Question 1

Graph the polar equation.
-$$r ^ { 2 } = 25 \cos 4 \theta$$

Accepted Answer

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

Question 2

Find the length of the curve.
-The spiral $\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pi$

Accepted Answer

A)  $\frac { 1 } { 8 } \left( e ^ { 4 \pi } - 1 \right)$ 
B)  $\frac { \sqrt { 5 } } { 2 } \left( e ^ { 2 \pi } - 1 \right)$ 
C)  $e ^ { 4 \pi } - \sqrt { 3 }$ 
D)  $\frac { 16 } { 3 } \left( \pi ^ { 2 } - 1 \right)$ 
A)  $\frac { 1 } { 8 } \left( e ^ { 4 \pi } - 1 \right)$ 
B)  $\frac { \sqrt { 5 } } { 2 } \left( e ^ { 2 \pi } - 1 \right)$ 
C)  $e ^ { 4 \pi } - \sqrt { 3 }$ 
D)  $\frac { 16 } { 3 } \left( \pi ^ { 2 } - 1 \right)$ B

Question 3

Find the focus and directrix of the parabola.
-$$x ^ { 2 } = - 20 y$$

Accepted Answer

A)  (0, -5); y = -5 
B)  (-5, 0); y = 5 
C)  (0, -5); y = 5 
D)  (0, -5); x = 5 
A)  (0, -5); y = -5 
B)  (-5, 0); y = 5 
C)  (0, -5); y = 5 
D)  (0, -5); x = 5 C

Question 4

Determine if the given polar coordinates represent the same point.
-$$( \mathrm { r } , \theta ) , ( - \mathrm { r } , \theta + \pi )$$

Accepted Answer

A) True 
 B)False

Question 5

Solve the problem.
-Find the foci and asymptotes of the following hyperbola: $16 y ^ { 2 } - x ^ { 2 } = 16$

Accepted Answer

A)  Foci: $( \sqrt { 17 } , 0 ) , ( - \sqrt { 17 } , 0 )$; Asymptotes: $y = \frac { 1 } { 4 } x , y = - \frac { 1 } { 4 } x$ 
B)  Foci: $( 0 , \sqrt { 17 } ) , ( 0 , - \sqrt { 17 } ) ;$ Asymptotes: $y = \frac { 1 } { 4 } x , y = - \frac { 1 } { 4 } x$ 
C)  Foci: $( 0,17 ) , ( 0 , - 17 ) ;$ Asymptotes: $y = 4 x , y = - 4 x$ 
D)  Foci: $( 0 , \sqrt { 17 } ) , ( 0 , - \sqrt { 17 } )$; Asymptotes: $y = 4 x , y = - 4 x$ 
A)  Foci: $( \sqrt { 17 } , 0 ) , ( - \sqrt { 17 } , 0 )$; Asymptotes: $y = \frac { 1 } { 4 } x , y = - \frac { 1 } { 4 } x$ 
B)  Foci: $( 0 , \sqrt { 17 } ) , ( 0 , - \sqrt { 17 } ) ;$ Asymptotes: $y = \frac { 1 } { 4 } x , y = - \frac { 1 } { 4 } x$ 
C)  Foci: $( 0,17 ) , ( 0 , - 17 ) ;$ Asymptotes: $y = 4 x , y = - 4 x$ 
D)  Foci: $( 0 , \sqrt { 17 } ) , ( 0 , - \sqrt { 17 } )$; Asymptotes: $y = 4 x , y = - 4 x$

Question 6

Graph.
-$$9 x ^ { 2 } + 6 y ^ { 2 } = 18$$

Accepted Answer

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

Question 7

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$
-$\sqrt { 3 } x + y = 4$

Accepted Answer

A)  $\mathrm { r } \cos \left( \theta - \frac { \pi } { 3 } \right) = 2$ 
B)  $r \cos \left( \theta + \frac { 2 \pi } { 3 } \right) = 2$ 
C)  $r \cos \left( \theta + \frac { \pi } { 6 } \right) = 2$ 
D)  $r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2$ 
A)  $\mathrm { r } \cos \left( \theta - \frac { \pi } { 3 } \right) = 2$ 
B)  $r \cos \left( \theta + \frac { 2 \pi } { 3 } \right) = 2$ 
C)  $r \cos \left( \theta + \frac { \pi } { 6 } \right) = 2$ 
D)  $r \cos \left( \theta - \frac { \pi } { 6 } \right) = 2$

Question 8

Graph the polar equation.
-$$r = 3 \theta$$

Accepted Answer

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

Question 9

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.
-$$2 x + 4 x ^ { 3 / 2 } = t ^ { 3 } + t , y ( t + 1 ) - 4 t \sqrt { y } = 4 , t = 0$$

Accepted Answer

A)  -8 
B)  2 
C)  8 
D)  4 
A)  -8 
B)  2 
C)  8 
D)  4

Question 10

Graph the parabola.
-$x=-4 y^{2}$

Accepted Answer

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

Question 11

Provide an appropriate response.
-Find any horizontal or vertical asymptotes for the curve $r = 4 \tan \theta , 0 \leq \theta \leq 2 \pi$. (Suggestion: Express the curve in parametric form $( \mathrm { x } = \mathrm { r } \cos \theta , \mathrm { y } = \mathrm { r } \sin \theta )$ and investigate what happens to $\mathrm { x }$ and $\mathrm { y }$ as $\theta$ varies.)

Accepted Answer

As @#LAT-DLM&or @#LAT-DLM& and @#LAT-DLM&. As @#LAT-DLM& and @#LAT-DLM&. As @#LAT-DLM& and

Question 12

Find the Cartesian coordinates of the given point.
-$$\left( 3 , \frac { 1 } { 2 } \pi \right)$$

Accepted Answer

A)  (-3, 0) 
B)  (0, -3) 
C)  (3, 0) 
D)  (0, 3) 
A)  (-3, 0) 
B)  (0, -3) 
C)  (3, 0) 
D)  (0, 3)

Question 13

Find the area of the specified region.
-Inside one leaf of the four-leaved rose $r = 3 \sin 2 \theta$

Accepted Answer

A)  $\frac { 3 \pi } { 2 }$ 
B)  $\frac { 9 \pi } { 8 }$ 
C)  $\frac { 9 \pi } { 2 }$ 
D)  $\frac { 9 \pi } { 8 } - \frac { 9 } { 16 }$ 
A)  $\frac { 3 \pi } { 2 }$ 
B)  $\frac { 9 \pi } { 8 }$ 
C)  $\frac { 9 \pi } { 2 }$ 
D)  $\frac { 9 \pi } { 8 } - \frac { 9 } { 16 }$

Question 14

Provide an appropriate response.
-Consider the inequality $x ^ { 2 } + y ^ { 2 } < 3 x + y$.
(a) Find an equivalent inequality in polar coordinates.
(b) Use an integral with respect to $\theta$ to find the area of the region defined by the inequality.
(c) Find the length of the curve that forms the boundary of the region defined by the inequality.
(d) Find the slope of this curve at the point $( 2,2 )$.

Accepted Answer

The answer of Provide an appropriate response.
-Consider the inequality \(x...

Question 15

Graph.
-$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 1$$

Accepted Answer

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

Question 16

Replace the polar equation with an equivalent Cartesian equation.
-$$\mathrm { r } ^ { 2 } \sin 2 \theta = 32$$

Accepted Answer

A)  $y = \frac { 32 } { x }$ 
B)  $y = 32 x ^ { 2 }$ 
C)  $y = 16 x$ 
D)  $y = \frac { 16 } { x }$ 
A)  $y = \frac { 32 } { x }$ 
B)  $y = 32 x ^ { 2 }$ 
C)  $y = 16 x$ 
D)  $y = \frac { 16 } { x }$

Question 17

Find the length of the curve.
-$x = 3 \cos t , y = 3 \sin t , 0 \leq t \leq \pi$

Accepted Answer

A)  $3 \pi$ 
B)  $9 \pi$ 
C)  $\pi$ 
D)  $6 \pi$ 
A)  $3 \pi$ 
B)  $9 \pi$ 
C)  $\pi$ 
D)  $6 \pi$

Question 18

Find the directrices of the hyperbola.
-$$9 x ^ { 2 } - 16 y ^ { 2 } = 144$$

Accepted Answer

A)  $y = \pm \frac { 16 } { 5 }$ 
B)  $x = \pm \frac { 9 } { 5 }$ 
C)  $x = \pm \frac { 16 } { 5 }$ 
D)  $y = \pm \frac { 9 } { 5 }$ 
A)  $y = \pm \frac { 16 } { 5 }$ 
B)  $x = \pm \frac { 9 } { 5 }$ 
C)  $x = \pm \frac { 16 } { 5 }$ 
D)  $y = \pm \frac { 9 } { 5 }$

Question 19

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

Accepted Answer

A)  $4 \sqrt { 3 }$ 
B)  $- 6 \sqrt { 3 }$ 
C)  $2 \sqrt { 3 }$ 
D)  $- 4 \sqrt { 3 }$ 
A)  $4 \sqrt { 3 }$ 
B)  $- 6 \sqrt { 3 }$ 
C)  $2 \sqrt { 3 }$ 
D)  $- 4 \sqrt { 3 }$

Question 20

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
-An ellipse with intercepts $( \pm 4,0 )$ and $( 0 , \pm 2 )$

Accepted Answer

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

Graph the polar equation. - $r ^ { 2 } = 25 \cos 4 \theta$ $Graph the polar equation. - r ^ { 2 } = 25 \cos 4 \theta$

Find the length of the curve. -The spiral $\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pi$

Find the focus and directrix of the parabola. - $x ^ { 2 } = - 20 y$

Determine if the given polar coordinates represent the same point. - $( \mathrm { r } , \theta ) , ( - \mathrm { r } , \theta + \pi )$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $16 y ^ { 2 } - x ^ { 2 } = 16$

Graph. - $9 x ^ { 2 } + 6 y ^ { 2 } = 18$ $Graph. - 9 x ^ { 2 } + 6 y ^ { 2 } = 18$

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$ - $\sqrt { 3 } x + y = 4$

Graph the polar equation. - $r = 3 \theta$ $Graph the polar equation. - r = 3 \theta$

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. - $2 x + 4 x ^ { 3 / 2 } = t ^ { 3 } + t , y ( t + 1 ) - 4 t \sqrt { y } = 4 , t = 0$

Graph the parabola. - $x=-4 y^{2}$ $Graph the parabola. - x=-4 y^{2}$

Find the Cartesian coordinates of the given point. - $\left( 3 , \frac { 1 } { 2 } \pi \right)$

Find the area of the specified region. -Inside one leaf of the four-leaved rose $r = 3 \sin 2 \theta$

Graph. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 1$ $Graph. - \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 1$

Replace the polar equation with an equivalent Cartesian equation. - $\mathrm { r } ^ { 2 } \sin 2 \theta = 32$

Find the length of the curve. - $x = 3 \cos t , y = 3 \sin t , 0 \leq t \leq \pi$

Find the directrices of the hyperbola. - $9 x ^ { 2 } - 16 y ^ { 2 } = 144$

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

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with intercepts $( \pm 4,0 )$ and $( 0 , \pm 2 )$

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 the polar equation. - r2=25cos⁡4θr ^ { 2 } = 25 \cos 4 \thetar2=25cos4θ

Find the length of the curve. -The spiral r=e2θ,0≤θ≤π\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pir=e2θ,0≤θ≤π

Find the focus and directrix of the parabola. - x2=−20yx ^ { 2 } = - 20 yx2=−20y

Determine if the given polar coordinates represent the same point. - (r,θ),(−r,θ+π)( \mathrm { r } , \theta ) , ( - \mathrm { r } , \theta + \pi )(r,θ),(−r,θ+π)

Solve the problem. -Find the foci and asymptotes of the following hyperbola: 16y2−x2=1616 y ^ { 2 } - x ^ { 2 } = 1616y2−x2=16

Graph. - 9x2+6y2=189 x ^ { 2 } + 6 y ^ { 2 } = 189x2+6y2=18

Graph the polar equation. - r=3θr = 3 \thetar=3θ

Graph the parabola. - x=−4y2x=-4 y^{2}x=−4y2

Find the Cartesian coordinates of the given point. - (3,12π)\left( 3 , \frac { 1 } { 2 } \pi \right)(3,21​π)

Find the area of the specified region. -Inside one leaf of the four-leaved rose r=3sin⁡2θr = 3 \sin 2 \thetar=3sin2θ

Graph. - x225+y236=1\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 125x2​+36y2​=1

Replace the polar equation with an equivalent Cartesian equation. - r2sin⁡2θ=32\mathrm { r } ^ { 2 } \sin 2 \theta = 32r2sin2θ=32

Find the length of the curve. - x=3cos⁡t,y=3sin⁡t,0≤t≤πx = 3 \cos t , y = 3 \sin t , 0 \leq t \leq \pix=3cost,y=3sint,0≤t≤π

Find the directrices of the hyperbola. - 9x2−16y2=1449 x ^ { 2 } - 16 y ^ { 2 } = 1449x2−16y2=144

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with intercepts (±4,0)( \pm 4,0 )(±4,0) and (0,±2)( 0 , \pm 2 )(0,±2)

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 the polar equation. - $r ^ { 2 } = 25 \cos 4 \theta$ $Graph the polar equation. - r ^ { 2 } = 25 \cos 4 \theta$

Find the length of the curve. -The spiral $\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pi$

Find the focus and directrix of the parabola. - $x ^ { 2 } = - 20 y$

Determine if the given polar coordinates represent the same point. - $( \mathrm { r } , \theta ) , ( - \mathrm { r } , \theta + \pi )$

Solve the problem. -Find the foci and asymptotes of the following hyperbola: $16 y ^ { 2 } - x ^ { 2 } = 16$

Graph. - $9 x ^ { 2 } + 6 y ^ { 2 } = 18$ $Graph. - 9 x ^ { 2 } + 6 y ^ { 2 } = 18$

Graph the polar equation. - $r = 3 \theta$ $Graph the polar equation. - r = 3 \theta$

Graph the parabola. - $x=-4 y^{2}$ $Graph the parabola. - x=-4 y^{2}$

Find the Cartesian coordinates of the given point. - $\left( 3 , \frac { 1 } { 2 } \pi \right)$

Find the area of the specified region. -Inside one leaf of the four-leaved rose $r = 3 \sin 2 \theta$

Graph. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 1$ $Graph. - \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 36 } = 1$

Replace the polar equation with an equivalent Cartesian equation. - $\mathrm { r } ^ { 2 } \sin 2 \theta = 32$

Find the length of the curve. - $x = 3 \cos t , y = 3 \sin t , 0 \leq t \leq \pi$

Find the directrices of the hyperbola. - $9 x ^ { 2 } - 16 y ^ { 2 } = 144$

Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions. -An ellipse with intercepts $( \pm 4,0 )$ and $( 0 , \pm 2 )$