Question 1

The polar equation that corresponds to the rectangular equation $x = - 2$ is

Accepted Answer

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

Question 2

Find the points on the curve $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ where the tangent line is horizontal.

Accepted Answer

A)  $( 0,0 )$ 
B)  $( - 3,3 )$ 
C)  $( 3,0 )$ 
D)  $( 0,3 )$ 
E)  $( 0 , - 3 )$ 
A)  $( 0,0 )$ 
B)  $( - 3,3 )$ 
C)  $( 3,0 )$ 
D)  $( 0,3 )$ 
E)  $( 0 , - 3 )$ D

Question 3

If A is the area of the intersection of the regions inside $r ^ { 2 } = 4 \sin 2 \theta$ and outside $r = \sqrt { 2 },$ then A is

Accepted Answer

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

Question 4

If the eccentricity of a conic section is less 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 5

The polar equation that corresponds to the rectangular equation x + y = 0 is

Accepted Answer

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

Question 6

The graph of the parametric equations $x ( t ) = 4 \sin t , y ( t ) = 4 \cos t,$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 4 from

Accepted Answer

A)  $( 0,4 ) \text { to } ( 0 , - 4 )$ 
B)  $( 0 , - 4 ) \text { to } ( 0,4 )$ 
C)  $( 0 , - 4 ) \text { to } ( 4,0 )$ 
D)  $( 4,0 ) \text { to } ( 0 , - 4 )$ 
E)  $( 0,4 ) \text { to } ( - 4,0 )$ 
A)  $( 0,4 ) \text { to } ( 0 , - 4 )$ 
B)  $( 0 , - 4 ) \text { to } ( 0,4 )$ 
C)  $( 0 , - 4 ) \text { to } ( 4,0 )$ 
D)  $( 4,0 ) \text { to } ( 0 , - 4 )$ 
E)  $( 0,4 ) \text { to } ( - 4,0 )$

Question 7

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

Accepted Answer

A)  $\pi \ln \left( \frac { \pi + 4 } { 4 } \right)$ 
B)  $6 \pi \ln \left( \frac { \pi - 2 } { 4 } \right)$ 
C)  $6 \pi \ln \left( \frac { \pi + 2 } { 4 } \right)$ 
D)  $6 \pi \ln \left( \frac { \pi + 2 } { 2 } \right)$ 
E)  $6 \pi \ln \left( \frac { \pi - 2 } { 2 } \right)$ 
A)  $\pi \ln \left( \frac { \pi + 4 } { 4 } \right)$ 
B)  $6 \pi \ln \left( \frac { \pi - 2 } { 4 } \right)$ 
C)  $6 \pi \ln \left( \frac { \pi + 2 } { 4 } \right)$ 
D)  $6 \pi \ln \left( \frac { \pi + 2 } { 2 } \right)$ 
E)  $6 \pi \ln \left( \frac { \pi - 2 } { 2 } \right)$

Question 8

The rectangular equation that corresponds to the polar equation $r = 3 \csc \theta$ is

Accepted Answer

A) y = 3 
B) x = 3 
C) y = x 
D) y = 3x 
E) y = -3 
A) y = 3 
B) x = 3 
C) y = x 
D) y = 3x 
E) y = -3

Question 9

The area of the surface generated by revolving the curve $x ( t ) = 4 t , \quad y = t ^ { 2 } - 2 \ln t$ with $t \in [ 0,1 ]$ about the y-axis is

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 3 }$ 
C)  $\frac { 8 \pi } { 3 }$ 
D)  $\frac { 16 \pi } { 3 }$ 
E)  $\frac { 64 \pi } { 3 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 3 }$ 
C)  $\frac { 8 \pi } { 3 }$ 
D)  $\frac { 16 \pi } { 3 }$ 
E)  $\frac { 64 \pi } { 3 }$

Question 10

The polar equation that corresponds to the rectangular equation $x - y = 0$ is

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 5$ is

Accepted Answer

A)  $r = 5$ 
B)  $r = 25$ 
C)  $r = \sqrt { 5 }$ 
D)  $r = \sqrt { 5 } \cos \theta$ 
E)  $r = 5 \cos \theta$ 
A)  $r = 5$ 
B)  $r = 25$ 
C)  $r = \sqrt { 5 }$ 
D)  $r = \sqrt { 5 } \cos \theta$ 
E)  $r = 5 \cos \theta$

Question 13

The length of $r = 1 + \sin \theta$ with $x \in \left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]$ is

Accepted Answer

A) 4 
B) 6 
C) 9 
D) 12 
E) 15 
A) 4 
B) 6 
C) 9 
D) 12 
E) 15

Question 14

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

Accepted Answer

A)  $- \frac { \cos t - \sin t } { \cos t + \sin t }$ 
B)  $\frac { \cos t + \sin t } { \cos t - \sin t }$ 
C)  $- \frac { \cos t + \sin t } { \cos t - \sin t }$ 
D)  $\frac { \cos t - \sin t } { \cos t + \sin t }$ 
E)  $\frac { \cos t - \sin t } { 2 ( \cos t + \sin t ) }$ 
A)  $- \frac { \cos t - \sin t } { \cos t + \sin t }$ 
B)  $\frac { \cos t + \sin t } { \cos t - \sin t }$ 
C)  $- \frac { \cos t + \sin t } { \cos t - \sin t }$ 
D)  $\frac { \cos t - \sin t } { \cos t + \sin t }$ 
E)  $\frac { \cos t - \sin t } { 2 ( \cos t + \sin t ) }$

Question 15

The rectangular equation of the conic section $r = \frac { 3 } { 4 - 5 \cos \theta }$ is

Accepted Answer

A)  $9 x ^ { 2 } - 16 y ^ { 2 } + 30 x = - 9$ 
B)  $9 x ^ { 2 } + 16 y ^ { 2 } + 30 y = 9$ 
C)  $9 x ^ { 2 } - 16 y ^ { 2 } - 30 x = 9$ 
D)  $9 x ^ { 2 } - 16 y ^ { 2 } + 30 y = 9$ 
E)  $16 x ^ { 2 } - 20 y = 9$ 
A)  $9 x ^ { 2 } - 16 y ^ { 2 } + 30 x = - 9$ 
B)  $9 x ^ { 2 } + 16 y ^ { 2 } + 30 y = 9$ 
C)  $9 x ^ { 2 } - 16 y ^ { 2 } - 30 x = 9$ 
D)  $9 x ^ { 2 } - 16 y ^ { 2 } + 30 y = 9$ 
E)  $16 x ^ { 2 } - 20 y = 9$

Question 16

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

Accepted Answer

A)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$ 
B)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$ 
C)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$ 
D)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$ 
E)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$ 
A)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 3 }$ 
B)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 18 }$ 
C)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 36 }$ 
D)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 9 }$ 
E)  $\frac { 8 ( 10 \sqrt { 10 } - 1 ) } { 27 }$

Question 17

For a > 0, the polar curve $r = - a \sin 2 \theta$ 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 18

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = e ^ { 2 t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

Accepted Answer

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

Question 19

The length of the curve $x ( t ) = 4 - 3 \cos t , y ( t ) = 3 + 3 \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

Accepted Answer

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

Question 20

For a $\neq$ 0, the polar curve $r = a ( 5 + 2 \sin \theta )$ 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

The polar equation that corresponds to the rectangular equation $x = - 2$ is

Find the points on the curve $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ where the tangent line is horizontal.

If A is the area of the intersection of the regions inside $r ^ { 2 } = 4 \sin 2 \theta$ and outside $r = \sqrt { 2 },$ then A is

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

The polar equation that corresponds to the rectangular equation x + y = 0 is

The graph of the parametric equations $x ( t ) = 4 \sin t , y ( t ) = 4 \cos t,$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 4 from

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

The rectangular equation that corresponds to the polar equation $r = 3 \csc \theta$ is

The area of the surface generated by revolving the curve $x ( t ) = 4 t , \quad y = t ^ { 2 } - 2 \ln t$ with $t \in [ 0,1 ]$ about the y-axis is

The polar equation that corresponds to the rectangular equation $x - y = 0$ is

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

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 5$ is

The length of $r = 1 + \sin \theta$ with $x \in \left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]$ is

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

The rectangular equation of the conic section $r = \frac { 3 } { 4 - 5 \cos \theta }$ is

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

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

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = e ^ { 2 t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The length of the curve $x ( t ) = 4 - 3 \cos t , y ( t ) = 3 + 3 \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

For a $\neq$ 0, the polar curve $r = a ( 5 + 2 \sin \theta )$ is a

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Exam 10: Parametric Equations; Polar Equations

The polar equation that corresponds to the rectangular equation x=−2x = - 2x=−2 is

Find the points on the curve x(t)=4t,y(t)=t2+3x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3x(t)=4t,y(t)=t2+3 where the tangent line is horizontal.

If A is the area of the intersection of the regions inside r2=4sin⁡2θr ^ { 2 } = 4 \sin 2 \thetar2=4sin2θ and outside r=2,r = \sqrt { 2 },r=2​, then A is

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

The polar equation that corresponds to the rectangular equation x + y = 0 is

The graph of the parametric equations x(t)=4sin⁡t,y(t)=4cos⁡t,x ( t ) = 4 \sin t , y ( t ) = 4 \cos t,x(t)=4sint,y(t)=4cost, with t∈[π,2π]t \in [ \pi , 2 \pi ]t∈[π,2π] is an arc of a circle centered at the origin with radius 4 from

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 x-axis is

The rectangular equation that corresponds to the polar equation r=3csc⁡θr = 3 \csc \thetar=3cscθ is

The area of the surface generated by revolving the curve x(t)=4t,y=t2−2ln⁡tx ( t ) = 4 t , \quad y = t ^ { 2 } - 2 \ln tx(t)=4t,y=t2−2lnt with t∈[0,1]t \in [ 0,1 ]t∈[0,1] about the y-axis is

The polar equation that corresponds to the rectangular equation x−y=0x - y = 0x−y=0 is

If A is the area of the intersection of the regions enclosed by the graphs of r=1r = 1r=1 and r=1−cos⁡θ,r = 1 - \cos \theta,r=1−cosθ, then A is

The polar equation that corresponds to the rectangular equation x2+y2=5x ^ { 2 } + y ^ { 2 } = 5x2+y2=5 is

The length of r=1+sin⁡θr = 1 + \sin \thetar=1+sinθ with x∈[π2,3π2]x \in \left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]x∈[2π​,23π​] is

Let x(t)=etsin⁡t,y(t)=etcos⁡tx ( t ) = e ^ { t } \sin t , y ( t ) = e ^ { t } \cos tx(t)=etsint,y(t)=etcost be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The rectangular equation of the conic section r=34−5cos⁡θr = \frac { 3 } { 4 - 5 \cos \theta }r=4−5cosθ3​ is

The length of the curve x(t)=t3,y(t)=t2x ( t ) = t ^ { 3 } , y ( t ) = t ^ { 2 }x(t)=t3,y(t)=t2 with t∈[0,2]t \in [ 0,2 ]t∈[0,2] is

For a > 0, the polar curve r=−asin⁡2θr = - a \sin 2 \thetar=−asin2θ is a

Let x(t)=1−et,y(t)=e2tx ( t ) = 1 - e ^ { t } , y ( t ) = e ^ { 2 t }x(t)=1−et,y(t)=e2t be the parametric equations of a curve. Then dydx\frac { d y } { d x }dxdy​ is

The length of the curve x(t)=4−3cos⁡t,y(t)=3+3sin⁡tx ( t ) = 4 - 3 \cos t , y ( t ) = 3 + 3 \sin tx(t)=4−3cost,y(t)=3+3sint with t∈[−π2,π2]t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]t∈[−2π​,2π​] is

For a ≠\neq= 0, the polar curve r=a(5+2sin⁡θ)r = a ( 5 + 2 \sin \theta )r=a(5+2sinθ) is a

Preparing for Calculus

Limits and Continuity

The Derivative

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Applications of the Integral

Techniques of Integration

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Vector Functions

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Directional Derivatives, Gradients, and Extrema

Multiple Integrals

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Differential Equations

Filters

The polar equation that corresponds to the rectangular equation $x = - 2$ is

Find the points on the curve $x ( t ) = 4 t , y ( t ) = t ^ { 2 } + 3$ where the tangent line is horizontal.

If A is the area of the intersection of the regions inside $r ^ { 2 } = 4 \sin 2 \theta$ and outside $r = \sqrt { 2 },$ then A is

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

The graph of the parametric equations $x ( t ) = 4 \sin t , y ( t ) = 4 \cos t,$ with $t \in [ \pi , 2 \pi ]$ is an arc of a circle centered at the origin with radius 4 from

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

The rectangular equation that corresponds to the polar equation $r = 3 \csc \theta$ is

The area of the surface generated by revolving the curve $x ( t ) = 4 t , \quad y = t ^ { 2 } - 2 \ln t$ with $t \in [ 0,1 ]$ about the y-axis is

The polar equation that corresponds to the rectangular equation $x - y = 0$ is

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

The polar equation that corresponds to the rectangular equation $x ^ { 2 } + y ^ { 2 } = 5$ is

The length of $r = 1 + \sin \theta$ with $x \in \left[ \frac { \pi } { 2 } , \frac { 3 \pi } { 2 } \right]$ is

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

The rectangular equation of the conic section $r = \frac { 3 } { 4 - 5 \cos \theta }$ is

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

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

Let $x ( t ) = 1 - e ^ { t } , y ( t ) = e ^ { 2 t }$ be the parametric equations of a curve. Then $\frac { d y } { d x }$ is

The length of the curve $x ( t ) = 4 - 3 \cos t , y ( t ) = 3 + 3 \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

For a $\neq$ 0, the polar curve $r = a ( 5 + 2 \sin \theta )$ is a