Exam 10: Parametric Equations; Polar Equations

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt [ 3 ] { 3 - t } , y ( t ) = \sqrt [ 3 ] { t }$ with $t \in ( - \infty , \infty ),$ is

(Multiple Choice)

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Question 61

The conic section $r = \frac { 5 } { 3 + 8 \sin \theta }$ is

(Multiple Choice)

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(39)

Question 62

The area of the surface generated by revolving the curve $x ( t ) = t + \sqrt { 2 } , \quad y = \frac { t ^ { 2 } } { 2 } + \sqrt { t }$ with $t \in [ - \sqrt { 2 } , \sqrt { 2 } ]$ about the y-axis is

(Multiple Choice)

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Question 63

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \tan t , y ( t ) = \sec ^ { 2 } t,$ is

(Multiple Choice)

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(35)

Question 64

The polar equation that corresponds to the rectangular equation $y = \sqrt { 3 } x$ is

(Multiple Choice)

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Question 65

If A is the area of the intersection of the regions enclosed by the graphs of r = 3 sin 2 $\theta$ and r = 3 cos 2 $\theta$ , then A is

(Multiple Choice)

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Question 66

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 \cos \theta$ and $r = 2 \sin \theta,$ then A is

(Multiple Choice)

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Question 67

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = 1 + e ^ { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

(Multiple Choice)

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Question 68

The points of intersections of $r = 2 - 2 \cos \theta$ and $r = 2 \cos \theta$ are

(Multiple Choice)

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Question 69

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 7 x$ is

(Multiple Choice)

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Question 70

The conic section $r = \frac { 3 } { 1 - \sin \theta }$ is

(Multiple Choice)

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Question 71

For a $\neq$ 0, the polar curve $r ^ { 2 } = a \cos 2 \theta$ is a

(Multiple Choice)

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Question 72

The rectangular equation that corresponds to the polar equation $r = 4 \cos \theta$ is

(Multiple Choice)

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Question 73

Let $x ( t ) = t + \sin t , y ( t ) = \cos t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

(Multiple Choice)

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Question 74

Find the points on the curve $x ( t ) = t ^ { 3 } - 9 t , y ( t ) = t ^ { 2 }$ where the tangent line is vertical.

(Multiple Choice)

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Question 75

Let $x ( t ) = \sin t , y ( t ) = \cos t - t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

(Multiple Choice)

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Question 76

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

(Multiple Choice)

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Question 77

Let $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

(Multiple Choice)

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Question 78

The area of the surface generated by revolving the curve $x ( t ) = 8 \sin t , y = 2 t - \sin ( 2 t )$ with $t \in \left[ 0 , \frac { \pi } { 2 } \right]$ about the y-axis is

(Multiple Choice)

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Question 79

The graph of the parametric equations $x ( t ) = 5 \cos t , y ( t ) = 5 \sin t$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 5 from

(Multiple Choice)

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Question 80

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt [ 3 ] { 3 - t } , y ( t ) = \sqrt [ 3 ] { t }$ with $t \in ( - \infty , \infty ),$ is

The conic section $r = \frac { 5 } { 3 + 8 \sin \theta }$ is

The area of the surface generated by revolving the curve $x ( t ) = t + \sqrt { 2 } , \quad y = \frac { t ^ { 2 } } { 2 } + \sqrt { t }$ with $t \in [ - \sqrt { 2 } , \sqrt { 2 } ]$ about the y-axis is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \tan t , y ( t ) = \sec ^ { 2 } t,$ is

The polar equation that corresponds to the rectangular equation $y = \sqrt { 3 } x$ is

If A is the area of the intersection of the regions enclosed by the graphs of r = 3 sin 2 $\theta$ and r = 3 cos 2 $\theta$ , then A is

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 \cos \theta$ and $r = 2 \sin \theta,$ then A is

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = 1 + e ^ { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The points of intersections of $r = 2 - 2 \cos \theta$ and $r = 2 \cos \theta$ are

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 7 x$ is

The conic section $r = \frac { 3 } { 1 - \sin \theta }$ is

For a $\neq$ 0, the polar curve $r ^ { 2 } = a \cos 2 \theta$ is a

The rectangular equation that corresponds to the polar equation $r = 4 \cos \theta$ is

Let $x ( t ) = t + \sin t , y ( t ) = \cos t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Find the points on the curve $x ( t ) = t ^ { 3 } - 9 t , y ( t ) = t ^ { 2 }$ where the tangent line is vertical.

Let $x ( t ) = \sin t , y ( t ) = \cos t - t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

Let $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The area of the surface generated by revolving the curve $x ( t ) = 8 \sin t , y = 2 t - \sin ( 2 t )$ with $t \in \left[ 0 , \frac { \pi } { 2 } \right]$ about the y-axis is

The graph of the parametric equations $x ( t ) = 5 \cos t , y ( t ) = 5 \sin t$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 5 from

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

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Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

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Exam 10: Parametric Equations; Polar Equations

The rectangular equation of the plane curve, with parametric equations x(t)=3−t3,y(t)=t3x ( t ) = \sqrt [ 3 ] { 3 - t } , y ( t ) = \sqrt [ 3 ] { t }x(t)=33−t​,y(t)=3t​ with t∈(−∞,∞),t \in ( - \infty , \infty ),t∈(−∞,∞), is

The conic section r=53+8sin⁡θr = \frac { 5 } { 3 + 8 \sin \theta }r=3+8sinθ5​ is

The area of the surface generated by revolving the curve x(t)=t+2,y=t22+tx ( t ) = t + \sqrt { 2 } , \quad y = \frac { t ^ { 2 } } { 2 } + \sqrt { t }x(t)=t+2​,y=2t2​+t​ with t∈[−2,2]t \in [ - \sqrt { 2 } , \sqrt { 2 } ]t∈[−2​,2​] about the y-axis is

The rectangular equation of the plane curve, with parametric equations x(t)=tan⁡t,y(t)=sec⁡2t,x ( t ) = \tan t , y ( t ) = \sec ^ { 2 } t,x(t)=tant,y(t)=sec2t, is

The polar equation that corresponds to the rectangular equation y=3xy = \sqrt { 3 } xy=3​x is

If A is the area of the intersection of the regions enclosed by the graphs of r = 3 sin 2 θ\thetaθ and r = 3 cos 2 θ\thetaθ , then A is

If A is the area of the intersection of the regions enclosed by the graphs of r=2cos⁡θr = 2 \cos \thetar=2cosθ and r=2sin⁡θ,r = 2 \sin \theta,r=2sinθ, then A is

Let x(t)=1−et,y(t)=1+etx ( t ) = 1 - e ^ { t } , y ( t ) = 1 + e ^ { t }x(t)=1−et,y(t)=1+et be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The points of intersections of r=2−2cos⁡θr = 2 - 2 \cos \thetar=2−2cosθ and r=2cos⁡θr = 2 \cos \thetar=2cosθ are

The polar equation that corresponds to the rectangular equation x2+y2=7xx ^ { 2 } + y ^ { 2 } = 7 xx2+y2=7x is

The conic section r=31−sin⁡θr = \frac { 3 } { 1 - \sin \theta }r=1−sinθ3​ is

For a ≠\neq= 0, the polar curve r2=acos⁡2θr ^ { 2 } = a \cos 2 \thetar2=acos2θ is a

The rectangular equation that corresponds to the polar equation r=4cos⁡θr = 4 \cos \thetar=4cosθ is

Let x(t)=t+sin⁡t,y(t)=cos⁡tx ( t ) = t + \sin t , y ( t ) = \cos tx(t)=t+sint,y(t)=cost be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

Find the points on the curve x(t)=t3−9t,y(t)=t2x ( t ) = t ^ { 3 } - 9 t , y ( t ) = t ^ { 2 }x(t)=t3−9t,y(t)=t2 where the tangent line is vertical.

Let x(t)=sin⁡t,y(t)=cos⁡t−tx ( t ) = \sin t , y ( t ) = \cos t - tx(t)=sint,y(t)=cost−t be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The area of the surface generated by revolving the curve x(t)=t,y(t)=t2x ( t ) = t , \quad y ( t ) = t ^ { 2 }x(t)=t,y(t)=t2 with t∈[0,2]t \in [ 0,2 ]t∈[0,2] about the y-axis is

Let x(t)=t31+t,y(t)=11+tx ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }x(t)=1+tt3​,y(t)=1+t1​ be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The area of the surface generated by revolving the curve x(t)=8sin⁡t,y=2t−sin⁡(2t)x ( t ) = 8 \sin t , y = 2 t - \sin ( 2 t )x(t)=8sint,y=2t−sin(2t) with t∈[0,π2]t \in \left[ 0 , \frac { \pi } { 2 } \right]t∈[0,2π​] about the y-axis is

The graph of the parametric equations x(t)=5cos⁡t,y(t)=5sin⁡tx ( t ) = 5 \cos t , y ( t ) = 5 \sin tx(t)=5cost,y(t)=5sint with t∈[π,2π]t \in [ \pi , 2 \pi ]t∈[π,2π] is an arc of a circle centered at the origin with radius 5 from

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt [ 3 ] { 3 - t } , y ( t ) = \sqrt [ 3 ] { t }$ with $t \in ( - \infty , \infty ),$ is

The conic section $r = \frac { 5 } { 3 + 8 \sin \theta }$ is

The area of the surface generated by revolving the curve $x ( t ) = t + \sqrt { 2 } , \quad y = \frac { t ^ { 2 } } { 2 } + \sqrt { t }$ with $t \in [ - \sqrt { 2 } , \sqrt { 2 } ]$ about the y-axis is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \tan t , y ( t ) = \sec ^ { 2 } t,$ is

The polar equation that corresponds to the rectangular equation $y = \sqrt { 3 } x$ is

If A is the area of the intersection of the regions enclosed by the graphs of r = 3 sin 2 $\theta$ and r = 3 cos 2 $\theta$ , then A is

If A is the area of the intersection of the regions enclosed by the graphs of $r = 2 \cos \theta$ and $r = 2 \sin \theta,$ then A is

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = 1 + e ^ { t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The points of intersections of $r = 2 - 2 \cos \theta$ and $r = 2 \cos \theta$ are

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 7 x$ is

The conic section $r = \frac { 3 } { 1 - \sin \theta }$ is

For a $\neq$ 0, the polar curve $r ^ { 2 } = a \cos 2 \theta$ is a

The rectangular equation that corresponds to the polar equation $r = 4 \cos \theta$ is

Let $x ( t ) = t + \sin t , y ( t ) = \cos t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Find the points on the curve $x ( t ) = t ^ { 3 } - 9 t , y ( t ) = t ^ { 2 }$ where the tangent line is vertical.

Let $x ( t ) = \sin t , y ( t ) = \cos t - t$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y ( t ) = t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

Let $x ( t ) = \frac { t ^ { 3 } } { 1 + t } , y ( t ) = \frac { 1 } { 1 + t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The area of the surface generated by revolving the curve $x ( t ) = 8 \sin t , y = 2 t - \sin ( 2 t )$ with $t \in \left[ 0 , \frac { \pi } { 2 } \right]$ about the y-axis is

The graph of the parametric equations $x ( t ) = 5 \cos t , y ( t ) = 5 \sin t$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 5 from