Question 1

The parametric equations of the line segment from (0,-4) to (4,0) with t $\in$[0,4] are

Accepted Answer

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

Question 2

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

Accepted Answer

A) (-4,-1) 
B) (-1,4) 
C) (-1,-4) 
D) (1,-4) 
E) (1,4) 
A) (-4,-1) 
B) (-1,4) 
C) (-1,-4) 
D) (1,-4) 
E) (1,4)

Question 3

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

Accepted Answer

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

Question 4

The conic section $r = \frac { 3 } { 4 + 5 \cos \theta }$ 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 length of the curve $x ( t ) = 1 - \cos t , y ( t ) = 1 + \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

Accepted Answer

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

Question 6

If A is the area of the region bounded by one petal of the four-petal rose $r = \sin 3 \theta,$ then A is

Accepted Answer

A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 8 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { \pi } { 2 }$ 
A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 8 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 4 }$ 
E)  $\frac { \pi } { 2 }$

Question 7

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

Accepted Answer

A)  $\left( 1 , - \frac { \pi } { 3 } \right) , \left( 1 , \frac { 5 \pi } { 3 } \right)$ 
B)  $\left( 1 , - \frac { \pi } { 3 } \right) , \left( 1 , \frac { 5 \pi } { 3 } \right) , ( 0,0 )$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } \right) , ( 0,0 )$ 
D)  $\left( 1 , \frac { 5 \pi } { 3 } \right) , ( 0,0 )$ 
E)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)$ 
A)  $\left( 1 , - \frac { \pi } { 3 } \right) , \left( 1 , \frac { 5 \pi } { 3 } \right)$ 
B)  $\left( 1 , - \frac { \pi } { 3 } \right) , \left( 1 , \frac { 5 \pi } { 3 } \right) , ( 0,0 )$ 
C)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } \right) , ( 0,0 )$ 
D)  $\left( 1 , \frac { 5 \pi } { 3 } \right) , ( 0,0 )$ 
E)  $\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)$

Question 8

For a $\neq$ 0, the polar curve $r = a \cos \theta$ 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 9

If A is the area of the region bounded by one loop of $r ^ { 2 } = 4 \sin 2 \theta,$ then A is

Accepted Answer

A) 8 
B) 6 
C) 4 
D) 2 
E) 1 
A) 8 
B) 6 
C) 4 
D) 2 
E) 1

Question 10

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

Accepted Answer

A)  $- \frac { 3 \cot t } { 4 }$ 
B)  $\frac { 3 \cot t } { 4 }$ 
C)  $- \frac { 3 \tan t } { 4 }$ 
D)  $\frac { 3 \tan t } { 4 }$ 
E)  $- \frac { 4 \cot t } { 3 }$ 
A)  $- \frac { 3 \cot t } { 4 }$ 
B)  $\frac { 3 \cot t } { 4 }$ 
C)  $- \frac { 3 \tan t } { 4 }$ 
D)  $\frac { 3 \tan t } { 4 }$ 
E)  $- \frac { 4 \cot t } { 3 }$

Question 11

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } + 1 , y ( t ) = e ^ { 2 t }$ with $t \in ( - \infty , \infty ),$ is

Accepted Answer

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

Question 12

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

Accepted Answer

A)  $\frac { 9 \pi - 18 } { 2 }$ 
B)  $\frac { \pi + 2 } { 3 }$ 
C)  $\frac { \pi - 1 } { 2 }$ 
D)  $\frac { 3 \pi + 1 } { 4 }$ 
E)  $\frac { 8 - \pi } { 4 }$ 
A)  $\frac { 9 \pi - 18 } { 2 }$ 
B)  $\frac { \pi + 2 } { 3 }$ 
C)  $\frac { \pi - 1 } { 2 }$ 
D)  $\frac { 3 \pi + 1 } { 4 }$ 
E)  $\frac { 8 - \pi } { 4 }$

Question 13

The conic section $r = \frac { 5 } { 8 + 3 \sin \theta }$ 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 conic section $r = \frac { 5 } { 6 + 2 \sin \theta }$ 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 15

The polar equation that corresponds to the rectangular equation $x ^ { 2 } - y ^ { 2 } = 4$ is

Accepted Answer

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

Question 16

If A is the area of the region bounded by one petal of the eight-petal rose $r = \sin 4 \theta,$ then A is

Accepted Answer

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

Question 17

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

The graph of the parametric equations $x ( t ) = 3 \cos t , y ( t ) = 3$ with $t \in \left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$ is an arc of a circle centered at the origin with radius 3 from

Accepted Answer

A)  $( 0 , - 3 ) \text { to } ( 3,0 )$ 
B)  $( 0,3 ) \text { to } ( 0 , - 3 )$ 
C)  $( - 3,0 ) \text { to } ( 3,0 )$ 
D)  $( 3,0 ) \text { to } ( 0 , - 3 )$ 
E)  $( 0,3 ) \text { to } ( - 3,0 )$ 
A)  $( 0 , - 3 ) \text { to } ( 3,0 )$ 
B)  $( 0,3 ) \text { to } ( 0 , - 3 )$ 
C)  $( - 3,0 ) \text { to } ( 3,0 )$ 
D)  $( 3,0 ) \text { to } ( 0 , - 3 )$ 
E)  $( 0,3 ) \text { to } ( - 3,0 )$

Question 19

The rectangular equation of the conic section $r = \frac { 4 } { 2 - 3 \sin \theta }$ is

Accepted Answer

A)  $4 y ^ { 2 } = 3 - 2 x$ 
B)  $4 x ^ { 2 } - 5 y ^ { 2 } - 24 y = 16$ 
C)  $5 x ^ { 2 } - 4 y ^ { 2 } - 24 y = 16$ 
D)  $5 x ^ { 2 } - 4 y ^ { 2 } + 24 y = 16$ 
E)  $4 x ^ { 2 } - 5 y ^ { 2 } + 24 x = 16$ 
A)  $4 y ^ { 2 } = 3 - 2 x$ 
B)  $4 x ^ { 2 } - 5 y ^ { 2 } - 24 y = 16$ 
C)  $5 x ^ { 2 } - 4 y ^ { 2 } - 24 y = 16$ 
D)  $5 x ^ { 2 } - 4 y ^ { 2 } + 24 y = 16$ 
E)  $4 x ^ { 2 } - 5 y ^ { 2 } + 24 x = 16$

Question 20

The graph of the parametric equations $x ( t ) = \sin t , y ( t ) = \cos t$ with $$t \in \left[ \frac { \pi } { 2 } , 2 \pi \right] t \in \left[ \frac { \pi } { 2 } , 2 \pi \right]$$ is an arc of the unit circle from

Accepted Answer

A) (1,0) to (-1,0) 
B) (0,1) to (1,0) 
C) (1,0) to (0,-1) 
D) (1,0) to (0,1) 
E) (0,1) to (-1,0) 
A) (1,0) to (-1,0) 
B) (0,1) to (1,0) 
C) (1,0) to (0,-1) 
D) (1,0) to (0,1) 
E) (0,1) to (-1,0)

The parametric equations of the line segment from (0,-4) to (4,0) with t $\in$ [0,4] are

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

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

The conic section $r = \frac { 3 } { 4 + 5 \cos \theta }$ is

The length of the curve $x ( t ) = 1 - \cos t , y ( t ) = 1 + \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

If A is the area of the region bounded by one petal of the four-petal rose $r = \sin 3 \theta,$ then A is

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

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

If A is the area of the region bounded by one loop of $r ^ { 2 } = 4 \sin 2 \theta,$ then A is

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

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } + 1 , y ( t ) = e ^ { 2 t }$ with $t \in ( - \infty , \infty ),$ is

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

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

The conic section $r = \frac { 5 } { 6 + 2 \sin \theta }$ is

The polar equation that corresponds to the rectangular equation $x ^ { 2 } - y ^ { 2 } = 4$ is

If A is the area of the region bounded by one petal of the eight-petal rose $r = \sin 4 \theta,$ then A is

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

The graph of the parametric equations $x ( t ) = 3 \cos t , y ( t ) = 3$ with $t \in \left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$ is an arc of a circle centered at the origin with radius 3 from

The rectangular equation of the conic section $r = \frac { 4 } { 2 - 3 \sin \theta }$ is

The graph of the parametric equations $x ( t ) = \sin t , y ( t ) = \cos t$ with $t \in \left[ \frac { \pi } { 2 } , 2 \pi \right] t \in \left[ \frac { \pi } { 2 } , 2 \pi \right]$ is an arc of the unit circle from

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Exam 10: Parametric Equations; Polar Equations

The parametric equations of the line segment from (0,-4) to (4,0) with t ∈\in∈ [0,4] are

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

For a ≠\neq= 0, the polar curve r=asin⁡2θr = a \sin 2 \thetar=asin2θ is a

The conic section r=34+5cos⁡θr = \frac { 3 } { 4 + 5 \cos \theta }r=4+5cosθ3​ is

The length of the curve x(t)=1−cos⁡t,y(t)=1+sin⁡tx ( t ) = 1 - \cos t , y ( t ) = 1 + \sin tx(t)=1−cost,y(t)=1+sint with t∈[−π2,π2]t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]t∈[−2π​,2π​] is

If A is the area of the region bounded by one petal of the four-petal rose r=sin⁡3θ,r = \sin 3 \theta,r=sin3θ, then A is

The points of intersections of r=2sin⁡θr = 2 \sin \thetar=2sinθ and r=2cos⁡θr = 2 \cos \thetar=2cosθ are

For a ≠\neq= 0, the polar curve r=acos⁡θr = a \cos \thetar=acosθ is a

If A is the area of the region bounded by one loop of r2=4sin⁡2θ,r ^ { 2 } = 4 \sin 2 \theta,r2=4sin2θ, then A is

Let x(t)=4cos⁡t,y(t)=3sin⁡tx ( t ) = 4 \cos t , y ( t ) = 3 \sin tx(t)=4cost,y(t)=3sint 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)=e−t+1,y(t)=e2tx ( t ) = e ^ { - t } + 1 , y ( t ) = e ^ { 2 t }x(t)=e−t+1,y(t)=e2t with t∈(−∞,∞),t \in ( - \infty , \infty ),t∈(−∞,∞), is

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

The conic section r=58+3sin⁡θr = \frac { 5 } { 8 + 3 \sin \theta }r=8+3sinθ5​ is

The conic section r=56+2sin⁡θr = \frac { 5 } { 6 + 2 \sin \theta }r=6+2sinθ5​ is

The polar equation that corresponds to the rectangular equation x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4 is

If A is the area of the region bounded by one petal of the eight-petal rose r=sin⁡4θ,r = \sin 4 \theta,r=sin4θ, then A is

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

The graph of the parametric equations x(t)=3cos⁡t,y(t)=3x ( t ) = 3 \cos t , y ( t ) = 3x(t)=3cost,y(t)=3 with t∈[3π2,2π]t \in \left[ \frac { 3 \pi } { 2 } , 2 \pi \right]t∈[23π​,2π] is an arc of a circle centered at the origin with radius 3 from

The rectangular equation of the conic section r=42−3sin⁡θr = \frac { 4 } { 2 - 3 \sin \theta }r=2−3sinθ4​ 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 parametric equations of the line segment from (0,-4) to (4,0) with t $\in$ [0,4] are

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

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

The conic section $r = \frac { 3 } { 4 + 5 \cos \theta }$ is

The length of the curve $x ( t ) = 1 - \cos t , y ( t ) = 1 + \sin t$ with $t \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is

If A is the area of the region bounded by one petal of the four-petal rose $r = \sin 3 \theta,$ then A is

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

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

If A is the area of the region bounded by one loop of $r ^ { 2 } = 4 \sin 2 \theta,$ then A is

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

The rectangular equation of the plane curve, with parametric equations $x ( t ) = e ^ { - t } + 1 , y ( t ) = e ^ { 2 t }$ with $t \in ( - \infty , \infty ),$ is

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

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

The conic section $r = \frac { 5 } { 6 + 2 \sin \theta }$ is

The polar equation that corresponds to the rectangular equation $x ^ { 2 } - y ^ { 2 } = 4$ is

If A is the area of the region bounded by one petal of the eight-petal rose $r = \sin 4 \theta,$ then A is

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

The graph of the parametric equations $x ( t ) = 3 \cos t , y ( t ) = 3$ with $t \in \left[ \frac { 3 \pi } { 2 } , 2 \pi \right]$ is an arc of a circle centered at the origin with radius 3 from

The rectangular equation of the conic section $r = \frac { 4 } { 2 - 3 \sin \theta }$ is