Question 1

Describe the conic: $x ^ { 2 } - 6 x + 4 y - 3 = 0$

Accepted Answer

A parabola with focus at (3, 2) and directrix y = 4

Question 2

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

A)  $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { counterclockwise }$ 
B)  $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { clockwise }$ 
C)  $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { counterclockwise }$ 
D)  $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { clockwise }$ 
A)  $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { counterclockwise }$ 
B)  $16 x ^ { 2 } + 9 y ^ { 2 } = 144 , \quad \text { clockwise }$ 
C)  $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { counterclockwise }$ 
D)  $9 x ^ { 2 } + 16 y ^ { 2 } = 144 , \quad \text { clockwise }$ C

Question 3

Find the equation of the hyperbola with foci ( $\pm 3$ , 0) and vertices $( \pm 2,0 )$

Accepted Answer

$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$

Question 4

Find the length of the parametric curve: $$x = e ^ { t } \sin t , \quad y = e ^ { t } \cos t , \quad \mathrm { t } \in [ 0,1 ]$$

Accepted Answer

A)  $\sqrt { 2 } ( e + 1 )$ 
B)  $\sqrt { 3 } ( e - 1 )$ 
C)  $\sqrt { 2 } e$ 
D)  $\sqrt { 2 } ( e - 1 )$ 
A)  $\sqrt { 2 } ( e + 1 )$ 
B)  $\sqrt { 3 } ( e - 1 )$ 
C)  $\sqrt { 2 } e$ 
D)  $\sqrt { 2 } ( e - 1 )$

Question 5

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$$

Accepted Answer

A)  $y = - 2 ( x + 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$ 
B)  $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$ 
C)  $y = - 2 ( x - 1 ) ^ { 2 } , x \leq 1 \text {, right to left }$ 
D)  $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$ 
A)  $y = - 2 ( x + 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$ 
B)  $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, right to left }$ 
C)  $y = - 2 ( x - 1 ) ^ { 2 } , x \leq 1 \text {, right to left }$ 
D)  $y = - 2 ( x - 1 ) ^ { 2 } , x \in ( - \infty , \infty ) \text {, left to right }$

Question 6

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { counterclockwise }$ 
B)  $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { clockwise }$ 
C)  $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { clockwise }$ 
D)  $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { counterclockwise }$ 
A)  $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { counterclockwise }$ 
B)  $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { clockwise }$ 
C)  $x ^ { 2 } + y ^ { 2 } = 3 , \quad \text { clockwise }$ 
D)  $x ^ { 2 } + y ^ { 2 } = 9 , \quad \text { counterclockwise }$

Question 7

Find the equation of the tangent line to the parametric curve: $$x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$$

Accepted Answer

A)  $y = \frac { 3 } { 2 } x - \frac { 5 } { 2 }$ 
B)  $y = \frac { 5 } { 2 } x + \frac { 3 } { 2 }$ 
C)  $y = \frac { 3 } { 2 } x + \frac { 5 } { 2 }$ 
D)  $y = - \frac { 3 } { 2 } x + \frac { 5 } { 2 }$ 
A)  $y = \frac { 3 } { 2 } x - \frac { 5 } { 2 }$ 
B)  $y = \frac { 5 } { 2 } x + \frac { 3 } { 2 }$ 
C)  $y = \frac { 3 } { 2 } x + \frac { 5 } { 2 }$ 
D)  $y = - \frac { 3 } { 2 } x + \frac { 5 } { 2 }$

Question 8

Convert the equation $r = 2 \sin \theta$ to rectangular coordinates.

Accepted Answer

A)  $x ^ { 2 } - 2 x + y ^ { 2 } = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + 2 y = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - 2 y = 0$ 
D)  $x ^ { 2 } + 2 x + y ^ { 2 } = 0$ 
A)  $x ^ { 2 } - 2 x + y ^ { 2 } = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + 2 y = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - 2 y = 0$ 
D)  $x ^ { 2 } + 2 x + y ^ { 2 } = 0$

Question 9

Find a definite integral expression that represents the area of the region between the loops of the limacon $r = 1 + 2 \cos \theta$ Then find the exact value of the integral.

Accepted Answer

The answer of Find a definite integral expression that represents...

Question 10

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 2 \cos 3 \theta$ Then find the exact value of the integral.

Accepted Answer

The answer of Find a definite integral expression that represents...

Question 11

Convert the equation $r = \sin 2 \theta$ to rectangular coordinates.

Accepted Answer

The answer of Convert the equation \(r = \sin 2...

Question 12

Find the equation of a parabola with focus (-2, 0) and vertex (0, 0).

Accepted Answer

The answer of Find the equation of a parabola with...

Question 13

Find the equation of the tangent line to the parametric curve: $$x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$$

Accepted Answer

A)  $y = - \frac { 4 } { 3 } x - \frac { 16 } { 3 }$ 
B)  $y = \frac { 4 } { 3 } x + \frac { 16 } { 3 }$ 
C)  $y = - \frac { 4 } { 3 } x + \frac { 16 } { 3 }$ 
D)  $y = \frac { 4 } { 3 } x - \frac { 16 } { 3 }$ 
A)  $y = - \frac { 4 } { 3 } x - \frac { 16 } { 3 }$ 
B)  $y = \frac { 4 } { 3 } x + \frac { 16 } { 3 }$ 
C)  $y = - \frac { 4 } { 3 } x + \frac { 16 } { 3 }$ 
D)  $y = \frac { 4 } { 3 } x - \frac { 16 } { 3 }$

Question 14

Convert the equation x = -2 to polar coordinates.

Accepted Answer

A)  $r = 2 \sec \theta$ 
B)  $r = - 2 \csc \theta$ 
C)  $r = 2 \csc \theta$ 
D)  $r = - 2 \sec \theta$ 
A)  $r = 2 \sec \theta$ 
B)  $r = - 2 \csc \theta$ 
C)  $r = 2 \csc \theta$ 
D)  $r = - 2 \sec \theta$

Question 15

Find the length of the parametric curve: $$x = t ^ { 2 } , \quad y = 3 - t ^ { 3 } , \quad \mathrm { t } \in [ 0,2 ]$$

Accepted Answer

The answer of Find the length of the parametric curve:...

Question 16

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 3 \sin 2 \theta$ Then find the exact value of the integral.

Accepted Answer

The answer of Find a definite integral expression that represents...

Question 17

Convert the equation $r ^ { 2 } = \cos \theta$ to rectangular coordinates.

Accepted Answer

The answer of Convert the equation \(r ^ { 2...

Question 18

Describe the conic: $4 x ^ { 2 } - 8 x + 2 y ^ { 2 } - 12 y - 3 = 0$

Accepted Answer

An ellipse with cent

Question 19

Convert the equation y = x to polar coordinates.

Accepted Answer

A)  $\theta = \frac { \pi } { 3 }$ 
B)  $\theta = \frac { \pi } { 6 }$ 
C)  $\theta = \frac { \pi } { 2 }$ 
D)  $\theta = \frac { \pi } { 4 }$ 
A)  $\theta = \frac { \pi } { 3 }$ 
B)  $\theta = \frac { \pi } { 6 }$ 
C)  $\theta = \frac { \pi } { 2 }$ 
D)  $\theta = \frac { \pi } { 4 }$

Question 20

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $$x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$$

Accepted Answer

A)  $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, left to right }$ 
B)  $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, right to left }$ 
C)  $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , x \in ( - \infty , \infty ) \text {, left to right }$ 
D)  $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , - 6 \leq x \leq 4 \text {, right to left }$ 
A)  $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, left to right }$ 
B)  $y = \frac { 3 } { 2 } x ^ { 2 } + 1 , x \in ( - \infty , \infty ) \text {, right to left }$ 
C)  $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , x \in ( - \infty , \infty ) \text {, left to right }$ 
D)  $y = \frac { 3 } { 2 } x ^ { 2 } - 1 , - 6 \leq x \leq 4 \text {, right to left }$

Describe the conic: $x ^ { 2 } - 6 x + 4 y - 3 = 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find the equation of the hyperbola with foci ( $\pm 3$ , 0) and vertices $( \pm 2,0 )$

Find the length of the parametric curve: $x = e ^ { t } \sin t , \quad y = e ^ { t } \cos t , \quad \mathrm { t } \in [ 0,1 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find the equation of the tangent line to the parametric curve: $x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$

Convert the equation $r = 2 \sin \theta$ to rectangular coordinates.

Find a definite integral expression that represents the area of the region between the loops of the limacon $r = 1 + 2 \cos \theta$ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 2 \cos 3 \theta$ Then find the exact value of the integral.

Convert the equation $r = \sin 2 \theta$ to rectangular coordinates.

Find the equation of a parabola with focus (-2, 0) and vertex (0, 0).

Find the equation of the tangent line to the parametric curve: $x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$

Convert the equation x = -2 to polar coordinates.

Find the length of the parametric curve: $x = t ^ { 2 } , \quad y = 3 - t ^ { 3 } , \quad \mathrm { t } \in [ 0,2 ]$

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 3 \sin 2 \theta$ Then find the exact value of the integral.

Convert the equation $r ^ { 2 } = \cos \theta$ to rectangular coordinates.

Describe the conic: $4 x ^ { 2 } - 8 x + 2 y ^ { 2 } - 12 y - 3 = 0$

Convert the equation y = x to polar coordinates.

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 9: Parametric Equations Polar Coordinates and Conic Sections

Describe the conic: x2−6x+4y−3=0x ^ { 2 } - 6 x + 4 y - 3 = 0x2−6x+4y−3=0

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=4cos⁡t,y=3sin⁡t,t∈[0,2π]x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]x=4cost,y=3sint,t∈[0,2π]

Find the equation of the hyperbola with foci ( ±3\pm 3±3 , 0) and vertices (±2,0)( \pm 2,0 )(±2,0)

Find the length of the parametric curve: x=etsin⁡t,y=etcos⁡t,t∈[0,1]x = e ^ { t } \sin t , \quad y = e ^ { t } \cos t , \quad \mathrm { t } \in [ 0,1 ]x=etsint,y=etcost,t∈[0,1]

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=1−t,y=−2t2,t≥0x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0x=1−t,y=−2t2,t≥0

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=3cos⁡t,y=3sin⁡t,t∈[0,2π]x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]x=3cost,y=3sint,t∈[0,2π]

Find the equation of the tangent line to the parametric curve: x=2t+1,y=3t+4, at t=−1x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1x=2t+1,y=3t+4, at t=−1

Convert the equation r=2sin⁡θr = 2 \sin \thetar=2sinθ to rectangular coordinates.

Find a definite integral expression that represents the area of the region between the loops of the limacon r=1+2cos⁡θr = 1 + 2 \cos \thetar=1+2cosθ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside one petal of the polar rose r=2cos⁡3θr = 2 \cos 3 \thetar=2cos3θ Then find the exact value of the integral.

Convert the equation r=sin⁡2θr = \sin 2 \thetar=sin2θ to rectangular coordinates.

Find the equation of a parabola with focus (-2, 0) and vertex (0, 0).

Find the equation of the tangent line to the parametric curve: x=t3,y=(3−t)2, at t=1x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1x=t3,y=(3−t)2, at t=1

Convert the equation x = -2 to polar coordinates.

Find the length of the parametric curve: x=t2,y=3−t3,t∈[0,2]x = t ^ { 2 } , \quad y = 3 - t ^ { 3 } , \quad \mathrm { t } \in [ 0,2 ]x=t2,y=3−t3,t∈[0,2]

Find a definite integral expression that represents the area of the region inside one petal of the polar rose r=3sin⁡2θr = 3 \sin 2 \thetar=3sin2θ Then find the exact value of the integral.

Convert the equation r2=cos⁡θr ^ { 2 } = \cos \thetar2=cosθ to rectangular coordinates.

Describe the conic: 4x2−8x+2y2−12y−3=04 x ^ { 2 } - 8 x + 2 y ^ { 2 } - 12 y - 3 = 04x2−8x+2y2−12y−3=0

Convert the equation y = x to polar coordinates.

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. x=−2t,y=6t2−1,t∈[−2,3]x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]x=−2t,y=6t2−1,t∈[−2,3]

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Describe the conic: $x ^ { 2 } - 6 x + 4 y - 3 = 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 4 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find the equation of the hyperbola with foci ( $\pm 3$ , 0) and vertices $( \pm 2,0 )$

Find the length of the parametric curve: $x = e ^ { t } \sin t , \quad y = e ^ { t } \cos t , \quad \mathrm { t } \in [ 0,1 ]$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 1 - t , \quad y = - 2 t ^ { 2 } , \quad t \geq 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = 3 \cos t , \quad y = 3 \sin t , \quad t \in [ 0,2 \pi ]$

Find the equation of the tangent line to the parametric curve: $x = 2 t + 1 , \quad y = 3 t + 4 , \quad \text { at } t = - 1$

Convert the equation $r = 2 \sin \theta$ to rectangular coordinates.

Find a definite integral expression that represents the area of the region between the loops of the limacon $r = 1 + 2 \cos \theta$ Then find the exact value of the integral.

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 2 \cos 3 \theta$ Then find the exact value of the integral.

Convert the equation $r = \sin 2 \theta$ to rectangular coordinates.

Find the equation of the tangent line to the parametric curve: $x = t ^ { 3 } , \quad y = ( 3 - t ) ^ { 2 } , \quad \text { at } t = 1$

Find the length of the parametric curve: $x = t ^ { 2 } , \quad y = 3 - t ^ { 3 } , \quad \mathrm { t } \in [ 0,2 ]$

Find a definite integral expression that represents the area of the region inside one petal of the polar rose $r = 3 \sin 2 \theta$ Then find the exact value of the integral.

Convert the equation $r ^ { 2 } = \cos \theta$ to rectangular coordinates.

Describe the conic: $4 x ^ { 2 } - 8 x + 2 y ^ { 2 } - 12 y - 3 = 0$

Find an equation of y as a function of x for the parametric equations given below by eliminating the parameter. Also, indicate the direction of the curve. $x = - 2 t , \quad y = 6 t ^ { 2 } - 1 , \quad t \in [ - 2,3 ]$