Question 1

For a $\neq$ 0, the polar curve $r = a ( 3 - 2 \cos \theta )$ is a

Accepted Answer

A) Circle 
B) Cardioid 
C) Limaçon with inner loop 
D) Dimpled limaçon 
E) Convex limaçon 
A) Circle 
B) Cardioid 
C) Limaçon with inner loop 
D) Dimpled limaçon 
E) Convex limaçon

Question 2

For a > 0, the polar curve $r = - a \cos 2 \theta$ is a

Accepted Answer

A) Circle 
B) Two-petal rose 
C) Four-petal rose 
D) Limaçon 
E) Lemniscate 
A) Circle 
B) Two-petal rose 
C) Four-petal rose 
D) Limaçon 
E) Lemniscate

Question 3

If A is the area of the intersection of the regions inside $r = - 2 \cos \theta$ and outside $r = 1,$ then A is

Accepted Answer

A)  $6 \pi - 16$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { 4 \pi - 3 \sqrt { 3 } } { 6 }$ 
D)  $\pi + 1 - \sqrt { 3 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $6 \pi - 16$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { 4 \pi - 3 \sqrt { 3 } } { 6 }$ 
D)  $\pi + 1 - \sqrt { 3 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 4

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

Accepted Answer

A) A straight line 
B) A circle 
C) A parabola 
D) An ellipse 
E) A hyperbola 
A) A straight line 
B) A circle 
C) A parabola 
D) An ellipse 
E) A hyperbola

Question 5

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

Accepted Answer

A)  $4 \pi ^ { 2 }$ 
B)  $8 \pi ^ { 2 }$ 
C)  $12 \pi ^ { 2 }$ 
D)  $16 \pi ^ { 2 }$ 
E)  $2 \pi ^ { 2 }$ 
A)  $4 \pi ^ { 2 }$ 
B)  $8 \pi ^ { 2 }$ 
C)  $12 \pi ^ { 2 }$ 
D)  $16 \pi ^ { 2 }$ 
E)  $2 \pi ^ { 2 }$

Question 6

The rectangular equation of the conic section $r = \frac { 1 } { 1 + \sin \theta }$ is

Accepted Answer

A)  $y ^ { 2 } = 1 - 2 x$ 
B)  $y ^ { 2 } = 2 x + 1$ 
C)  $x ^ { 2 } = 1 - 2 y$ 
D)  $x ^ { 2 } = 2 y + 1$ 
E)  $y ^ { 2 } = 1 - x$ 
A)  $y ^ { 2 } = 1 - 2 x$ 
B)  $y ^ { 2 } = 2 x + 1$ 
C)  $x ^ { 2 } = 1 - 2 y$ 
D)  $x ^ { 2 } = 2 y + 1$ 
E)  $y ^ { 2 } = 1 - x$

Question 7

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

Accepted Answer

A)  $x - y = 6$ 
B)  $x + y = - 6$ 
C)  $x - y = - 6$ 
D)  $x + y = 6$ 
E)  $x + y = - 3$ 
A)  $x - y = 6$ 
B)  $x + y = - 6$ 
C)  $x - y = - 6$ 
D)  $x + y = 6$ 
E)  $x + y = - 3$

Question 8

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 2 t + 5 , y ( t ) = 4 t - 7,$ is

Accepted Answer

A)  $y = 4 x - 3$ 
B)  $y = 2 x - 7$ 
C)  $y = 2 x - 17$ 
D)  $y = 4 x + 3$ 
E)  $y = 2 x - 2$ 
A)  $y = 4 x - 3$ 
B)  $y = 2 x - 7$ 
C)  $y = 2 x - 17$ 
D)  $y = 4 x + 3$ 
E)  $y = 2 x - 2$

Question 9

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y = 3 t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

Accepted Answer

A)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 60 }$ 
B)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 54 }$ 
C)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 48 }$ 
D)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 42 }$ 
E)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 36 }$ 
A)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 60 }$ 
B)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 54 }$ 
C)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 48 }$ 
D)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 42 }$ 
E)  $\frac { \pi ( 145 \sqrt { 145 } - 1 ) } { 36 }$

Question 10

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sin 3 t , y ( t ) = \cos ^ { 2 } 3 t$ with $t \in \left[ 0 , \frac { \pi } { 6 } \right],$ is

Accepted Answer

A)  $y = 1 - x ^ { 2 }$ 
B)  $y = x ^ { 2 } - 1$ 
C)  $y = x ^ { 2 } + 1$ 
D)  $y = - \sqrt { 1 - x ^ { 2 } }$ 
E)  $y = \sqrt { 1 - x ^ { 2 } }$ 
A)  $y = 1 - x ^ { 2 }$ 
B)  $y = x ^ { 2 } - 1$ 
C)  $y = x ^ { 2 } + 1$ 
D)  $y = - \sqrt { 1 - x ^ { 2 } }$ 
E)  $y = \sqrt { 1 - x ^ { 2 } }$

Question 11

The polar equation that corresponds to the rectangular equation xy = 1 is

Accepted Answer

A)  $r ^ { 2 } = - \tan \theta \sec \theta$ 
B)  $r ^ { 2 } = \tan \theta \sec \theta$ 
C)  $r ^ { 2 } = \csc \theta \sec \theta$ 
D)  $r ^ { 2 } = - \csc \theta \sec \theta$ 
E)  $r ^ { 2 } = \cos \theta$ 
A)  $r ^ { 2 } = - \tan \theta \sec \theta$ 
B)  $r ^ { 2 } = \tan \theta \sec \theta$ 
C)  $r ^ { 2 } = \csc \theta \sec \theta$ 
D)  $r ^ { 2 } = - \csc \theta \sec \theta$ 
E)  $r ^ { 2 } = \cos \theta$

Question 12

If A is the area of the region bounded by $r = 2 + 2 \cos \theta,$ then A is

Accepted Answer

A)  $8 \pi$ 
B)  $6 \pi$ 
C)  $4 \pi$ 
D)  $2 \pi$ 
E)  $\pi$ 
A)  $8 \pi$ 
B)  $6 \pi$ 
C)  $4 \pi$ 
D)  $2 \pi$ 
E)  $\pi$

Question 13

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \csc t , y ( t ) = \cot t,$ is

Accepted Answer

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

Question 14

The rectangular equation of the conic section $r = \frac { 2 } { 1 + 2 \cos \theta }$ is

Accepted Answer

A)  $2 y ^ { 2 } = 1 - 2 x$ 
B)  $x ^ { 2 } - 3 y ^ { 2 } - 8 x = 4$ 
C)  $3 x ^ { 2 } - y ^ { 2 } - 8 y = 4$ 
D)  $x ^ { 2 } + 3 y ^ { 2 } + 8 y = 4$ 
E)  $3 x ^ { 2 } - y ^ { 2 } - 8 x = - 4$ 
A)  $2 y ^ { 2 } = 1 - 2 x$ 
B)  $x ^ { 2 } - 3 y ^ { 2 } - 8 x = 4$ 
C)  $3 x ^ { 2 } - y ^ { 2 } - 8 y = 4$ 
D)  $x ^ { 2 } + 3 y ^ { 2 } + 8 y = 4$ 
E)  $3 x ^ { 2 } - y ^ { 2 } - 8 x = - 4$

Question 15

The polar equation that corresponds to the rectangular equation $2 x + y = 4$ is

Accepted Answer

A)  $r = \frac { 4 } { 2 \cos \theta + \sin \theta }$ 
B)  $r = \frac { 4 } { 2 \cos \theta - \sin \theta }$ 
C)  $r = \frac { 4 } { \cos \theta + 2 \sin \theta }$ 
D)  $r = \frac { 4 } { \cos \theta - 2 \sin \theta }$ 
E)  $r = \frac { - 4 } { 2 \cos \theta + \sin \theta }$ 
A)  $r = \frac { 4 } { 2 \cos \theta + \sin \theta }$ 
B)  $r = \frac { 4 } { 2 \cos \theta - \sin \theta }$ 
C)  $r = \frac { 4 } { \cos \theta + 2 \sin \theta }$ 
D)  $r = \frac { 4 } { \cos \theta - 2 \sin \theta }$ 
E)  $r = \frac { - 4 } { 2 \cos \theta + \sin \theta }$

Question 16

The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

Accepted Answer

A)  $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$ 
B)  $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$ 
C)  $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$ 
D)  $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$ 
E)  $\left( - x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$ 
A)  $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$ 
B)  $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$ 
C)  $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$ 
D)  $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = y$ 
E)  $\left( - x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = x$

Question 17

For a $\neq$ 0, the polar curve $r = a ( 1 - 3 \sin \theta )$ is a

Accepted Answer

A) Circle 
B) Cardioid 
C) Limaçon with inner loop 
D) Dimpled limaçon 
E) Convex limaçon 
A) Circle 
B) Cardioid 
C) Limaçon with inner loop 
D) Dimpled limaçon 
E) Convex limaçon

Question 18

For a$\neq$ 0, the polar curve $r = a \sin 3 \theta$ is a

Accepted Answer

A) Straight line 
B) Circle 
C) Three-petal rose 
D) Cardioid 
E) Limaçon 
A) Straight line 
B) Circle 
C) Three-petal rose 
D) Cardioid 
E) Limaçon

Question 19

Let $x ( t ) = 2 t ^ { 3 } , y ( t ) = 3 t ^ { 2 }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Accepted Answer

A)  $- \frac { 1 } { t }$ 
B)  $\frac { 2 } { t }$ 
C)  $\frac { t ^ { 2 } } { 2 }$ 
D)  $\frac { 1 } { t }$ 
E)  $- \frac { t } { 2 }$ 
A)  $- \frac { 1 } { t }$ 
B)  $\frac { 2 } { t }$ 
C)  $\frac { t ^ { 2 } } { 2 }$ 
D)  $\frac { 1 } { t }$ 
E)  $- \frac { t } { 2 }$

Question 20

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 4 \sin t , y ( t ) = 3 \cos t,$ is

Accepted Answer

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

For a $\neq$ 0, the polar curve $r = a ( 3 - 2 \cos \theta )$ is a

For a > 0, the polar curve $r = - a \cos 2 \theta$ is a

If A is the area of the intersection of the regions inside $r = - 2 \cos \theta$ and outside $r = 1,$ then A is

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

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

The rectangular equation of the conic section $r = \frac { 1 } { 1 + \sin \theta }$ is

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

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 2 t + 5 , y ( t ) = 4 t - 7,$ is

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y = 3 t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sin 3 t , y ( t ) = \cos ^ { 2 } 3 t$ with $t \in \left[ 0 , \frac { \pi } { 6 } \right],$ is

The polar equation that corresponds to the rectangular equation xy = 1 is

If A is the area of the region bounded by $r = 2 + 2 \cos \theta,$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \csc t , y ( t ) = \cot t,$ is

The rectangular equation of the conic section $r = \frac { 2 } { 1 + 2 \cos \theta }$ is

The polar equation that corresponds to the rectangular equation $2 x + y = 4$ is

The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

For a $\neq$ 0, the polar curve $r = a ( 1 - 3 \sin \theta )$ is a

For a $\neq$ 0, the polar curve $r = a \sin 3 \theta$ is a

Let $x ( t ) = 2 t ^ { 3 } , y ( t ) = 3 t ^ { 2 }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 4 \sin t , y ( t ) = 3 \cos t,$ is

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

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Directional Derivatives, Gradients, and Extrema

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Exam 10: Parametric Equations; Polar Equations

For a ≠\neq= 0, the polar curve r=a(3−2cos⁡θ)r = a ( 3 - 2 \cos \theta )r=a(3−2cosθ) is a

For a > 0, the polar curve r=−acos⁡2θr = - a \cos 2 \thetar=−acos2θ is a

If A is the area of the intersection of the regions inside r=−2cos⁡θr = - 2 \cos \thetar=−2cosθ and outside r=1,r = 1,r=1, then A is

The conic section r=11+sin⁡θr = \frac { 1 } { 1 + \sin \theta }r=1+sinθ1​ is

The area of the surface generated by revolving the curve x(t)=cos⁡t,y(t)=2+sin⁡tx ( t ) = \cos t , \quad y ( t ) = 2 + \sin tx(t)=cost,y(t)=2+sint with t∈[0,2π]t \in [ 0,2 \pi ]t∈[0,2π] about the x-axis is

The rectangular equation of the conic section r=11+sin⁡θr = \frac { 1 } { 1 + \sin \theta }r=1+sinθ1​ is

The rectangular equation of the plane curve, with parametric equations x(t)=3−t,y(t)=t+3x ( t ) = 3 - t , y ( t ) = t + 3x(t)=3−t,y(t)=t+3 with t∈(−∞,∞),t \in ( - \infty , \infty ),t∈(−∞,∞), is

The rectangular equation of the plane curve, with parametric equations x(t)=2t+5,y(t)=4t−7,x ( t ) = 2 t + 5 , y ( t ) = 4 t - 7,x(t)=2t+5,y(t)=4t−7, is

The area of the surface generated by revolving the curve x(t)=t,y=3t2x ( t ) = t , \quad y = 3 t ^ { 2 }x(t)=t,y=3t2 with t∈[0,2]t \in [ 0,2 ]t∈[0,2] about the y-axis is

The rectangular equation of the plane curve, with parametric equations x(t)=sin⁡3t,y(t)=cos⁡23tx ( t ) = \sin 3 t , y ( t ) = \cos ^ { 2 } 3 tx(t)=sin3t,y(t)=cos23t with t∈[0,π6],t \in \left[ 0 , \frac { \pi } { 6 } \right],t∈[0,6π​], is

The polar equation that corresponds to the rectangular equation xy = 1 is

If A is the area of the region bounded by r=2+2cos⁡θ,r = 2 + 2 \cos \theta,r=2+2cosθ, then A is

The rectangular equation of the plane curve, with parametric equations x(t)=csc⁡t,y(t)=cot⁡t,x ( t ) = \csc t , y ( t ) = \cot t,x(t)=csct,y(t)=cott, is

The rectangular equation of the conic section r=21+2cos⁡θr = \frac { 2 } { 1 + 2 \cos \theta }r=1+2cosθ2​ is

The polar equation that corresponds to the rectangular equation 2x+y=42 x + y = 42x+y=4 is

The rectangular equation that corresponds to the polar equation r2=cos⁡θr ^ { 2 } = \cos \thetar2=cosθ is

For a ≠\neq= 0, the polar curve r=a(1−3sin⁡θ)r = a ( 1 - 3 \sin \theta )r=a(1−3sinθ) is a

For a ≠\neq= 0, the polar curve r=asin⁡3θr = a \sin 3 \thetar=asin3θ is a

Let x(t)=2t3,y(t)=3t2x ( t ) = 2 t ^ { 3 } , y ( t ) = 3 t ^ { 2 }x(t)=2t3,y(t)=3t2 be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The rectangular equation of the plane curve, with parametric equations x(t)=4sin⁡t,y(t)=3cos⁡t,x ( t ) = 4 \sin t , y ( t ) = 3 \cos t,x(t)=4sint,y(t)=3cost, is

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

For a $\neq$ 0, the polar curve $r = a ( 3 - 2 \cos \theta )$ is a

For a > 0, the polar curve $r = - a \cos 2 \theta$ is a

If A is the area of the intersection of the regions inside $r = - 2 \cos \theta$ and outside $r = 1,$ then A is

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

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

The rectangular equation of the conic section $r = \frac { 1 } { 1 + \sin \theta }$ is

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

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 2 t + 5 , y ( t ) = 4 t - 7,$ is

The area of the surface generated by revolving the curve $x ( t ) = t , \quad y = 3 t ^ { 2 }$ with $t \in [ 0,2 ]$ about the y-axis is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sin 3 t , y ( t ) = \cos ^ { 2 } 3 t$ with $t \in \left[ 0 , \frac { \pi } { 6 } \right],$ is

If A is the area of the region bounded by $r = 2 + 2 \cos \theta,$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \csc t , y ( t ) = \cot t,$ is

The rectangular equation of the conic section $r = \frac { 2 } { 1 + 2 \cos \theta }$ is

The polar equation that corresponds to the rectangular equation $2 x + y = 4$ is

The rectangular equation that corresponds to the polar equation $r ^ { 2 } = \cos \theta$ is

For a $\neq$ 0, the polar curve $r = a ( 1 - 3 \sin \theta )$ is a

For a $\neq$ 0, the polar curve $r = a \sin 3 \theta$ is a

Let $x ( t ) = 2 t ^ { 3 } , y ( t ) = 3 t ^ { 2 }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = 4 \sin t , y ( t ) = 3 \cos t,$ is