 } } { 4 }$ 
D)  $8 ( 2 \pi - 3 \sqrt { 3 } )$ 
A)  $16 \pi$ 
B)  $4 ( 3 \sqrt { 2 } - \pi )$ 
C)  $\frac { \pi ^ { 2 } } { 4 }$ 
D)  $8 ( 2 \pi - 3 \sqrt { 3 } )$

Question 5

Solve the problem.
-The parabola $y ^ { 2 } = 14 x$ is shifted up 7 units and left 7 units. Find an equation for the new parabola and find the new vertex.

Accepted Answer

A)  $( \mathrm { y } + 7 ) ^ { 2 } = 14 ( \mathrm { x } - 7 )$; vertex; $( 7 , - 7 )$ 
B)  $( y - 7 ) ^ { 2 } = 14 ( x + 7 )$; vertex; $( - 7,7 )$ 
C)  $( y - 7 ) ^ { 2 } = 14 ( x + 7 )$; vertex; $( 7 , - 7 )$ 
D)  $( \mathrm { y } + 7 ) ^ { 2 } = 14 ( \mathrm { x } - 7 )$; vertex; $( - 7,7 )$ 
A)  $( \mathrm { y } + 7 ) ^ { 2 } = 14 ( \mathrm { x } - 7 )$; vertex; $( 7 , - 7 )$ 
B)  $( y - 7 ) ^ { 2 } = 14 ( x + 7 )$; vertex; $( - 7,7 )$ 
C)  $( y - 7 ) ^ { 2 } = 14 ( x + 7 )$; vertex; $( 7 , - 7 )$ 
D)  $( \mathrm { y } + 7 ) ^ { 2 } = 14 ( \mathrm { x } - 7 )$; vertex; $( - 7,7 )$

Question 6

Choose the equation that matches the graph.
-

Accepted Answer

A)  $- 4 x ^ { 2 } = y$ 
B)  $y ^ { 2 } = - 4 x$ 
C)  $x ^ { 2 } = - 4 y$ 
D)  $x ^ { 2 } = 4 y$ 
A)  $- 4 x ^ { 2 } = y$ 
B)  $y ^ { 2 } = - 4 x$ 
C)  $x ^ { 2 } = - 4 y$ 
D)  $x ^ { 2 } = 4 y$

Question 7

Determine the symmetries of the curve.
-$$r = 1 + 5 \cos \theta$$

Accepted Answer

A)  x-axis only 
B)  No symmetry 
C)  y-axis only 
D)  Origin only 
A)  x-axis only 
B)  No symmetry 
C)  y-axis only 
D)  Origin only

Question 8

Graph the set of points whose polar coordinates satisfy the given equation or inequality.
-$$\mathrm { r } \leq 4$$

Accepted Answer

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

Question 9

Graph the parabola.
-$x^{2}=-2 y$

Accepted Answer

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

Question 10

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 = 2, x = -8

Accepted Answer

A)  $r = \frac { 16 } { 1 - 2 \cos \theta }$ 
B)  $r = \frac { 16 } { 1 + 2 \cos \theta }$ 
C)  $r = \frac { 16 } { 1 + 8 \cos \theta }$ 
D)  $r = \frac { 16 } { 1 - 2 \sin \theta }$ 
A)  $r = \frac { 16 } { 1 - 2 \cos \theta }$ 
B)  $r = \frac { 16 } { 1 + 2 \cos \theta }$ 
C)  $r = \frac { 16 } { 1 + 8 \cos \theta }$ 
D)  $r = \frac { 16 } { 1 - 2 \sin \theta }$

Question 11

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$
-x = -4

Accepted Answer

A)  $r \cos ( \theta + \pi ) = 4$ 
B)  $\mathrm { r } \cos \left( \theta - \frac { \pi } { 2 } \right) = 4$ 
C)  $r \cos \theta = 4$ 
D)  $r \cos \left( \theta + \frac { \pi } { 2 } \right) = 4$ 
A)  $r \cos ( \theta + \pi ) = 4$ 
B)  $\mathrm { r } \cos \left( \theta - \frac { \pi } { 2 } \right) = 4$ 
C)  $r \cos \theta = 4$ 
D)  $r \cos \left( \theta + \frac { \pi } { 2 } \right) = 4$

Question 12

Provide an appropriate response.
-Assuming $\mathrm { r } = \mathrm { f } ( \theta )$ is continuous for $\alpha \leq \theta \leq \beta$ and $\alpha < \beta \leq \alpha + 2 \pi$, what can be said about the relative areas betweel the origin and the polar curves and
$\begin{array} { l } 
\mathrm { r } = \mathrm { f } ( \theta ) , \quad \alpha \leq \theta \leq \beta \
\mathrm { r } = 2 \mathrm { f } ( \theta ) , \quad \alpha \leq \theta \leq \beta ?
\end{array}$
Give reasons for your answer.

Accepted Answer

The area between the origin an

Question 13

Solve the problem.
-The hyperbola $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 4 } = 1$ is shifted horizontally and vertically to obtain the hyperbola $\frac { ( y + 5 ) ^ { 2 } } { 25 } - \frac { ( x + 3 ) ^ { 2 } } { 4 } = 1$. Graph the new hyperbola.

Accepted Answer

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

Question 14

Solve the problem.
-The ellipse $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$ is shifted horizontally and vertically to obtain the ellipse $\frac { ( \mathrm { x } - 1 ) ^ { 2 } } { 36 } + \frac { ( \mathrm { y } + 1 ) ^ { 2 } } { 4 } = 1 .$ Graph the new ellipse.

Accepted Answer

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

Question 15

Describe the graph of the polar equation.
-$$\mathrm { r } ^ { 2 } \sin 2 \theta = 22$$

Accepted Answer

A)  Hyperbola with the $x$ - and $y$-axes as asymptotes 
B)  Hyperbola with asymptotes $y = \pm 11 x$ 
C)  Parabola centered at the origin opening to the right 
D)  Parabola centered at the origin opening upward 
A)  Hyperbola with the $x$ - and $y$-axes as asymptotes 
B)  Hyperbola with asymptotes $y = \pm 11 x$ 
C)  Parabola centered at the origin opening to the right 
D)  Parabola centered at the origin opening upward

Question 16

Find the foci of the ellipse.
-$196 \mathrm { x } ^ { 2 } + 81 \mathrm { y } ^ { 2 } = 15,876$

Accepted Answer

A)  $( \pm \sqrt { 115 } , 0 )$ 
B)  $( 0 , \pm \sqrt { 115 } )$ 
C)  $( 0 , \pm 14 )$ 
D)  $( 0 , \pm 9 )$ 
A)  $( \pm \sqrt { 115 } , 0 )$ 
B)  $( 0 , \pm \sqrt { 115 } )$ 
C)  $( 0 , \pm 14 )$ 
D)  $( 0 , \pm 9 )$

Question 17

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.
-$$25 x ^ { 2 } + 4 y ^ { 2 } - 300 x - 40 y + 900 = 0$$

Accepted Answer

A)  C: (6, 5); V: (8, 5), (4, 5); F: (11.4, 5), (0.6, 5) 
B)  C: (6, 5); V: (6, 10), (6, 0); F: (6, 9.6), (6, 0.4) 
C)  C: (6, 5); V: (11, 5), (1, 5); F: (10.6, 5), (1.4, 5) 
D)  C: (6,5); V: (6, 7), (6, 3); F: (6, 10.4), (6, -0.4) 
A)  C: (6, 5); V: (8, 5), (4, 5); F: (11.4, 5), (0.6, 5) 
B)  C: (6, 5); V: (6, 10), (6, 0); F: (6, 9.6), (6, 0.4) 
C)  C: (6, 5); V: (11, 5), (1, 5); F: (10.6, 5), (1.4, 5) 
D)  C: (6,5); V: (6, 7), (6, 3); F: (6, 10.4), (6, -0.4)

Question 18

Find the length of the curve.
-The curve $r = 7 \sqrt { 1 + \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 4 }$

Accepted Answer

A)  $2 \pi$ 
B)  $\frac { 7 } { 4 } \pi \sqrt { 2 }$ 
C)  $\frac { 49 } { 8 } ( \pi + 2 )$ 
D)  $7 ( \pi + \ln ( \pi + \sqrt { 2 } ) )$ 
A)  $2 \pi$ 
B)  $\frac { 7 } { 4 } \pi \sqrt { 2 }$ 
C)  $\frac { 49 } { 8 } ( \pi + 2 )$ 
D)  $7 ( \pi + \ln ( \pi + \sqrt { 2 } ) )$

Question 19

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

Accepted Answer

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

Question 20

Solve the problem.
-Find the volume generated by revolving about the $y$-axis the area bounded by the curves $2 x ^ { 2 } - y ^ { 2 } = 50 , x = 8$

Accepted Answer

A)  $52 \pi \sqrt { 39 }$ 
B)  $26 \pi \sqrt { 78 }$ 
C)  $52 \pi \sqrt { 78 }$ 
D)  $52 \pi \sqrt { 26 }$ 
A)  $52 \pi \sqrt { 39 }$ 
B)  $26 \pi \sqrt { 78 }$ 
C)  $52 \pi \sqrt { 78 }$ 
D)  $52 \pi \sqrt { 26 }$

Graph. - $48 x^{2}+27 y^{2}=432$ $Graph. - 48 x^{2}+27 y^{2}=432$

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

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = 10 t ^ { 2 } - 7 , y = t ^ { 3 } , t = 1$

Find the area of the specified region. -Inside the smaller loop of the limacon $r = 4 + 8 \sin \theta$

Solve the problem. -The parabola $y ^ { 2 } = 14 x$ is shifted up 7 units and left 7 units. Find an equation for the new parabola and find the new vertex.

Choose the equation that matches the graph. -

Determine the symmetries of the curve. - $r = 1 + 5 \cos \theta$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $\mathrm { r } \leq 4$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - \mathrm { r } \leq 4$

Graph the parabola. - $x^{2}=-2 y$ $Graph the parabola. - x^{2}=-2 y$

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 = 2, x = -8

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ -x = -4

Describe the graph of the polar equation. - $\mathrm { r } ^ { 2 } \sin 2 \theta = 22$

Find the foci of the ellipse. - $196 \mathrm { x } ^ { 2 } + 81 \mathrm { y } ^ { 2 } = 15,876$

Find the length of the curve. -The curve $r = 7 \sqrt { 1 + \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 4 }$

Solve the problem. -Find the volume generated by revolving about the $y$ -axis the area bounded by the curves $2 x ^ { 2 } - y ^ { 2 } = 50 , x = 8$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 12: Parametric Equations and Polar Coordinates

Graph. - 48x2+27y2=43248 x^{2}+27 y^{2}=43248x2+27y2=432

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - r=248−4sin⁡θr = \frac { 24 } { 8 - 4 \sin \theta }r=8−4sinθ24​

Find an equation for the line tangent to the curve at the point defined by the given value of t. - x=10t2−7,y=t3,t=1x = 10 t ^ { 2 } - 7 , y = t ^ { 3 } , t = 1x=10t2−7,y=t3,t=1

Find the area of the specified region. -Inside the smaller loop of the limacon r=4+8sin⁡θr = 4 + 8 \sin \thetar=4+8sinθ

Solve the problem. -The parabola y2=14xy ^ { 2 } = 14 xy2=14x is shifted up 7 units and left 7 units. Find an equation for the new parabola and find the new vertex.

Choose the equation that matches the graph. -

Determine the symmetries of the curve. - r=1+5cos⁡θr = 1 + 5 \cos \thetar=1+5cosθ

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - r≤4\mathrm { r } \leq 4r≤4

Graph the parabola. - x2=−2yx^{2}=-2 yx2=−2y

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 = 2, x = -8

Find a polar equation in the form rcos⁡(θ−θ0)=r0 for the given line. \text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. } Find a polar equation in the form rcos(θ−θ0​)=r0​ for the given line. -x = -4

Describe the graph of the polar equation. - r2sin⁡2θ=22\mathrm { r } ^ { 2 } \sin 2 \theta = 22r2sin2θ=22

Find the foci of the ellipse. - 196x2+81y2=15,876196 \mathrm { x } ^ { 2 } + 81 \mathrm { y } ^ { 2 } = 15,876196x2+81y2=15,876

Find the length of the curve. -The curve r=71+sin⁡2θ,0≤θ≤π4r = 7 \sqrt { 1 + \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 4 }r=71+sin2θ​,0≤θ≤4π​

Graph the pair of parametric equations with the aid of a graphing calculator. - x=2cos⁡t+cos⁡2t,y=2sin⁡t−sin⁡2t,0≤t≤2πx = 2 \cos t + \cos 2 t , y = 2 \sin t - \sin 2 t , 0 \leq t \leq 2 \pix=2cost+cos2t,y=2sint−sin2t,0≤t≤2π

Solve the problem. -Find the volume generated by revolving about the yyy -axis the area bounded by the curves 2x2−y2=50,x=82 x ^ { 2 } - y ^ { 2 } = 50 , x = 82x2−y2=50,x=8

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Graph. - $48 x^{2}+27 y^{2}=432$ $Graph. - 48 x^{2}+27 y^{2}=432$

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

Find an equation for the line tangent to the curve at the point defined by the given value of t. - $x = 10 t ^ { 2 } - 7 , y = t ^ { 3 } , t = 1$

Find the area of the specified region. -Inside the smaller loop of the limacon $r = 4 + 8 \sin \theta$

Solve the problem. -The parabola $y ^ { 2 } = 14 x$ is shifted up 7 units and left 7 units. Find an equation for the new parabola and find the new vertex.

Determine the symmetries of the curve. - $r = 1 + 5 \cos \theta$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $\mathrm { r } \leq 4$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - \mathrm { r } \leq 4$

Graph the parabola. - $x^{2}=-2 y$ $Graph the parabola. - x^{2}=-2 y$

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ -x = -4

Describe the graph of the polar equation. - $\mathrm { r } ^ { 2 } \sin 2 \theta = 22$

Find the foci of the ellipse. - $196 \mathrm { x } ^ { 2 } + 81 \mathrm { y } ^ { 2 } = 15,876$

Find the length of the curve. -The curve $r = 7 \sqrt { 1 + \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 4 }$

Solve the problem. -Find the volume generated by revolving about the $y$ -axis the area bounded by the curves $2 x ^ { 2 } - y ^ { 2 } = 50 , x = 8$