Question 1

Find the eccentricity of the ellipse.
-$$64 x ^ { 2 } + 4 y ^ { 2 } = 256$$

Accepted Answer

A)  $\sqrt { 15 }$ 
B)  $\frac { \sqrt { 15 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 17 } } { 4 }$ 
A)  $\sqrt { 15 }$ 
B)  $\frac { \sqrt { 15 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 17 } } { 4 }$

Question 2

Solve the problem.
-Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$, closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$, (Hint: Minimize the square of the distance as a function of $\mathrm { t }$.)

Accepted Answer

A)  $( 0,0 )$ 
B)  $( 36,6 )$ 
C)  $( 9,3 )$ 
D)  $\left( \frac { 9 } { 4 } , \frac { 3 } { 2 } \right)$ 
A)  $( 0,0 )$ 
B)  $( 36,6 )$ 
C)  $( 9,3 )$ 
D)  $\left( \frac { 9 } { 4 } , \frac { 3 } { 2 } \right)$

Question 3

Graph the polar equation.
-$r=-1-\sin \theta$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)

Question 4

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

Accepted Answer

A)  $- \frac { 1 } { 6 \sqrt { 6 } }$ 
B)  $- \frac { 1 } { \sqrt { 6 } }$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
A)  $- \frac { 1 } { 6 \sqrt { 6 } }$ 
B)  $- \frac { 1 } { \sqrt { 6 } }$ 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$

Question 5

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

Accepted Answer

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

Question 6

Solve the problem.
-The parabola $y ^ { 2 } = - 20 x$ is shifted 7 units down and 3 units to the right. Graph the new parabola.

Accepted Answer

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

Question 7

Find the vertices and foci of the ellipse.
-$$4 x ^ { 2 } + 25 y ^ { 2 } = 100$$

Accepted Answer

A)  Vertices: $( \pm 5,0 )$; Foci: $( \pm \sqrt { 21 } , 0 )$ 
B)  Vertices: $( 0 , \pm 2 )$; Foci $( 0 , \pm \sqrt { 21 } )$ 
C)  Vertices: $( \pm 2,0 )$; Foci: $( \pm \sqrt { 21 } , 0 )$ 
D)  Vertices: $( 0 , \pm 5 )$; Foci $( 0 , \pm \sqrt { 21 } )$ 
A)  Vertices: $( \pm 5,0 )$; Foci: $( \pm \sqrt { 21 } , 0 )$ 
B)  Vertices: $( 0 , \pm 2 )$; Foci $( 0 , \pm \sqrt { 21 } )$ 
C)  Vertices: $( \pm 2,0 )$; Foci: $( \pm \sqrt { 21 } , 0 )$ 
D)  Vertices: $( 0 , \pm 5 )$; Foci $( 0 , \pm \sqrt { 21 } )$

Question 8

Find the eccentricity of the hyperbola.
-$$8 x ^ { 2 } - 9 y ^ { 2 } = 72$$

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 3 }$ 
D)  $\frac { \sqrt { 17 } } { 3 }$ 
A)  $\frac { 3 } { 2 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { \sqrt { 13 } } { 3 }$ 
D)  $\frac { \sqrt { 17 } } { 3 }$

Question 9

Graph the set of points whose polar coordinates satisfy the given equation or inequality.
-$$2 \leq r \leq 5$$

Accepted Answer

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

Question 10

Find the focus and directrix of the parabola.
-$- \frac { 1 } { 28 } x ^ { 2 } = y$

Accepted Answer

A)  (0, -7); y = 7 
B)  (-14, 0); x = 7 
C)  (0, 7); y = -7 
D)  (0, -7); y = -7 
A)  (0, -7); y = 7 
B)  (-14, 0); x = 7 
C)  (0, 7); y = -7 
D)  (0, -7); y = -7

Question 11

Find the directrices of the ellipse.
-$64 x ^ { 2 } + 49 y ^ { 2 } = 3136$

Accepted Answer

A)  $y = \pm \frac { 64 } { \sqrt { 15 } }$ 
B)  $x = \pm \frac { 64 } { \sqrt { 15 } }$ 
C)  $x = \pm \frac { 64 } { \sqrt { 113 } }$ 
D)  $y = \pm \frac { 49 } { \sqrt { 15 } }$ 
A)  $y = \pm \frac { 64 } { \sqrt { 15 } }$ 
B)  $x = \pm \frac { 64 } { \sqrt { 15 } }$ 
C)  $x = \pm \frac { 64 } { \sqrt { 113 } }$ 
D)  $y = \pm \frac { 49 } { \sqrt { 15 } }$

Question 12

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin.
-$r = \frac { 4 } { 1 - \cos \theta }$

Accepted Answer

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

Question 13

Replace the polar equation with an equivalent Cartesian equation.
-$$\mathbf { r } \cos \theta = 11$$

Accepted Answer

A)  y = 11 
B)  x = 11 
C)  11y = 1 
D)  11x = 1 
A)  y = 11 
B)  x = 11 
C)  11y = 1 
D)  11x = 1

Question 14

Determine the symmetries of the curve.
-$$\mathrm { r } = \frac { 7 } { \theta } , \theta > 0$$

Accepted Answer

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

Question 15

Graph the set of points whose polar coordinates satisfy the given equation or inequality.
-$$0 \leq \theta \leq \frac { \pi } { 4 } , 0 \leq r \leq 4$$

Accepted Answer

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

Question 16

Choose the equation that matches the graph.
-

Accepted Answer

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

Question 17

Find the area of the specified region.
-Shared by the circle $r = 9$ and the cardioid $r = 9 ( 1 + \sin \theta )$

Accepted Answer

A)  $\frac { 81 } { 4 } ( 5 \pi + 8 )$ 
B)  0 
C)  81 
D)  $\frac { 81 } { 4 } ( 5 \pi - 8 )$ 
A)  $\frac { 81 } { 4 } ( 5 \pi + 8 )$ 
B)  0 
C)  81 
D)  $\frac { 81 } { 4 } ( 5 \pi - 8 )$

Question 18

Find a polar equation for the circle.
-$$x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25$$

Accepted Answer

A)  $r = 10 \cos \theta$ 
B)  $r = - 10 \sin \theta$ 
C)  $r \cos \theta = - 10$ 
D)  $r = 10 \sin \theta$ 
A)  $r = 10 \cos \theta$ 
B)  $r = - 10 \sin \theta$ 
C)  $r \cos \theta = - 10$ 
D)  $r = 10 \sin \theta$

Question 19

Determine the symmetries of the curve.
-$$r = 8 \cos 5 \theta$$

Accepted Answer

A)  x-axis, y-axis, origin 
B)  x-axis only 
C)  Origin only 
D)  y-axis only 
A)  x-axis, y-axis, origin 
B)  x-axis only 
C)  Origin only 
D)  y-axis only

Question 20

$$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$$
-$$x = \tan t , y = 9 \sec t , t = \frac { 3 \pi } { 4 }$$

Accepted Answer

A)  $- \frac { 9 } { 2 }$ 
B)  $- \frac { 9 \sqrt { 2 } } { 4 }$ 
C)  $9 \sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 4 }$ 
A)  $- \frac { 9 } { 2 }$ 
B)  $- \frac { 9 \sqrt { 2 } } { 4 }$ 
C)  $9 \sqrt { 2 }$ 
D)  $\frac { \sqrt { 2 } } { 4 }$

Find the eccentricity of the ellipse. - $64 x ^ { 2 } + 4 y ^ { 2 } = 256$

Solve the problem. -Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$ , closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$ , (Hint: Minimize the square of the distance as a function of $\mathrm { t }$ .)

Graph the polar equation. - $r=-1-\sin \theta$ $Graph the polar equation. - r=-1-\sin \theta$

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

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

Solve the problem. -The parabola $y ^ { 2 } = - 20 x$ is shifted 7 units down and 3 units to the right. Graph the new parabola. $Solve the problem. -The parabola y ^ { 2 } = - 20 x is shifted 7 units down and 3 units to the right. Graph the new parabola.$

Find the vertices and foci of the ellipse. - $4 x ^ { 2 } + 25 y ^ { 2 } = 100$

Find the eccentricity of the hyperbola. - $8 x ^ { 2 } - 9 y ^ { 2 } = 72$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $2 \leq r \leq 5$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - 2 \leq r \leq 5$

Find the focus and directrix of the parabola. - $- \frac { 1 } { 28 } x ^ { 2 } = y$

Find the directrices of the ellipse. - $64 x ^ { 2 } + 49 y ^ { 2 } = 3136$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 4 } { 1 - \cos \theta }$ $Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - r = \frac { 4 } { 1 - \cos \theta }$

Replace the polar equation with an equivalent Cartesian equation. - $\mathbf { r } \cos \theta = 11$

Determine the symmetries of the curve. - $\mathrm { r } = \frac { 7 } { \theta } , \theta > 0$

Choose the equation that matches the graph. -

Find the area of the specified region. -Shared by the circle $r = 9$ and the cardioid $r = 9 ( 1 + \sin \theta )$

Find a polar equation for the circle. - $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25$

Determine the symmetries of the curve. - $r = 8 \cos 5 \theta$

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$ - $x = \tan t , y = 9 \sec t , t = \frac { 3 \pi } { 4 }$

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Exam 12: Parametric Equations and Polar Coordinates

Find the eccentricity of the ellipse. - 64x2+4y2=25664 x ^ { 2 } + 4 y ^ { 2 } = 25664x2+4y2=256

Solve the problem. -Find the point on the parabola x=t2,y=t,∗<t<∞x = t ^ { 2 } , y = t , * < t < \inftyx=t2,y=t,∗<t<∞ , closest to the point (12,54)\left( \frac { 1 } { 2 } , 54 \right)(21​,54) , (Hint: Minimize the square of the distance as a function of t\mathrm { t }t .)

Graph the polar equation. - r=−1−sin⁡θr=-1-\sin \thetar=−1−sinθ

Solve the problem. -The parabola y2=−20xy ^ { 2 } = - 20 xy2=−20x is shifted 7 units down and 3 units to the right. Graph the new parabola.

Find the vertices and foci of the ellipse. - 4x2+25y2=1004 x ^ { 2 } + 25 y ^ { 2 } = 1004x2+25y2=100

Find the eccentricity of the hyperbola. - 8x2−9y2=728 x ^ { 2 } - 9 y ^ { 2 } = 728x2−9y2=72

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - 2≤r≤52 \leq r \leq 52≤r≤5

Find the focus and directrix of the parabola. - −128x2=y- \frac { 1 } { 28 } x ^ { 2 } = y−281​x2=y

Find the directrices of the ellipse. - 64x2+49y2=313664 x ^ { 2 } + 49 y ^ { 2 } = 313664x2+49y2=3136

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - r=41−cos⁡θr = \frac { 4 } { 1 - \cos \theta }r=1−cosθ4​

Replace the polar equation with an equivalent Cartesian equation. - rcos⁡θ=11\mathbf { r } \cos \theta = 11rcosθ=11

Determine the symmetries of the curve. - r=7θ,θ>0\mathrm { r } = \frac { 7 } { \theta } , \theta > 0r=θ7​,θ>0

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - 0≤θ≤π4,0≤r≤40 \leq \theta \leq \frac { \pi } { 4 } , 0 \leq r \leq 40≤θ≤4π​,0≤r≤4

Choose the equation that matches the graph. -

Find the area of the specified region. -Shared by the circle r=9r = 9r=9 and the cardioid r=9(1+sin⁡θ)r = 9 ( 1 + \sin \theta )r=9(1+sinθ)

Find a polar equation for the circle. - x2+(y−5)2=25x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25x2+(y−5)2=25

Determine the symmetries of the curve. - r=8cos⁡5θr = 8 \cos 5 \thetar=8cos5θ

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

Find the eccentricity of the ellipse. - $64 x ^ { 2 } + 4 y ^ { 2 } = 256$

Solve the problem. -Find the point on the parabola $x = t ^ { 2 } , y = t , * < t < \infty$ , closest to the point $\left( \frac { 1 } { 2 } , 54 \right)$ , (Hint: Minimize the square of the distance as a function of $\mathrm { t }$ .)

Graph the polar equation. - $r=-1-\sin \theta$ $Graph the polar equation. - r=-1-\sin \theta$

Solve the problem. -The parabola $y ^ { 2 } = - 20 x$ is shifted 7 units down and 3 units to the right. Graph the new parabola. $Solve the problem. -The parabola y ^ { 2 } = - 20 x is shifted 7 units down and 3 units to the right. Graph the new parabola.$

Find the vertices and foci of the ellipse. - $4 x ^ { 2 } + 25 y ^ { 2 } = 100$

Find the eccentricity of the hyperbola. - $8 x ^ { 2 } - 9 y ^ { 2 } = 72$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $2 \leq r \leq 5$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - 2 \leq r \leq 5$

Find the focus and directrix of the parabola. - $- \frac { 1 } { 28 } x ^ { 2 } = y$

Find the directrices of the ellipse. - $64 x ^ { 2 } + 49 y ^ { 2 } = 3136$

Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - $r = \frac { 4 } { 1 - \cos \theta }$ $Graph the parabola or ellipse. Include the directrix that corresponds to the focus at the origin. - r = \frac { 4 } { 1 - \cos \theta }$

Replace the polar equation with an equivalent Cartesian equation. - $\mathbf { r } \cos \theta = 11$

Determine the symmetries of the curve. - $\mathrm { r } = \frac { 7 } { \theta } , \theta > 0$

Find the area of the specified region. -Shared by the circle $r = 9$ and the cardioid $r = 9 ( 1 + \sin \theta )$

Find a polar equation for the circle. - $x ^ { 2 } + ( y - 5 ) ^ { 2 } = 25$

Determine the symmetries of the curve. - $r = 8 \cos 5 \theta$