Question 1

Write the corresponding rectangular equation for the curve represented by the parametric equation $x = 9 t - 2 , y = 4 t + 3$ by eliminating the parameter.

Accepted Answer

A)  $4 x - y + 35 = 0$ 
B)  $4 x - 9 y + 35 = 0$ 
C)  $4 x - 9 y + 11 = 0$ 
D)  $4 x + y - 35 = 0$ 
E)  $4 x + 9 y - 11 = 0$ 
A)  $4 x - y + 35 = 0$ 
B)  $4 x - 9 y + 35 = 0$ 
C)  $4 x - 9 y + 11 = 0$ 
D)  $4 x + y - 35 = 0$ 
E)  $4 x + 9 y - 11 = 0$

Question 2

Match the equation with its graph. $$\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E)  none of these 
A) 
  
B) 
  
C) 
  
D) 
  
E)  none of these

Question 3

Find the length of the curve over the given interval. $$r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$

Accepted Answer

A)  $24 \pi$ 
B)  $12 \pi$ 
C)  $48 \pi$ 
D)  $72 \pi$ 
E)  $6 \pi$ 
A)  $24 \pi$ 
B)  $12 \pi$ 
C)  $48 \pi$ 
D)  $72 \pi$ 
E)  $6 \pi$

Question 4

Pluto moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is $5.906 \times 10 ^ { 9 }$ kilometers, and the eccentricity is $0.2488$. Find the perihelion distance of Pluto from the sun. Round your answer to the nearest kilometers.

Accepted Answer

A)  $7,375,412,800$ kilometers 
B)  $4,436,587,201$ kilometers 
C)  $4,436,587,200$ kilometers 
D)  $8,873,174,399$ kilometers 
E)  $8,873,174,401$ kilometers 
A)  $7,375,412,800$ kilometers 
B)  $4,436,587,201$ kilometers 
C)  $4,436,587,200$ kilometers 
D)  $8,873,174,399$ kilometers 
E)  $8,873,174,401$ kilometers

Question 5

Find the length of the curve over the given interval. $$r = 2 ( 1 + \cos \theta ) , 0 \leq \theta \leq 2 \pi$$

Accepted Answer

A)  2 
B)  20 
C)  16 
D)  8 
E)  4 
A)  2 
B)  20 
C)  16 
D)  8 
E)  4

Question 6

Find a set of parametric equations for the rectangular equation $y = 2 x - 3 \text { that }$ satisfies the condition $t = 0$ at the point $( 2,1 )$.

Accepted Answer

A)  $x = t , y = 3 t ^ { 2 } - t - 3$ 
B)  $x = t + 2 , y = 3 t ^ { 2 } + t - 5$ 
C)  $x = t - 2 , y = 2 t + 1$ 
D)  $x = t + 2 , y = 2 t + 1$ 
E)  $x = t , y = 2 t - 3$ 
A)  $x = t , y = 3 t ^ { 2 } - t - 3$ 
B)  $x = t + 2 , y = 3 t ^ { 2 } + t - 5$ 
C)  $x = t - 2 , y = 2 t + 1$ 
D)  $x = t + 2 , y = 2 t + 1$ 
E)  $x = t , y = 2 t - 3$

Question 7

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ point corresponding to $\theta = \frac { \pi } { 4 }$.
$\begin{array} { l } 
x = 4 \cos \theta \
y = 4 \sin \theta
\end{array}$

Accepted Answer

A) $\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta } ;$ at $\theta = \frac { \pi } { 4 } ;$ slope 1 and concave down 
B)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 } :$ slope $- 1$ and concave down 
C)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta } ;$ at $\theta = \frac { \pi } { 4 }$ : slope $- 1$ and concave up 
D)  $\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 } :$ slope 1 and concave up 
E)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 }$ : slope of $- 1$ and concave down 
A) $\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta } ;$ at $\theta = \frac { \pi } { 4 } ;$ slope 1 and concave down 
B)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 } :$ slope $- 1$ and concave down 
C)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta } ;$ at $\theta = \frac { \pi } { 4 }$ : slope $- 1$ and concave up 
D)  $\frac { d y } { d x } = \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 } :$ slope 1 and concave up 
E)  $\frac { d y } { d x } = - \cot \theta , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 1 } { 4 \sin ^ { 3 } \theta }$; at $\theta = \frac { \pi } { 4 }$ : slope of $- 1$ and concave down

Question 8

$\text { Find the area of the surface formed by revolving about the } \theta = \frac { \pi } { 2 } \text { axis the following }$ curve over the given interval. $r = e ^ { 8 \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

Accepted Answer

A)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 385 }$ 
B)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 257 }$ 
C)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 129 }$ 
D)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 8 \right) \pi } { 129 }$ 
E)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 8 \right) \pi } { 257 }$ 
A)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 385 }$ 
B)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 257 }$ 
C)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 16 \right) \pi } { 129 }$ 
D)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 8 \right) \pi } { 129 }$ 
E)  $\frac { 2 \sqrt { 65 } \left( e ^ { 8 x } - 8 \right) \pi } { 257 }$

Question 9

Identify any points at which the Folium of Descartes $x = \frac { 8 t } { 1 + t ^ { 3 } } , y = \frac { 8 t ^ { 2 } } { 1 + t ^ { 3 } } \text { is not }$ smooth. Round your answer to two decimal places, if necessary.

Accepted Answer

A)  not smooth when $t = - 1.00$ 
B)  not smooth when $t = 0$ 
C)  not smooth when $t = 2.67 , - 2.67$ 
D)  smooth everywhere 
E)  not smooth when $t = 2.67$ 
A)  not smooth when $t = - 1.00$ 
B)  not smooth when $t = 0$ 
C)  not smooth when $t = 2.67 , - 2.67$ 
D)  smooth everywhere 
E)  not smooth when $t = 2.67$

Question 10

Identify the choice that best completes the statement or answers the question.
$\text { Find } \frac { d y } { d x } \text {. }$ $x = t ^ { 2 }$
$y = 10 - 10 t$

Accepted Answer

A)  $\frac { d y } { d x } = - \frac { 20 } { t }$ 
B)  $\frac { d y } { d x } = - \frac { 10 } { t }$ 
C)  $\frac { d y } { d x } = - \frac { 5 } { t }$ 
D)  $\frac { d y } { d x } = - 20 t$ 
E)  $\frac { d y } { d x } = - \frac { t } { 10 }$ 
A)  $\frac { d y } { d x } = - \frac { 20 } { t }$ 
B)  $\frac { d y } { d x } = - \frac { 10 } { t }$ 
C)  $\frac { d y } { d x } = - \frac { 5 } { t }$ 
D)  $\frac { d y } { d x } = - 20 t$ 
E)  $\frac { d y } { d x } = - \frac { t } { 10 }$

Question 11

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d t ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ point corresponding to t = 5. 
$\begin{array} { l } 
x = t + 10 \
y = t ^ { 2 } + 4 t
\end{array}$

Accepted Answer

A)  $\frac { d y } { d x } = \frac { 1 } { 3 } t ^ { 3 } + 2 t ^ { 2 } , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 3 t ^ { 2 } + 4 t$, at $t = 5$ : slope $- \frac { 325 } { 6 }$ and concave up 
B)  $\frac { d y } { d x } = - 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - 2 ;$ at $t = 5$ : slope $- 6$ and concave down 
C)  $\frac { d y } { d x } = 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 2$; at $t = 5$ : slope 14 and concave up 
D)  $\frac { d y } { d x } = - 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - 2$; at $t = 5$ : slope 6 and concave down 
E)  $\frac { d y } { d x } = \frac { 1 } { 3 } t ^ { 3 } + 2 t ^ { 2 } , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 3 t ^ { 2 } + 4 t$, at $t = 5 :$ slope $\frac { 325 } { 6 }$ and concave up 
A)  $\frac { d y } { d x } = \frac { 1 } { 3 } t ^ { 3 } + 2 t ^ { 2 } , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 3 t ^ { 2 } + 4 t$, at $t = 5$ : slope $- \frac { 325 } { 6 }$ and concave up 
B)  $\frac { d y } { d x } = - 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - 2 ;$ at $t = 5$ : slope $- 6$ and concave down 
C)  $\frac { d y } { d x } = 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 2$; at $t = 5$ : slope 14 and concave up 
D)  $\frac { d y } { d x } = - 2 t + 4 , \frac { d ^ { 2 } y } { d x ^ { 2 } } = - 2$; at $t = 5$ : slope 6 and concave down 
E)  $\frac { d y } { d x } = \frac { 1 } { 3 } t ^ { 3 } + 2 t ^ { 2 } , \frac { d ^ { 2 } y } { d x ^ { 2 } } = 3 t ^ { 2 } + 4 t$, at $t = 5 :$ slope $\frac { 325 } { 6 }$ and concave up

Question 12

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 t , y = 8 t - 3 \text {. }$ Round your answer to two decimal places, if necessary.

Accepted Answer

A)  $0$ 
B)  $2.00$ 
C)  $- \infty$ 
D)  $0.50$ 
E)  $\infty$ 
A)  $0$ 
B)  $2.00$ 
C)  $- \infty$ 
D)  $0.50$ 
E)  $\infty$

Question 13

Write the corresponding rectangular equation for the curve represented by the parametric equations \)x = 7 + \frac { 2 } { t } , y = t - 9 	ext { by eliminating the parameter. }\)

Accepted Answer

A)  $y = \frac { x - 7 } { 2 } - 9$ 
B)  $y = \frac { 2 } { x + 7 } + 9$ 
C)  $y = \frac { 2 } { x - 7 } - 9$ 
D)  $y = \frac { 2 } { x - 7 } + 9$ 
E)  $y = \frac { 2 } { x + 7 } - 9$ 
A)  $y = \frac { x - 7 } { 2 } - 9$ 
B)  $y = \frac { 2 } { x + 7 } + 9$ 
C)  $y = \frac { 2 } { x - 7 } - 9$ 
D)  $y = \frac { 2 } { x - 7 } + 9$ 
E)  $y = \frac { 2 } { x + 7 } - 9$

Question 14

Find a polar equation for the ellipse with its focus at the pole and vertices $( 6,0 ) , ( 24 , \pi )$

Accepted Answer

A)  $r = \frac { 48 } { 5 + 3 \cos \theta }$ 
B)  $r = \frac { 24 } { 3 + 2 \cos \theta }$ 
C)  $r = \frac { 48 } { 3 + 2 \cos \theta }$ 
D)  $r = \frac { 48 } { 3 + 2 \sin \theta }$ 
E)  $r = \frac { 24 } { 5 + 3 \cos \theta }$ 
A)  $r = \frac { 48 } { 5 + 3 \cos \theta }$ 
B)  $r = \frac { 24 } { 3 + 2 \cos \theta }$ 
C)  $r = \frac { 48 } { 3 + 2 \cos \theta }$ 
D)  $r = \frac { 48 } { 3 + 2 \sin \theta }$ 
E)  $r = \frac { 24 } { 5 + 3 \cos \theta }$

Question 15

Find the arc length of the curve on the given interval. $x = t ^ { 2 } + 3 , y = 8 t ^ { 3 } + 9 , - 1 \leq t \leq 0$

Accepted Answer

A) 
$\frac { ( 770 ) ^ { \frac { 3 } { 2 } } - 8 } { 864 }$ 
B) 
$\frac { ( 145 ) ^ { \frac { 3 } { 2 } } - 1 } { 216 }$ 
C) 
$\frac { ( 2,308 ) ^ { \frac { 9 } { 4 } } - 8 } { 1728 }$ 
D) 
$\frac { ( 2,308 ) ^ { \frac { 9 } { 4 } } - 1 } { 864 }$ 
E) 
$\frac { ( 769 ) ^ { \frac { 3 } { 2 } } - 1 } { 1728 }$ 
A) 
$\frac { ( 770 ) ^ { \frac { 3 } { 2 } } - 8 } { 864 }$ 
B) 
$\frac { ( 145 ) ^ { \frac { 3 } { 2 } } - 1 } { 216 }$ 
C) 
$\frac { ( 2,308 ) ^ { \frac { 9 } { 4 } } - 8 } { 1728 }$ 
D) 
$\frac { ( 2,308 ) ^ { \frac { 9 } { 4 } } - 1 } { 864 }$ 
E) 
$\frac { ( 769 ) ^ { \frac { 3 } { 2 } } - 1 } { 1728 }$

Question 16

Use the result, "the set of parametric equations for the circle is $x = h + r \cos \theta , y = k + r \sin \theta "$ to find a set of parametric equations for the circle with a center $( - 2,6 )$ and radius 5 .

Accepted Answer

A)  $x = 5 + \cos \theta , y = 5 + \sin \theta$ 
B)  $x = 4 + 25 \cos \theta , y = 36 + 25 \sin \theta$ 
C)  $x = 2 - 5 \cos \theta , y = 6 + 5 \sin \theta$ 
D) $x = - \frac { 2 } { 5 } \cos \theta , y = \frac { 6 } { 5 } \sin \theta$ 
E)  $x = - 2 + 5 \cos \theta , y = 6 + 5 \sin \theta$ 
A)  $x = 5 + \cos \theta , y = 5 + \sin \theta$ 
B)  $x = 4 + 25 \cos \theta , y = 36 + 25 \sin \theta$ 
C)  $x = 2 - 5 \cos \theta , y = 6 + 5 \sin \theta$ 
D) $x = - \frac { 2 } { 5 } \cos \theta , y = \frac { 6 } { 5 } \sin \theta$ 
E)  $x = - 2 + 5 \cos \theta , y = 6 + 5 \sin \theta$

Question 17

Find the length of the curve $r = 6 \text { over the interval } 0 \leq \theta \leq 2 \pi \text {. }$

Accepted Answer

A)  $2 \pi$ 
B)  $\pi$ 
C)  $36 \pi$ 
D)  $12 \pi$ 
E)  $6 \pi$ 
A)  $2 \pi$ 
B)  $\pi$ 
C)  $36 \pi$ 
D)  $12 \pi$ 
E)  $6 \pi$

Question 18

Identify the graph for the polar equation $r = \frac { 2 } { 2 + 3 \sin \theta }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 19

$\text { Find } \frac { d y } { d x }$ $\begin{array} { l } 
x = \sqrt [ 12 ] { t } \
y = 6 - t
\end{array}$

Accepted Answer

A) $\frac { d y } { d x } = - 12 t ^ { \frac { 13 } { 12 } }$ 
B) $\frac { d y } { d x } = - 12 t ^ { \frac { 11 } { 12 } }$ 
C) $\frac { d y } { d x } = - 6 t ^ { \frac { 11 } { 12 } }$ 
D) $\frac { d y } { d x } = 6 t ^ { \frac { 13 } { 12 } }$ 
E) $\frac { d y } { d x } = 12 t ^ { \frac { 11 } { 12 } }$ 
A) $\frac { d y } { d x } = - 12 t ^ { \frac { 13 } { 12 } }$ 
B) $\frac { d y } { d x } = - 12 t ^ { \frac { 11 } { 12 } }$ 
C) $\frac { d y } { d x } = - 6 t ^ { \frac { 11 } { 12 } }$ 
D) $\frac { d y } { d x } = 6 t ^ { \frac { 13 } { 12 } }$ 
E) $\frac { d y } { d x } = 12 t ^ { \frac { 11 } { 12 } }$

Question 20

$\text { Find the area of the interior of } r = 12 \sin \theta \text {. }$

Accepted Answer

A)  $48 \pi$ 
B)  $12 \pi$ 
C)  $36 \pi$ 
D)  $24 \pi$ 
E)  $30 \pi$ 
A)  $48 \pi$ 
B)  $12 \pi$ 
C)  $36 \pi$ 
D)  $24 \pi$ 
E)  $30 \pi$

Write the corresponding rectangular equation for the curve represented by the parametric equation $x = 9 t - 2 , y = 4 t + 3$ by eliminating the parameter.

Match the equation with its graph. $\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$

Find the length of the curve over the given interval. $r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

Pluto moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is $5.906 \times 10 ^ { 9 }$ kilometers, and the eccentricity is $0.2488$ . Find the perihelion distance of Pluto from the sun. Round your answer to the nearest kilometers.

Find the length of the curve over the given interval. $r = 2 ( 1 + \cos \theta ) , 0 \leq \theta \leq 2 \pi$

Find a set of parametric equations for the rectangular equation $y = 2 x - 3 \text { that }$ satisfies the condition $t = 0$ at the point $( 2,1 )$ .

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ point corresponding to $\theta = \frac { \pi } { 4 }$ . x=4\theta y=4\theta

$\text { Find the area of the surface formed by revolving about the } \theta = \frac { \pi } { 2 } \text { axis the following }$ curve over the given interval. $r = e ^ { 8 \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

Identify any points at which the Folium of Descartes $x = \frac { 8 t } { 1 + t ^ { 3 } } , y = \frac { 8 t ^ { 2 } } { 1 + t ^ { 3 } } \text { is not }$ smooth. Round your answer to two decimal places, if necessary.

Identify the choice that best completes the statement or answers the question. $\text { Find } \frac { d y } { d x } \text {. }$ $x = t ^ { 2 }$ $y = 10 - 10 t$

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d t ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ point corresponding to t = 5. x=t+10 y=+4t

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 t , y = 8 t - 3 \text {. }$ Round your answer to two decimal places, if necessary.

Write the corresponding rectangular equation for the curve represented by the parametric equations \)x = 7 + \frac { 2 } { t } , y = t - 9 \text { by eliminating the parameter. }\)

Find a polar equation for the ellipse with its focus at the pole and vertices $( 6,0 ) , ( 24 , \pi )$

Find the arc length of the curve on the given interval. $x = t ^ { 2 } + 3 , y = 8 t ^ { 3 } + 9 , - 1 \leq t \leq 0$

Use the result, "the set of parametric equations for the circle is $x = h + r \cos \theta , y = k + r \sin \theta "$ to find a set of parametric equations for the circle with a center $( - 2,6 )$ and radius 5 .

Find the length of the curve $r = 6 \text { over the interval } 0 \leq \theta \leq 2 \pi \text {. }$

Identify the graph for the polar equation $r = \frac { 2 } { 2 + 3 \sin \theta }$

$\text { Find } \frac { d y } { d x }$ x= y=6-t

$\text { Find the area of the interior of } r = 12 \sin \theta \text {. }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 10: Conics and Calculus

Write the corresponding rectangular equation for the curve represented by the parametric equation x=9t−2,y=4t+3x = 9 t - 2 , y = 4 t + 3x=9t−2,y=4t+3 by eliminating the parameter.

Match the equation with its graph. (x−3)24−y29=1\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 14(x−3)2​−9y2​=1

Find the length of the curve over the given interval. r=24cos⁡θ,−π2≤θ≤π2r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }r=24cosθ,−2π​≤θ≤2π​

Find the length of the curve over the given interval. r=2(1+cos⁡θ),0≤θ≤2πr = 2 ( 1 + \cos \theta ) , 0 \leq \theta \leq 2 \pir=2(1+cosθ),0≤θ≤2π

Find a set of parametric equations for the rectangular equation y=2x−3 that y = 2 x - 3 \text { that }y=2x−3 that satisfies the condition t=0t = 0t=0 at the point (2,1)( 2,1 )(2,1) .

Identify any points at which the Folium of Descartes x=8t1+t3,y=8t21+t3 is not x = \frac { 8 t } { 1 + t ^ { 3 } } , y = \frac { 8 t ^ { 2 } } { 1 + t ^ { 3 } } \text { is not }x=1+t38t​,y=1+t38t2​ is not smooth. Round your answer to two decimal places, if necessary.

Identify the choice that best completes the statement or answers the question. Find dydx. \text { Find } \frac { d y } { d x } \text {. } Find dxdy​. x=t2x = t ^ { 2 }x=t2 y=10−10ty = 10 - 10 ty=10−10t

Write the corresponding rectangular equation for the curve represented by the parametric equations \)x = 7 + \frac { 2 } { t } , y = t - 9 \text { by eliminating the parameter. }\)

Find a polar equation for the ellipse with its focus at the pole and vertices (6,0),(24,π)( 6,0 ) , ( 24 , \pi )(6,0),(24,π)

Find the arc length of the curve on the given interval. x=t2+3,y=8t3+9,−1≤t≤0x = t ^ { 2 } + 3 , y = 8 t ^ { 3 } + 9 , - 1 \leq t \leq 0x=t2+3,y=8t3+9,−1≤t≤0

Use the result, "the set of parametric equations for the circle is x=h+rcos⁡θ,y=k+rsin⁡θ"x = h + r \cos \theta , y = k + r \sin \theta "x=h+rcosθ,y=k+rsinθ" to find a set of parametric equations for the circle with a center (−2,6)( - 2,6 )(−2,6) and radius 5 .

Find the length of the curve r=6 over the interval 0≤θ≤2π. r = 6 \text { over the interval } 0 \leq \theta \leq 2 \pi \text {. }r=6 over the interval 0≤θ≤2π.

Identify the graph for the polar equation r=22+3sin⁡θr = \frac { 2 } { 2 + 3 \sin \theta }r=2+3sinθ2​

Find dydx\text { Find } \frac { d y } { d x } Find dxdy​ x= y=6-t

Find the area of the interior of r=12sin⁡θ. \text { Find the area of the interior of } r = 12 \sin \theta \text {. } Find the area of the interior of r=12sinθ.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Write the corresponding rectangular equation for the curve represented by the parametric equation $x = 9 t - 2 , y = 4 t + 3$ by eliminating the parameter.

Match the equation with its graph. $\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 9 } = 1$

Find the length of the curve over the given interval. $r = 24 \cos \theta , - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$

Find the length of the curve over the given interval. $r = 2 ( 1 + \cos \theta ) , 0 \leq \theta \leq 2 \pi$

Find a set of parametric equations for the rectangular equation $y = 2 x - 3 \text { that }$ satisfies the condition $t = 0$ at the point $( 2,1 )$ .

Identify any points at which the Folium of Descartes $x = \frac { 8 t } { 1 + t ^ { 3 } } , y = \frac { 8 t ^ { 2 } } { 1 + t ^ { 3 } } \text { is not }$ smooth. Round your answer to two decimal places, if necessary.

Identify the choice that best completes the statement or answers the question. $\text { Find } \frac { d y } { d x } \text {. }$ $x = t ^ { 2 }$ $y = 10 - 10 t$

Find a polar equation for the ellipse with its focus at the pole and vertices $( 6,0 ) , ( 24 , \pi )$

Find the arc length of the curve on the given interval. $x = t ^ { 2 } + 3 , y = 8 t ^ { 3 } + 9 , - 1 \leq t \leq 0$

Use the result, "the set of parametric equations for the circle is $x = h + r \cos \theta , y = k + r \sin \theta "$ to find a set of parametric equations for the circle with a center $( - 2,6 )$ and radius 5 .

Find the length of the curve $r = 6 \text { over the interval } 0 \leq \theta \leq 2 \pi \text {. }$

Identify the graph for the polar equation $r = \frac { 2 } { 2 + 3 \sin \theta }$

$\text { Find } \frac { d y } { d x }$ x= y=6-t

$\text { Find the area of the interior of } r = 12 \sin \theta \text {. }$