Question 1

The length of the curve $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \frac { t ^ { 3 } } { 2 } + 3$ with $t \in [ 0,2 ]$ is

Accepted Answer

A)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 3 }$ 
B)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 9 }$ 
C)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 27 }$ 
D)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 36 }$ 
E)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 18 }$ 
A)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 3 }$ 
B)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 9 }$ 
C)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 27 }$ 
D)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 36 }$ 
E)  $\frac { 4 ( 13 \sqrt { 13 } - 8 ) } { 18 }$

Question 2

The rectangular equation that corresponds to the polar equation $r = \frac { 2 } { 1 - \cos \theta }$ is

Accepted Answer

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

Question 3

If the eccentricity of a conic section is 1, it must be
​

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 4

The rectangular equation that corresponds to the polar equation $r = 5 \sec \theta$ is

Accepted Answer

A)  $y = 5$ 
B)  $y = 5 x$ 
C)  $x = 5$ 
D)  $y = - 5 x$ 
E)  $5 y = x$ 
A)  $y = 5$ 
B)  $y = 5 x$ 
C)  $x = 5$ 
D)  $y = - 5 x$ 
E)  $5 y = x$

Question 5

The area of the surface generated by revolving the curve with $x ( t ) = 4 - 3 \cos t,$ about the y-axis is

Accepted Answer

A)  $6 \pi ( 2 \pi - 3 )$ 
B)  $6 \pi ( 2 \pi + 3 )$ 
C)  $8 \pi ( 2 \pi - 3 )$ 
D)  $8 \pi ( 2 \pi + 3 )$ 
E)  $9 \pi ( 2 \pi - 3 )$ 
A)  $6 \pi ( 2 \pi - 3 )$ 
B)  $6 \pi ( 2 \pi + 3 )$ 
C)  $8 \pi ( 2 \pi - 3 )$ 
D)  $8 \pi ( 2 \pi + 3 )$ 
E)  $9 \pi ( 2 \pi - 3 )$

Question 6

Let $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Accepted Answer

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

Question 7

The rectangular equation that corresponds to the polar equation $\theta = \frac { \pi } { 4 }$ is

Accepted Answer

A)  $y = 4 x$ 
B)  $4 y = x$ 
C)  $y = - 4 x$ 
D)  $y = x$ 
E)  $y = - x$ 
A)  $y = 4 x$ 
B)  $4 y = x$ 
C)  $y = - 4 x$ 
D)  $y = x$ 
E)  $y = - x$

Question 8

Which one of the following sets of parametric equations does not correspond to the rectangular equation $y = 4 x - 2 ?$

Accepted Answer

A)  $x ( t ) = t , y ( t ) = 4 t - 2$ 
B)  $x ( t ) = \frac { t } { 4 } + \frac { 1 } { 2 } , y ( t ) = t$ 
C)  $x ( t ) = \frac { t + 1 } { 2 } , y ( t ) = 2 t$ 
D)  $x ( t ) = 2 t , y ( t ) = 8 t - 2$ 
E)  $x ( t ) = 2 t , y ( t ) = 8 t - 4$ 
A)  $x ( t ) = t , y ( t ) = 4 t - 2$ 
B)  $x ( t ) = \frac { t } { 4 } + \frac { 1 } { 2 } , y ( t ) = t$ 
C)  $x ( t ) = \frac { t + 1 } { 2 } , y ( t ) = 2 t$ 
D)  $x ( t ) = 2 t , y ( t ) = 8 t - 2$ 
E)  $x ( t ) = 2 t , y ( t ) = 8 t - 4$

Question 9

For a $\neq$ 0, the polar curve $r = a \sin \theta$ 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 10

For a $\neq$ 0, the polar curve $r = a ( 1 + \cos \theta )$ 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 11

If A is the area of the intersection of the regions enclosed by the graphs of $r = 1 + \cos \theta$ and $r = 2 \cos \theta,$ then A is

Accepted Answer

A)  $\frac { 9 \pi - 18 } { 2 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi + 1 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $\frac { 9 \pi - 18 } { 2 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi + 1 } { 2 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 12

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt { t } , y ( t ) = 3 t ^ { \frac { 5 } { 2 } }$ is

Accepted Answer

A)  $x = 3 y ^ { 3 }$ 
B)  $y = 3 y ^ { 3 }$ 
C)  $x = 3 y ^ { 5 }$ 
D)  $y = 3 x ^ { 5 }$ 
E)  $y = - 3 x ^ { 5 }$ 
A)  $x = 3 y ^ { 3 }$ 
B)  $y = 3 y ^ { 3 }$ 
C)  $x = 3 y ^ { 5 }$ 
D)  $y = 3 x ^ { 5 }$ 
E)  $y = - 3 x ^ { 5 }$

Question 13

The conic section $r = \frac { 8 } { 5 - 5 \sin \theta }$ 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 14

The rectangular equation of the conic section $r = \frac { 2 } { 6 + 2 \sin \theta }$ is

Accepted Answer

A)  $3 y ^ { 2 } = 1 - 2 x$ 
B)  $36 x ^ { 2 } + 32 y ^ { 2 } - 4 y = 1$ 
C)  $36 x ^ { 2 } + 32 y ^ { 2 } + 4 x = 1$ 
D)  $36 x ^ { 2 } + 32 y ^ { 2 } + 8 y = 4$ 
E)  $36 x ^ { 2 } + 32 y ^ { 2 } - 4 x = 1$ 
A)  $3 y ^ { 2 } = 1 - 2 x$ 
B)  $36 x ^ { 2 } + 32 y ^ { 2 } - 4 y = 1$ 
C)  $36 x ^ { 2 } + 32 y ^ { 2 } + 4 x = 1$ 
D)  $36 x ^ { 2 } + 32 y ^ { 2 } + 8 y = 4$ 
E)  $36 x ^ { 2 } + 32 y ^ { 2 } - 4 x = 1$

Question 15

If A is the area of the intersection of the regions inside $r = 6$ and above $r = 3 \csc \theta,$ then A is

Accepted Answer

A)  $12 \pi - 9 \sqrt { 3 }$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { 2 \pi + 3 \sqrt { 3 } } { 6 }$ 
D)  $\pi + 1 - \sqrt { 3 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $12 \pi - 9 \sqrt { 3 }$ 
B)  $3 \sqrt { 3 } - \pi$ 
C)  $\frac { 2 \pi + 3 \sqrt { 3 } } { 6 }$ 
D)  $\pi + 1 - \sqrt { 3 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 16

The rectangular equation that corresponds to the polar equation $r = 1 + \sin \theta$ is

Accepted Answer

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

Question 17

The parametric equations of the line segment from (-3,0) to (0,-3) with t$\in$[0,3] are

Accepted Answer

A)  $x ( t ) = t - 3 , y ( t ) = t - 3$ 
B)  $x ( t ) = 3 - t , y ( t ) = t$ 
C)  $x ( t ) = 3 - t , y ( t ) = - t$ 
D)  $x ( t ) = t - 3 , y ( t ) = - t$ 
E)  $x ( t ) = t - 3 , y ( t ) = 3 - t$ 
A)  $x ( t ) = t - 3 , y ( t ) = t - 3$ 
B)  $x ( t ) = 3 - t , y ( t ) = t$ 
C)  $x ( t ) = 3 - t , y ( t ) = - t$ 
D)  $x ( t ) = t - 3 , y ( t ) = - t$ 
E)  $x ( t ) = t - 3 , y ( t ) = 3 - t$

Question 18

If the eccentricity of a conic section is greater than 1, it must be
​

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 19

If A is the area of the region bounded by $r = 1 - \sin \theta,$ then A is

Accepted Answer

A)  $\frac { 3 \pi } { 4 }$ 
B)  $\frac { 5 \pi } { 4 }$ 
C)  $\frac { 3 \pi + 8 } { 2 }$ 
D)  $\frac { 3 \pi - 8 } { 2 }$ 
E)  $\frac { 3 \pi } { 2 }$ 
A)  $\frac { 3 \pi } { 4 }$ 
B)  $\frac { 5 \pi } { 4 }$ 
C)  $\frac { 3 \pi + 8 } { 2 }$ 
D)  $\frac { 3 \pi - 8 } { 2 }$ 
E)  $\frac { 3 \pi } { 2 }$

Question 20

The area of the surface generated by revolving the curve $x ( t ) = 3 t ^ { 2 } , y = 2 t ^ { 3 }$ with $t \in [ 0,1 ]$ about the y-axis is

Accepted Answer

A)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 7 }$ 
B)  $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 3 }$ 
C)  $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 5 }$ 
D)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 5 }$ 
E)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 3 }$ 
A)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 7 }$ 
B)  $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 3 }$ 
C)  $\frac { 24 \pi ( \sqrt { 2 } - 1 ) } { 5 }$ 
D)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 5 }$ 
E)  $\frac { 24 \pi ( 1 + \sqrt { 2 } ) } { 3 }$

The length of the curve $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \frac { t ^ { 3 } } { 2 } + 3$ with $t \in [ 0,2 ]$ is

The rectangular equation that corresponds to the polar equation $r = \frac { 2 } { 1 - \cos \theta }$ is

If the eccentricity of a conic section is 1, it must be

The rectangular equation that corresponds to the polar equation $r = 5 \sec \theta$ is

The area of the surface generated by revolving the curve with $x ( t ) = 4 - 3 \cos t,$ about the y-axis is

Let $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The rectangular equation that corresponds to the polar equation $\theta = \frac { \pi } { 4 }$ is

Which one of the following sets of parametric equations does not correspond to the rectangular equation $y = 4 x - 2 ?$

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

For a $\neq$ 0, the polar curve $r = a ( 1 + \cos \theta )$ is a

If A is the area of the intersection of the regions enclosed by the graphs of $r = 1 + \cos \theta$ and $r = 2 \cos \theta,$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt { t } , y ( t ) = 3 t ^ { \frac { 5 } { 2 } }$ is

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

The rectangular equation of the conic section $r = \frac { 2 } { 6 + 2 \sin \theta }$ is

If A is the area of the intersection of the regions inside $r = 6$ and above $r = 3 \csc \theta,$ then A is

The rectangular equation that corresponds to the polar equation $r = 1 + \sin \theta$ is

The parametric equations of the line segment from (-3,0) to (0,-3) with t $\in$ [0,3] are

If the eccentricity of a conic section is greater than 1, it must be

If A is the area of the region bounded by $r = 1 - \sin \theta,$ then A is

The area of the surface generated by revolving the curve $x ( t ) = 3 t ^ { 2 } , y = 2 t ^ { 3 }$ with $t \in [ 0,1 ]$ about the y-axis 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

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Filters

Exam 10: Parametric Equations; Polar Equations

The length of the curve x(t)=t2−2,y(t)=t32+3x ( t ) = t ^ { 2 } - 2 , y ( t ) = \frac { t ^ { 3 } } { 2 } + 3x(t)=t2−2,y(t)=2t3​+3 with t∈[0,2]t \in [ 0,2 ]t∈[0,2] is

The rectangular equation that corresponds to the polar equation r=21−cos⁡θr = \frac { 2 } { 1 - \cos \theta }r=1−cosθ2​ is

If the eccentricity of a conic section is 1, it must be ​

The rectangular equation that corresponds to the polar equation r=5sec⁡θr = 5 \sec \thetar=5secθ is

The area of the surface generated by revolving the curve with x(t)=4−3cos⁡t,x ( t ) = 4 - 3 \cos t,x(t)=4−3cost, about the y-axis is

Let x(t)=4t,y(t)=t2+3x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3x(t)=4t,y(t)=t2+3 be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The rectangular equation that corresponds to the polar equation θ=π4\theta = \frac { \pi } { 4 }θ=4π​ is

Which one of the following sets of parametric equations does not correspond to the rectangular equation y=4x−2?y = 4 x - 2 ?y=4x−2?

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

For a ≠\neq= 0, the polar curve r=a(1+cos⁡θ)r = a ( 1 + \cos \theta )r=a(1+cosθ) is a

If A is the area of the intersection of the regions enclosed by the graphs of r=1+cos⁡θr = 1 + \cos \thetar=1+cosθ and r=2cos⁡θ,r = 2 \cos \theta,r=2cosθ, then A is

The rectangular equation of the plane curve, with parametric equations x(t)=t,y(t)=3t52x ( t ) = \sqrt { t } , y ( t ) = 3 t ^ { \frac { 5 } { 2 } }x(t)=t​,y(t)=3t25​ is

The conic section r=85−5sin⁡θr = \frac { 8 } { 5 - 5 \sin \theta }r=5−5sinθ8​ is

The rectangular equation of the conic section r=26+2sin⁡θr = \frac { 2 } { 6 + 2 \sin \theta }r=6+2sinθ2​ is

If A is the area of the intersection of the regions inside r=6r = 6r=6 and above r=3csc⁡θ,r = 3 \csc \theta,r=3cscθ, then A is

The rectangular equation that corresponds to the polar equation r=1+sin⁡θr = 1 + \sin \thetar=1+sinθ is

The parametric equations of the line segment from (-3,0) to (0,-3) with t ∈\in∈ [0,3] are

If the eccentricity of a conic section is greater than 1, it must be ​

If A is the area of the region bounded by r=1−sin⁡θ,r = 1 - \sin \theta,r=1−sinθ, then A is

The area of the surface generated by revolving the curve x(t)=3t2,y=2t3x ( t ) = 3 t ^ { 2 } , y = 2 t ^ { 3 }x(t)=3t2,y=2t3 with t∈[0,1]t \in [ 0,1 ]t∈[0,1] about the y-axis 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

The length of the curve $x ( t ) = t ^ { 2 } - 2 , y ( t ) = \frac { t ^ { 3 } } { 2 } + 3$ with $t \in [ 0,2 ]$ is

The rectangular equation that corresponds to the polar equation $r = \frac { 2 } { 1 - \cos \theta }$ is

If the eccentricity of a conic section is 1, it must be

The rectangular equation that corresponds to the polar equation $r = 5 \sec \theta$ is

The area of the surface generated by revolving the curve with $x ( t ) = 4 - 3 \cos t,$ about the y-axis is

Let $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The rectangular equation that corresponds to the polar equation $\theta = \frac { \pi } { 4 }$ is

Which one of the following sets of parametric equations does not correspond to the rectangular equation $y = 4 x - 2 ?$

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

For a $\neq$ 0, the polar curve $r = a ( 1 + \cos \theta )$ is a

If A is the area of the intersection of the regions enclosed by the graphs of $r = 1 + \cos \theta$ and $r = 2 \cos \theta,$ then A is

The rectangular equation of the plane curve, with parametric equations $x ( t ) = \sqrt { t } , y ( t ) = 3 t ^ { \frac { 5 } { 2 } }$ is

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

The rectangular equation of the conic section $r = \frac { 2 } { 6 + 2 \sin \theta }$ is

If A is the area of the intersection of the regions inside $r = 6$ and above $r = 3 \csc \theta,$ then A is

The rectangular equation that corresponds to the polar equation $r = 1 + \sin \theta$ is

The parametric equations of the line segment from (-3,0) to (0,-3) with t $\in$ [0,3] are

If the eccentricity of a conic section is greater than 1, it must be

If A is the area of the region bounded by $r = 1 - \sin \theta,$ then A is

The area of the surface generated by revolving the curve $x ( t ) = 3 t ^ { 2 } , y = 2 t ^ { 3 }$ with $t \in [ 0,1 ]$ about the y-axis is