Question 1

Find the Cartesian coordinates of the given point.
-$\left( 18 \sqrt { 2 } , \frac { \pi } { 4 } \right)$

Accepted Answer

A)  $( 9 \sqrt { 3 } , 9 )$ 
B)  $( 18 \sqrt { 2 } , 18 \sqrt { 2 } )$ 
C)  $( 18,18 )$ 
D)  $( 9 \sqrt { 2 } , 9 \sqrt { 2 } )$ 
A)  $( 9 \sqrt { 3 } , 9 )$ 
B)  $( 18 \sqrt { 2 } , 18 \sqrt { 2 } )$ 
C)  $( 18,18 )$ 
D)  $( 9 \sqrt { 2 } , 9 \sqrt { 2 } )$

Question 2

Find the Cartesian coordinates of the given point.
-$( - 7,5 \pi / 6 )$

Accepted Answer

A)  $\left( \frac { - 7 } { 2 } , \frac { - 7 } { 2 } \right)$ 
B)  $\left( \frac { 7 \sqrt { 3 } } { 2 } , \frac { - 7 } { 2 } \right)$ 
C)  $\left( \frac { - 7 } { 2 } , \frac { 7 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { 7 \sqrt { 3 } } { 2 } , \frac { 7 } { 2 } \right)$ 
A)  $\left( \frac { - 7 } { 2 } , \frac { - 7 } { 2 } \right)$ 
B)  $\left( \frac { 7 \sqrt { 3 } } { 2 } , \frac { - 7 } { 2 } \right)$ 
C)  $\left( \frac { - 7 } { 2 } , \frac { 7 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( \frac { 7 \sqrt { 3 } } { 2 } , \frac { 7 } { 2 } \right)$

Question 3

Determine the symmetries of the curve.
-$$r = - 3 - 4 \sin \theta$$

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 4

Graph the set of points whose polar coordinates satisfy the given equation or inequality.
-$$\pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$$

Accepted Answer

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

Question 5

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.
-$$2 \mathrm { tx } + 4 \mathrm { t } ^ { 2 } = 4,2 \mathrm { y } - \mathrm { t } ^ { 2 } - \mathrm { t } = 0 , \mathrm { t } = 1$$

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  $- \frac { 3 } { 8 }$ 
C)  $- 6$ 
D)  $\frac { 3 } { 8 }$ 
A)  $\frac { 3 } { 2 }$ 
B)  $- \frac { 3 } { 8 }$ 
C)  $- 6$ 
D)  $\frac { 3 } { 8 }$

Question 6

Graph the pair of parametric equations with the aid of a graphing calculator.
-$x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$

Accepted Answer

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

Question 7

Find the slope of the polar curve at the indicated point.
-$$\mathrm { r } = 1 - \sin \theta , \theta = \frac { \pi } { 2 }$$

Accepted Answer

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

Question 8

Graph the set of points whose polar coordinates satisfy the given equation or inequality.
-$$r = 3$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D)

Question 9

Provide an appropriate response.
-$\text { Find the area of the region enclosed by the rose } r = a \cos n \theta \text { for } n = 1,2,3 , \ldots$

Accepted Answer

The answer of Provide an appropriate response.
-\(\text { Find the...

Question 10

Choose the equation that matches the graph.
-

Accepted Answer

A)  $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$ 
B)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
A)  $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$ 
B)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$

Question 11

Find the directrices of the hyperbola.
-$$81 x ^ { 2 } - 16 y ^ { 2 } = 1296$$

Accepted Answer

A)  $\mathrm { y } = \pm \frac { 81 } { \sqrt { 97 } }$ 
B)  $y = \pm \frac { 16 } { \sqrt { 97 } }$ 
C)  $x = \pm \frac { 16 } { \sqrt { 97 } }$ 
D)  $x = \pm \frac { 81 } { \sqrt { 97 } }$ 
A)  $\mathrm { y } = \pm \frac { 81 } { \sqrt { 97 } }$ 
B)  $y = \pm \frac { 16 } { \sqrt { 97 } }$ 
C)  $x = \pm \frac { 16 } { \sqrt { 97 } }$ 
D)  $x = \pm \frac { 81 } { \sqrt { 97 } }$

Question 12

Graph.
-$36 y^{2}-25 x^{2}=900$

Accepted Answer

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

Question 13

Determine if the given polar coordinates represent the same point.
-$$( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )$$

Accepted Answer

A) True 
 B)False

Question 14

$$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$$
-$$x = 6 t ^ { 2 } - 3 , y = t ^ { 3 } , t = 1$$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 48 }$ 
D)  $- \frac { 1 } { 48 }$ 
A)  $\frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 48 }$ 
D)  $- \frac { 1 } { 48 }$

Question 15

Solve the problem.
-Find the point on the curve $x = 3 \sin t , y = \cos t , - \frac { \pi } { 2 } \leq t \leq \frac { \pi } { 2 }$, closest to the point $\left( \frac { 4 } { 3 } , 0 \right)$. (Hint: Minimize the square of the distance as a function of $t$.)

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
B)  $( - 3,0 )$ 
C)  $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$ 
D)  $( 3,0 )$ 
A)  $\left( \frac { 3 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)$ 
B)  $( - 3,0 )$ 
C)  $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$ 
D)  $( 3,0 )$

Question 16

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions.
-Vertices at (2, 0) and (-2, 0); foci at (4, 0) and (-4, 0)

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 12 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 12 } - \frac { y ^ { 2 } } { 4 } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 12 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 12 } - \frac { y ^ { 2 } } { 4 } = 1$

Question 17

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

Accepted Answer

A)  $( \bar { x } , \bar { y } ) = \left( \frac { 60 } { \pi } - \frac { 120 } { \pi ^ { 2 } } , \frac { 120 } { \pi ^ { 2 } } - 10 \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 60 } { \pi } , \frac { 30 } { \pi ^ { 2 } } \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 120 } { \pi ^ { 2 } } , \frac { 15 } { \pi } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 60 } { \pi ^ { 2 } } , \frac { 30 } { \pi } \right)$ 
A)  $( \bar { x } , \bar { y } ) = \left( \frac { 60 } { \pi } - \frac { 120 } { \pi ^ { 2 } } , \frac { 120 } { \pi ^ { 2 } } - 10 \right)$ 
B)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 60 } { \pi } , \frac { 30 } { \pi ^ { 2 } } \right)$ 
C)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 120 } { \pi ^ { 2 } } , \frac { 15 } { \pi } \right)$ 
D)  $( \bar { x } , \bar { y } ) = \left( 10 - \frac { 60 } { \pi ^ { 2 } } , \frac { 30 } { \pi } \right)$

Question 18

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

Accepted Answer

A)  $\left( 3 \sqrt { 2 } , \frac { 3 \pi } { 4 } \right)$ 
B)  $\left( 3 \sqrt { 2 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( 3 , \frac { \pi } { 2 } \right)$ 
D)  $\left( 3 , \frac { \pi } { 4 } \right)$ 
A)  $\left( 3 \sqrt { 2 } , \frac { 3 \pi } { 4 } \right)$ 
B)  $\left( 3 \sqrt { 2 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( 3 , \frac { \pi } { 2 } \right)$ 
D)  $\left( 3 , \frac { \pi } { 4 } \right)$

Question 19

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions.
-Foci at $( 0 , - 5 ) , ( 0,5 )$; asymptotes: $y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ 
D)  $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$

Question 20

Find the vertices and foci of the ellipse.
-$36 x ^ { 2 } + 100 y ^ { 2 } = 3600$

Accepted Answer

A)  Vertices: $( \pm 10,0 )$; Foci: $( \pm 8,0 )$ 
B)  Vertices: $( \pm 36,0 )$; Foci: $( \pm 8,0 )$ 
C)  Vertices: $( 0 , \pm 10 )$; Foci: $( 0 , \pm 8 )$ 
D)  Vertices: $( 0 , \pm 36 )$; Foci: $( 0 , \pm 10 )$ 
A)  Vertices: $( \pm 10,0 )$; Foci: $( \pm 8,0 )$ 
B)  Vertices: $( \pm 36,0 )$; Foci: $( \pm 8,0 )$ 
C)  Vertices: $( 0 , \pm 10 )$; Foci: $( 0 , \pm 8 )$ 
D)  Vertices: $( 0 , \pm 36 )$; Foci: $( 0 , \pm 10 )$

Find the Cartesian coordinates of the given point. - $\left( 18 \sqrt { 2 } , \frac { \pi } { 4 } \right)$

Find the Cartesian coordinates of the given point. - $( - 7,5 \pi / 6 )$

Determine the symmetries of the curve. - $r = - 3 - 4 \sin \theta$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $\pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - \pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 1 - \sin \theta , \theta = \frac { \pi } { 2 }$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $r = 3$

Provide an appropriate response. - $\text { Find the area of the region enclosed by the rose } r = a \cos n \theta \text { for } n = 1,2,3 , \ldots$

Choose the equation that matches the graph. -

Find the directrices of the hyperbola. - $81 x ^ { 2 } - 16 y ^ { 2 } = 1296$

Graph. - $36 y^{2}-25 x^{2}=900$ $Graph. - 36 y^{2}-25 x^{2}=900$

Determine if the given polar coordinates represent the same point. - $( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )$

$\text { Find the value of } d ^ { 2 } y / d x ^ { 2 } \text { at the point defined by the given value of } t \text {. }$ - $x = 6 t ^ { 2 } - 3 , y = t ^ { 3 } , t = 1$

Solve the problem. -Find the point on the curve $x = 3 \sin t , y = \cos t , - \frac { \pi } { 2 } \leq t \leq \frac { \pi } { 2 }$ , closest to the point $\left( \frac { 4 } { 3 } , 0 \right)$ . (Hint: Minimize the square of the distance as a function of $t$ .)

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Vertices at (2, 0) and (-2, 0); foci at (4, 0) and (-4, 0)

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

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

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Foci at $( 0 , - 5 ) , ( 0,5 )$ ; asymptotes: $y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$

Find the vertices and foci of the ellipse. - $36 x ^ { 2 } + 100 y ^ { 2 } = 3600$

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

Find the Cartesian coordinates of the given point. - (182,π4)\left( 18 \sqrt { 2 } , \frac { \pi } { 4 } \right)(182​,4π​)

Find the Cartesian coordinates of the given point. - (−7,5π/6)( - 7,5 \pi / 6 )(−7,5π/6)

Determine the symmetries of the curve. - r=−3−4sin⁡θr = - 3 - 4 \sin \thetar=−3−4sinθ

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - π≤θ≤3π2,r=−4\pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4π≤θ≤23π​,r=−4

Graph the pair of parametric equations with the aid of a graphing calculator. - x=5tan⁡t,y=2sec⁡t;−π/2≤t≤π/2x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2x=5tant,y=2sect;−π/2≤t≤π/2

Find the slope of the polar curve at the indicated point. - r=1−sin⁡θ,θ=π2\mathrm { r } = 1 - \sin \theta , \theta = \frac { \pi } { 2 }r=1−sinθ,θ=2π​

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - r=3r = 3r=3

Provide an appropriate response. - Find the area of the region enclosed by the rose r=acos⁡nθ for n=1,2,3,…\text { Find the area of the region enclosed by the rose } r = a \cos n \theta \text { for } n = 1,2,3 , \ldots Find the area of the region enclosed by the rose r=acosnθ for n=1,2,3,…

Choose the equation that matches the graph. -

Find the directrices of the hyperbola. - 81x2−16y2=129681 x ^ { 2 } - 16 y ^ { 2 } = 129681x2−16y2=1296

Graph. - 36y2−25x2=90036 y^{2}-25 x^{2}=90036y2−25x2=900

Determine if the given polar coordinates represent the same point. - (5,π/3),(−5,−2π/3)( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )(5,π/3),(−5,−2π/3)

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Vertices at (2, 0) and (-2, 0); foci at (4, 0) and (-4, 0)

Find the coordinates of the centroid of the curve. -Find the coordinates of the centroid of the curve x=5cos⁡t+5tsin⁡t,y=5sin⁡t−5tcos⁡t,0≤t≤πx = 5 \cos t + 5 t \sin t , y = 5 \sin t - 5 t \cos t , 0 \leq t \leq \pix=5cost+5tsint,y=5sint−5tcost,0≤t≤π .

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Foci at (0,−5),(0,5)( 0 , - 5 ) , ( 0,5 )(0,−5),(0,5) ; asymptotes: y=43x,y=−43xy = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } xy=34​x,y=−34​x

Find the vertices and foci of the ellipse. - 36x2+100y2=360036 x ^ { 2 } + 100 y ^ { 2 } = 360036x2+100y2=3600

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 Cartesian coordinates of the given point. - $\left( 18 \sqrt { 2 } , \frac { \pi } { 4 } \right)$

Find the Cartesian coordinates of the given point. - $( - 7,5 \pi / 6 )$

Determine the symmetries of the curve. - $r = - 3 - 4 \sin \theta$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $\pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$ $Graph the set of points whose polar coordinates satisfy the given equation or inequality. - \pi \leq \theta \leq \frac { 3 \pi } { 2 } , r = - 4$

Graph the pair of parametric equations with the aid of a graphing calculator. - $x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$ $Graph the pair of parametric equations with the aid of a graphing calculator. - x=5 \tan t, y=2 \sec t ;-\pi / 2 \leq t \leq \pi / 2$

Find the slope of the polar curve at the indicated point. - $\mathrm { r } = 1 - \sin \theta , \theta = \frac { \pi } { 2 }$

Graph the set of points whose polar coordinates satisfy the given equation or inequality. - $r = 3$

Provide an appropriate response. - $\text { Find the area of the region enclosed by the rose } r = a \cos n \theta \text { for } n = 1,2,3 , \ldots$

Find the directrices of the hyperbola. - $81 x ^ { 2 } - 16 y ^ { 2 } = 1296$

Graph. - $36 y^{2}-25 x^{2}=900$ $Graph. - 36 y^{2}-25 x^{2}=900$

Determine if the given polar coordinates represent the same point. - $( 5 , \pi / 3 ) , ( - 5 , - 2 \pi / 3 )$

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

Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions. -Foci at $( 0 , - 5 ) , ( 0,5 )$ ; asymptotes: $y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$

Find the vertices and foci of the ellipse. - $36 x ^ { 2 } + 100 y ^ { 2 } = 3600$