Question 1

Find the area of the region enclosed by one loop of the curve. $r = 7 \cos 8 \theta$

Accepted Answer

A)  $\frac { 49 \pi } { 11 }$ 
B)  $\frac { 49 \pi } { 32 }$ 
C)  $\pi$ 
D)  $49 \pi$ 
E)  $\frac { \pi } { 8 }$ 
A)  $\frac { 49 \pi } { 11 }$ 
B)  $\frac { 49 \pi } { 32 }$ 
C)  $\pi$ 
D)  $49 \pi$ 
E)  $\frac { \pi } { 8 }$

Question 2

Find the area bounded by the curve $x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t }$ and the line y = 2.5.

Accepted Answer

The answer of Find the area bounded by the curve...

Question 3

Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

Accepted Answer

The answer of Eliminate the parameter to find a Cartesian...

Question 4

Find the vertex, focus, and directrix of the parabola. $y ^ { 2 } - 2 y - 20 x + 81 = 0$

Accepted Answer

Vertex: @#LAT-DLM&
F

Question 5

Find parametric equations for the path of a particle that moves once clockwise along the circle $x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4$ , starting at $( 2,7 )$ .

Accepted Answer

The answer of Find parametric equations for the path of...

Question 6

Find the vertex, focus, and directrix fo the parabola. $9 y ^ { 2 } = 2 x$

Accepted Answer

Vertex: @#LAT-DLM&
F

Question 7

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Accepted Answer

The answer of Find \(\frac { d ^ { 2...

Question 8

Find an equation of the ellipse that satisfies the given conditions. Foci: (0, ± 1), vertices (0, ± 6)

Accepted Answer

A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 35 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 35 } = 1$ 
C)  $\frac { x ^ { 2 } } { 35 } + \frac { y ^ { 2 } } { 36 } = 1$ 
D)  $\frac { x ^ { 2 } } { 35 } - \frac { y ^ { 2 } } { 36 } = 1$ 
A)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 35 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 35 } = 1$ 
C)  $\frac { x ^ { 2 } } { 35 } + \frac { y ^ { 2 } } { 36 } = 1$ 
D)  $\frac { x ^ { 2 } } { 35 } - \frac { y ^ { 2 } } { 36 } = 1$

Question 9

Find the area enclosed by the curve $r ^ { 2 } = 3 \cos 5 \theta$ .

Accepted Answer

The answer of Find the area enclosed by the curve...

Question 10

Find the area enclosed by the curve $r ^ { 2 } = 6 \cos 5 \theta$ .

Accepted Answer

The answer of Find the area enclosed by the curve...

Question 11

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ± $\frac { 7 } { 4 }$ x

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

Question 12

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Accepted Answer

A)  $6 x ^ { 2 } = y$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$ 
C)  $\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$ 
E)  $36 x ^ { 2 } = y$ 
A)  $6 x ^ { 2 } = y$ 
B)  $\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$ 
C)  $\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$ 
D)  $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$ 
E)  $36 x ^ { 2 } = y$

Question 13

Find the length of the curve. $x = 3 t ^ { 2 } + 8$ , $y = 2 t ^ { 3 } + 8$ , $0 \leq t \leq 1$

Accepted Answer

A)  $2 \sqrt { 2 } - 2$ 
B)  $4 \sqrt { 2 }$ 
C)  $4 \sqrt { 2 } - 1$ 
D)  $4 \sqrt { 2 } - 2$ 
E)  None of these 
A)  $2 \sqrt { 2 } - 2$ 
B)  $4 \sqrt { 2 }$ 
C)  $4 \sqrt { 2 } - 1$ 
D)  $4 \sqrt { 2 } - 2$ 
E)  None of these

Question 14

The graph of the following curve is given. Find the area that it encloses. $$r = 1 + 5 \sin 6 \theta$$

Accepted Answer

A)  $A = 18 \pi$ 
B)  $A = 18 \pi + 27 / 2$ 
C)  $27 / 2 \pi$ 
D)  $A = \frac { 27 \sqrt { 5 } } { 2 }$ 
E)  A = $\pi + \frac { 27 \sqrt { 5 } } { 2 }$ 
A)  $A = 18 \pi$ 
B)  $A = 18 \pi + 27 / 2$ 
C)  $27 / 2 \pi$ 
D)  $A = \frac { 27 \sqrt { 5 } } { 2 }$ 
E)  A = $\pi + \frac { 27 \sqrt { 5 } } { 2 }$

Question 15

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 4 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 4$

Accepted Answer

A)  $( 4,4 ) , ( 6,0 )$ 
B)  $( 0,0 ) , ( 4 , - 4 )$ 
C)  $( - 1,1 ) , ( - 4 , - 4 )$ 
D)  $( 4 , - 4 )$ 
E)  None of these 
A)  $( 4,4 ) , ( 6,0 )$ 
B)  $( 0,0 ) , ( 4 , - 4 )$ 
C)  $( - 1,1 ) , ( - 4 , - 4 )$ 
D)  $( 4 , - 4 )$ 
E)  None of these

Question 16

The exact length of the parametric curve $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 }$ is $\sqrt { 2 } e ^ { \pi / 7 }$ .

Accepted Answer

A) True 
 B)False

Question 17

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $x = 4 \sin \theta , \quad y = 2 \cos \theta$

Accepted Answer

The answer of Using the arc length formula, set up,...

Question 18

Find the equation of the directrix of the conic. $r = \frac { 14 } { 7 + \sin \theta }$

Accepted Answer

A)  $x = 7$ 
B)  $x = - 7$ 
C)  $x = 2$ 
D)  $y = - 14$ 
E)  $y = 14$ 
A)  $x = 7$ 
B)  $x = - 7$ 
C)  $x = 2$ 
D)  $y = - 14$ 
E)  $y = 14$

Question 19

Find the slope of the tangent line to the given polar curve at the point specified by the value of $a$  . $r = \frac { 1 } { a } , a = \pi$

Accepted Answer

A)  $- \pi$ 
B)  $\frac { 3 } { \pi }$ 
C)  $- 2 \pi$ 
D)  $\frac { 1 } { 4 }$ 
E)  3 
A)  $- \pi$ 
B)  $\frac { 3 } { \pi }$ 
C)  $- 2 \pi$ 
D)  $\frac { 1 } { 4 }$ 
E)  3

Question 20

Write a polar equation in r and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity $5$ and directrix $r = - 10 \csc \theta$ .

Accepted Answer

A)  $r = \frac { 50 } { 1 - 5 \sin \theta }$ 
B)  $r = \frac { 50 } { 1 - 50 \sin \theta }$ 
C)  $r = \frac { 50 } { 1 - \sin \theta }$ 
D)  $r = \frac { 10 } { 1 + 10 \sin \theta }$ 
E)  $r = \frac { 50 } { 1 + 5 \sin \theta }$ 
A)  $r = \frac { 50 } { 1 - 5 \sin \theta }$ 
B)  $r = \frac { 50 } { 1 - 50 \sin \theta }$ 
C)  $r = \frac { 50 } { 1 - \sin \theta }$ 
D)  $r = \frac { 10 } { 1 + 10 \sin \theta }$ 
E)  $r = \frac { 50 } { 1 + 5 \sin \theta }$

Find the area of the region enclosed by one loop of the curve. $r = 7 \cos 8 \theta$

Find the area bounded by the curve $x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t }$ and the line y = 2.5.

Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

Find the vertex, focus, and directrix of the parabola. $y ^ { 2 } - 2 y - 20 x + 81 = 0$

Find parametric equations for the path of a particle that moves once clockwise along the circle $x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4$ , starting at $( 2,7 )$ .

Find the vertex, focus, and directrix fo the parabola. $9 y ^ { 2 } = 2 x$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Find an equation of the ellipse that satisfies the given conditions. Foci: (0, ± 1), vertices (0, ± 6)

Find the area enclosed by the curve $r ^ { 2 } = 3 \cos 5 \theta$ .

Find the area enclosed by the curve $r ^ { 2 } = 6 \cos 5 \theta$ .

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ± $\frac { 7 } { 4 }$ x

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Find the length of the curve. $x = 3 t ^ { 2 } + 8$ , $y = 2 t ^ { 3 } + 8$ , $0 \leq t \leq 1$

The graph of the following curve is given. Find the area that it encloses. $r = 1 + 5 \sin 6 \theta$ $The graph of the following curve is given. Find the area that it encloses. r = 1 + 5 \sin 6 \theta$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 4 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 4$

The exact length of the parametric curve $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 }$ is $\sqrt { 2 } e ^ { \pi / 7 }$ .

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $x = 4 \sin \theta , \quad y = 2 \cos \theta$

Find the equation of the directrix of the conic. $r = \frac { 14 } { 7 + \sin \theta }$

Find the slope of the tangent line to the given polar curve at the point specified by the value of $a$ . $r = \frac { 1 } { a } , a = \pi$

Write a polar equation in r and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity $5$ and directrix $r = - 10 \csc \theta$ .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 10: Parametric Equations and Polar Coordinates

Find the area of the region enclosed by one loop of the curve. r=7cos⁡8θr = 7 \cos 8 \thetar=7cos8θ

Find the area bounded by the curve x=t−1t,y=t+1tx = t - \frac { 1 } { t } , y = t + \frac { 1 } { t }x=t−t1​,y=t+t1​ and the line y = 2.5.

Eliminate the parameter to find a Cartesian equation of the curve. x=e4t−5,y=e8tx = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }x=e4t−5,y=e8t

Find the vertex, focus, and directrix of the parabola. y2−2y−20x+81=0y ^ { 2 } - 2 y - 20 x + 81 = 0y2−2y−20x+81=0

Find parametric equations for the path of a particle that moves once clockwise along the circle x2+(y−7)2=4x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4x2+(y−7)2=4 , starting at (2,7)( 2,7 )(2,7) .

Find the vertex, focus, and directrix fo the parabola. 9y2=2x9 y ^ { 2 } = 2 x9y2=2x

Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ . x=5+t2,y=t−t3x = 5 + t ^ { 2 } , y = t - t ^ { 3 }x=5+t2,y=t−t3

Find an equation of the ellipse that satisfies the given conditions. Foci: (0, ± 1), vertices (0, ± 6)

Find the area enclosed by the curve r2=3cos⁡5θr ^ { 2 } = 3 \cos 5 \thetar2=3cos5θ .

Find the area enclosed by the curve r2=6cos⁡5θr ^ { 2 } = 6 \cos 5 \thetar2=6cos5θ .

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ± 74\frac { 7 } { 4 }47​ x

Find an equation for the conic that satisfies the given conditions. ellipse, foci (±1,6)( \pm 1,6 )(±1,6) , length of major axis 8

Find the length of the curve. x=3t2+8x = 3 t ^ { 2 } + 8x=3t2+8 , y=2t3+8y = 2 t ^ { 3 } + 8y=2t3+8 , 0≤t≤10 \leq t \leq 10≤t≤1

The graph of the following curve is given. Find the area that it encloses. r=1+5sin⁡6θr = 1 + 5 \sin 6 \thetar=1+5sin6θ

Find the point(s) on the curve where the tangent is horizontal. x=t3−3t+4,y=t3−3t2+4x = t ^ { 3 } - 3 t + 4 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 4x=t3−3t+4,y=t3−3t2+4

The exact length of the parametric curve x=etcos⁡t,y=etsin⁡t,0≤t≤π7x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 }x=etcost,y=etsint,0≤t≤7π​ is 2eπ/7\sqrt { 2 } e ^ { \pi / 7 }2​eπ/7 .

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. x=4sin⁡θ,y=2cos⁡θx = 4 \sin \theta , \quad y = 2 \cos \thetax=4sinθ,y=2cosθ

Find the equation of the directrix of the conic. r=147+sin⁡θr = \frac { 14 } { 7 + \sin \theta }r=7+sinθ14​

Find the slope of the tangent line to the given polar curve at the point specified by the value of aaa . r=1a,a=πr = \frac { 1 } { a } , a = \pir=a1​,a=π

Write a polar equation in r and θ\thetaθ of a hyperbola with the focus at the origin, with the eccentricity 555 and directrix r=−10csc⁡θr = - 10 \csc \thetar=−10cscθ .

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the area of the region enclosed by one loop of the curve. $r = 7 \cos 8 \theta$

Find the area bounded by the curve $x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t }$ and the line y = 2.5.

Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

Find the vertex, focus, and directrix of the parabola. $y ^ { 2 } - 2 y - 20 x + 81 = 0$

Find parametric equations for the path of a particle that moves once clockwise along the circle $x ^ { 2 } + ( y - 7 ) ^ { 2 } = 4$ , starting at $( 2,7 )$ .

Find the vertex, focus, and directrix fo the parabola. $9 y ^ { 2 } = 2 x$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Find the area enclosed by the curve $r ^ { 2 } = 3 \cos 5 \theta$ .

Find the area enclosed by the curve $r ^ { 2 } = 6 \cos 5 \theta$ .

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ± $\frac { 7 } { 4 }$ x

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Find the length of the curve. $x = 3 t ^ { 2 } + 8$ , $y = 2 t ^ { 3 } + 8$ , $0 \leq t \leq 1$

The graph of the following curve is given. Find the area that it encloses. $r = 1 + 5 \sin 6 \theta$ $The graph of the following curve is given. Find the area that it encloses. r = 1 + 5 \sin 6 \theta$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 4 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 4$

The exact length of the parametric curve $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \frac { \pi } { 7 }$ is $\sqrt { 2 } e ^ { \pi / 7 }$ .

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $x = 4 \sin \theta , \quad y = 2 \cos \theta$

Find the equation of the directrix of the conic. $r = \frac { 14 } { 7 + \sin \theta }$

Find the slope of the tangent line to the given polar curve at the point specified by the value of $a$ . $r = \frac { 1 } { a } , a = \pi$

Write a polar equation in r and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity $5$ and directrix $r = - 10 \csc \theta$ .