Question 1

Eliminate the parameter to find a Cartesian equation of the curve. $$x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$$

Accepted Answer

$$y = ( x + 5 ) ^ { 2 }$$

Question 2

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$
$$x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$$

Accepted Answer

$$- \frac { 3 t ^ { 2 } + 1 } { 4 t ^ { 3 } }$$

Question 3

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$, length of major axis 8

Accepted Answer

A)  $6 x ^ { 2 } = y$ 
B) $\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$ 
C) $\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$ 
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$ 
E)  $36 x ^ { 2 } = y$ 
A)  $6 x ^ { 2 } = y$ 
B) $\frac { x ^ { 2 } } { 16 } + \frac { ( y - 6 ) ^ { 2 } } { 15 } = 1$ 
C) $\frac { ( x - 6 ) ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 15 } = 1$ 
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 15 } = 1$ 
E)  $36 x ^ { 2 } = y$ B

Question 4

Find the eccentricity of the conic. $$r = \frac { 5 } { 8 - 5 \sin \theta }$$

Accepted Answer

A)  $e = \frac { 8 } { 5 }$ 
B)  $e = 8$ 
C)  $e = \frac { 5 } { 8 }$ 
D)  $e = - 5$ 
E)  $e = 5$ 
A)  $e = \frac { 8 } { 5 }$ 
B)  $e = 8$ 
C)  $e = \frac { 5 } { 8 }$ 
D)  $e = - 5$ 
E)  $e = 5$

Question 5

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x = t \cos t , \quad y = t \sin t , \quad t = 5 \pi$$

Accepted Answer

The answer of Find an equation of the tangent to...

Question 6

Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to four decimal places. $$x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pi$$

Accepted Answer

The answer of Set up an integral that represents the...

Question 7

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x = 4 t \cos t , y = 4 t \sin t , t = - \pi$$

Accepted Answer

The answer of Find an equation of the tangent to...

Question 8

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 7 } { 2 }$, and directrix $y = - 7$. Identify the conic.

Accepted Answer

A) 
$r = \frac { 49 } { 2 - 7 \cos \theta }$, hyperbola 
B) 
$r = \frac { 49 } { 2 - 7 \sin \theta }$, hyperbola 
C)  $r = \frac { 49 } { 2 - \sin \theta }$, ellipse 
D)  $r = \frac { 49 } { 2 - \cos \theta }$, ellipse 
A) 
$r = \frac { 49 } { 2 - 7 \cos \theta }$, hyperbola 
B) 
$r = \frac { 49 } { 2 - 7 \sin \theta }$, hyperbola 
C)  $r = \frac { 49 } { 2 - \sin \theta }$, ellipse 
D)  $r = \frac { 49 } { 2 - \cos \theta }$, ellipse

Question 9

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude $106 \mathrm {~km}$ and apolune altitude $318 \mathrm {~km}$ (above the moon). Find an equation of this ellipse if the radius of the moon is $1728 \mathrm {~km}$ and the center of the moon is at one focus.

Accepted Answer

The answer of The point in a lunar orbit nearest...

Question 10

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is $18 \mathrm {~cm}$. Find an equation of the parabola. Let $V$ be the origin. Find the diameter of the opening $| C D | , 19 \mathrm {~cm}$ from the vertex.

Accepted Answer

A)  $| C D | = 6 \sqrt { 38 }$ 
B)  $| C D | = 18$ 
C)  The equation is $y = 18 x ^ { 2 }$ 
D)  $| C D | = 4 \sqrt { 414 }$ 
E)  The equation is $y ^ { 2 } = 18 x$
F) The equation is $y ^ { 2 } - \frac { x ^ { 2 } } { 18 } = 1$ 
A)  $| C D | = 6 \sqrt { 38 }$ 
B)  $| C D | = 18$ 
C)  The equation is $y = 18 x ^ { 2 }$ 
D)  $| C D | = 4 \sqrt { 414 }$ 
E)  The equation is $y ^ { 2 } = 18 x$
F) The equation is $y ^ { 2 } - \frac { x ^ { 2 } } { 18 } = 1$

Question 11

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $366.5 \mathrm { AU }$. [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$.) Find the answer in $\mathrm { AU }$ and round to the nearest hundredth.

Accepted Answer

A)  $372.58 \mathrm { AU }$ 
B)  $370.58 \mathrm { AU }$ 
C)  $365.58 \mathrm { AU }$ 
D)  $368.58 \mathrm { AU }$ 
E)  $374.58 \mathrm { AU }$ 
A)  $372.58 \mathrm { AU }$ 
B)  $370.58 \mathrm { AU }$ 
C)  $365.58 \mathrm { AU }$ 
D)  $368.58 \mathrm { AU }$ 
E)  $374.58 \mathrm { AU }$

Question 12

Use a graph to estimate the values of $\theta$ for which the curves $r = 9 + 3 \sin 5 \theta$ and $r = 18 \sin \theta$ intersect. Round your answer to two decimal places.

Accepted Answer

A)  $\theta = 1.48$ 
B)  $\theta = 0.58$ 
C)  $\theta = 4.7$ 
D)  $\theta = 0.49$ 
E)  $\theta = 2.57$ 
A)  $\theta = 1.48$ 
B)  $\theta = 0.58$ 
C)  $\theta = 4.7$ 
D)  $\theta = 0.49$ 
E)  $\theta = 2.57$

Question 13

Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Accepted Answer

A)  $\frac { 24 \pi } { 5 }$ 
B)  $\frac { 18 \pi } { 5 }$ 
C)  $\frac { 12 \pi } { 5 }$ 
D)  $\frac { 2 \pi } { 4 }$ 
E)  None of these 
A)  $\frac { 24 \pi } { 5 }$ 
B)  $\frac { 18 \pi } { 5 }$ 
C)  $\frac { 12 \pi } { 5 }$ 
D)  $\frac { 2 \pi } { 4 }$ 
E)  None of these

Question 14

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$.

Accepted Answer

A)  $\left( 2 , \frac { \pi } { 6 } \right) , \left( 2 , - \frac { \pi } { 6 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } \right) , \left( 2 , - \frac { \pi } { 4 } \right)$ 
C)  $\left( 2 , \frac { \pi } { 3 } \right) , \left( 2 , - \frac { \pi } { 3 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 6 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } \right)$ 
A)  $\left( 2 , \frac { \pi } { 6 } \right) , \left( 2 , - \frac { \pi } { 6 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } \right) , \left( 2 , - \frac { \pi } { 4 } \right)$ 
C)  $\left( 2 , \frac { \pi } { 3 } \right) , \left( 2 , - \frac { \pi } { 3 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 6 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 3 } \right)$

Question 15

Find the area of the region that lies inside the first curve and outside the second curve.
$$r = 3 \cos \theta , \quad r = 1 + \cos \theta$$

Accepted Answer

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

Question 16

Set up, but do not evaluate, an integral that represents the length of the parametric curve. $$x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , \quad 8 \leq \mathrm { t } \leq 18$$

Accepted Answer

The answer of Set up, but do not evaluate, an...

Question 17

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $$x = 4 \sin \theta , \quad y = 2 \cos \theta$$

Accepted Answer

The answer of Using the arc length formula, set up,...

Question 18

$\text { Find the area bounded by the curve } x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t } \text { and the line } y = 2.5 \text {. }$

Accepted Answer

The answer of \(\text { Find the area bounded by...

Question 19

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$, asymptotes: $y = \pm \frac { 7 } { 4 } x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 49 } = 1$ 
B)  $\frac { y ^ { 2 } } { 49 } - \frac { x ^ { 2 } } { 16 } = 1$ 
C)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$ 
D)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

Question 20

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2$

Accepted Answer

A)  $( 0 , - 0 )$ 
B)  $( 0,0 ) , ( 2 , - 2 )$ 
C)  $( - 1,1 ) , ( - 2 , - 2 )$ 
D)  $( 0,0 ) , ( 4 , - 2 )$ 
E)  None of these 
A)  $( 0 , - 0 )$ 
B)  $( 0,0 ) , ( 2 , - 2 )$ 
C)  $( - 1,1 ) , ( - 2 , - 2 )$ 
D)  $( 0,0 ) , ( 4 , - 2 )$ 
E)  None of these

Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Find the eccentricity of the conic. $r = \frac { 5 } { 8 - 5 \sin \theta }$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = t \cos t , \quad y = t \sin t , \quad t = 5 \pi$

Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pi$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = 4 t \cos t , y = 4 t \sin t , t = - \pi$

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 7 } { 2 }$ , and directrix $y = - 7$ . Identify the conic.

Use a graph to estimate the values of $\theta$ for which the curves $r = 9 + 3 \sin 5 \theta$ and $r = 18 \sin \theta$ intersect. Round your answer to two decimal places.

Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$ .

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , \quad r = 1 + \cos \theta$

Set up, but do not evaluate, an integral that represents the length of the parametric curve. $x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , \quad 8 \leq \mathrm { t } \leq 18$

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $x = 4 \sin \theta , \quad y = 2 \cos \theta$

$\text { Find the area bounded by the curve } x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t } \text { and the line } y = 2.5 \text {. }$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2$

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Exam 10: Parametric Equations and Polar Coordinates

Eliminate the parameter to find a Cartesian equation of the curve. x=e4t−5,y=e8tx = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }x=e4t−5,y=e8t

Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ x=5+t2,y=t−t3x = 5 + t ^ { 2 } , y = t - t ^ { 3 }x=5+t2,y=t−t3

Find an equation for the conic that satisfies the given conditions. ellipse, foci (±1,6)( \pm 1,6 )(±1,6) , length of major axis 8

Find the eccentricity of the conic. r=58−5sin⁡θr = \frac { 5 } { 8 - 5 \sin \theta }r=8−5sinθ5​

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=tcos⁡t,y=tsin⁡t,t=5πx = t \cos t , \quad y = t \sin t , \quad t = 5 \pix=tcost,y=tsint,t=5π

Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to four decimal places. x=t−2sin⁡t,y=1−2cos⁡t,0≤t≤5πx = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pix=t−2sint,y=1−2cost,0≤t≤5π

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=4tcos⁡t,y=4tsin⁡t,t=−πx = 4 t \cos t , y = 4 t \sin t , t = - \pix=4tcost,y=4tsint,t=−π

Write a polar equation of the conic that has a focus at the origin, eccentricity 72\frac { 7 } { 2 }27​ , and directrix y=−7y = - 7y=−7 . Identify the conic.

Use a graph to estimate the values of θ\thetaθ for which the curves r=9+3sin⁡5θr = 9 + 3 \sin 5 \thetar=9+3sin5θ and r=18sin⁡θr = 18 \sin \thetar=18sinθ intersect. Round your answer to two decimal places.

Find the exact area of the surface obtained by rotating the given curve about the x-axis. x=2cos⁡3θ,y=2sin⁡3θ,0≤θ≤π/2x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2x=2cos3θ,y=2sin3θ,0≤θ≤π/2

Find the point(s) of intersection of the curves r=2r = 2r=2 and r=4cos⁡θr = 4 \cos \thetar=4cosθ .

Find the area of the region that lies inside the first curve and outside the second curve. r=3cos⁡θ,r=1+cos⁡θr = 3 \cos \theta , \quad r = 1 + \cos \thetar=3cosθ,r=1+cosθ

Set up, but do not evaluate, an integral that represents the length of the parametric curve. x=t−t10,y=109t9/8,8≤t≤18x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , \quad 8 \leq \mathrm { t } \leq 18x=t−t10,y=910​t9/8,8≤t≤18

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. x=4sin⁡θ,y=2cos⁡θx = 4 \sin \theta , \quad y = 2 \cos \thetax=4sinθ,y=2cosθ

Find the area bounded by the curve x=t−1t,y=t+1t and the line y=2.5. \text { Find the area bounded by the curve } x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t } \text { and the line } y = 2.5 \text {. } Find the area bounded by the curve x=t−t1​,y=t+t1​ and the line y=2.5.

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (±4,0)( \pm 4,0 )(±4,0) , asymptotes: y=±74xy = \pm \frac { 7 } { 4 } xy=±47​x

Find the point(s) on the curve where the tangent is horizontal. x=t3−3t+2,y=t3−3t2+2x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2x=t3−3t+2,y=t3−3t2+2

Functions and Models

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Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

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Vectors and the Geometry of Space

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Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

Find the eccentricity of the conic. $r = \frac { 5 } { 8 - 5 \sin \theta }$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = t \cos t , \quad y = t \sin t , \quad t = 5 \pi$

Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pi$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = 4 t \cos t , y = 4 t \sin t , t = - \pi$

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 7 } { 2 }$ , and directrix $y = - 7$ . Identify the conic.

Use a graph to estimate the values of $\theta$ for which the curves $r = 9 + 3 \sin 5 \theta$ and $r = 18 \sin \theta$ intersect. Round your answer to two decimal places.

Find the exact area of the surface obtained by rotating the given curve about the x-axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$ .

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , \quad r = 1 + \cos \theta$

Set up, but do not evaluate, an integral that represents the length of the parametric curve. $x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , \quad 8 \leq \mathrm { t } \leq 18$

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse. $x = 4 \sin \theta , \quad y = 2 \cos \theta$

$\text { Find the area bounded by the curve } x = t - \frac { 1 } { t } , y = t + \frac { 1 } { t } \text { and the line } y = 2.5 \text {. }$

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: $( \pm 4,0 )$ , asymptotes: $y = \pm \frac { 7 } { 4 } x$

Find the point(s) on the curve where the tangent is horizontal. $x = t ^ { 3 } - 3 t + 2 , \quad y = t ^ { 3 } - 3 t ^ { 2 } + 2$