Question 1

Find the arc length of the curve on the given interval. $$x = t ^ { 2 } , y = 8 t , 0 \leq t \leq 4$$

Accepted Answer

A)  $4 ( \sqrt { 2 } + \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
B)  $4 ( \sqrt { 2 } - \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
C)  $4 ( \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
D)  $16 ( \sqrt { 2 } - \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
E)  $16 ( \sqrt { 2 } + \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
A)  $4 ( \sqrt { 2 } + \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
B)  $4 ( \sqrt { 2 } - \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
C)  $4 ( \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
D)  $16 ( \sqrt { 2 } - \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ 
E)  $16 ( \sqrt { 2 } + \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$

Question 2

Find the center of the ellipse given by $$16 x ^ { 2 } + y ^ { 2 } = 16$$

Accepted Answer

A)  $( 0,4 )$ 
B)  $( 0 , - 4 )$ 
C)  $( 0,0 )$ 
D)  $( - 4,0 )$ 
E)  $( 4,0 )$ 
A)  $( 0,4 )$ 
B)  $( 0 , - 4 )$ 
C)  $( 0,0 )$ 
D)  $( - 4,0 )$ 
E)  $( 4,0 )$

Question 3

Find the distance from the pole to the directrix for the conic $$r = \frac { 42 } { 14 + \cos \theta }$$

Accepted Answer

A)  $\frac { 1 } { 14 }$ 
B)  14 
C)  $\frac { 1 } { 42 }$ 
D)  42 
E)  7 
A)  $\frac { 1 } { 14 }$ 
B)  14 
C)  $\frac { 1 } { 42 }$ 
D)  42 
E)  7

Question 4

Sketch the curve represented by the parametric equations $$x = e ^ { t } , y = e ^ { 3 t } + 2$$

Accepted Answer

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

Question 5

Find the area of the interior of $r = 10 + 2 \sin \theta$

Accepted Answer

A)  $109 \pi$ 
B)  $105 \pi$ 
C)  $102 \pi$ 
D)  $100 \pi$ 
E)  $114 \pi$ 
A)  $109 \pi$ 
B)  $105 \pi$ 
C)  $102 \pi$ 
D)  $100 \pi$ 
E)  $114 \pi$

Question 6

Find the area of inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$ equations using the graphing utility.

Accepted Answer

A)  $\frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$ 
B)  $- \frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$ 
C)  $\frac { 1936 \pi } { 3 } + \frac { 968 \sqrt { 3 } } { 1 }$ 
D)  $\frac { 968 \pi } { 1 } + \frac { 1936 \sqrt { 3 } } { 3 }$ 
E)  $\frac { 968 \pi } { 1 } - \frac { 1936 \sqrt { 3 } } { 3 }$ 
A)  $\frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$ 
B)  $- \frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$ 
C)  $\frac { 1936 \pi } { 3 } + \frac { 968 \sqrt { 3 } } { 1 }$ 
D)  $\frac { 968 \pi } { 1 } + \frac { 1936 \sqrt { 3 } } { 3 }$ 
E)  $\frac { 968 \pi } { 1 } - \frac { 1936 \sqrt { 3 } } { 3 }$

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

Sketch the curve represented by the parametric equations $$x = \sqrt { t } , y = t - 5$$

Accepted Answer

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

Question 9

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x ^ { 2 } + 4 y ^ { 2 } + 3 x + 7 y - 6 = 0$$

Accepted Answer

A)  parabola 
B)  circle 
C)  ellipse 
D)  hyperbola 
A)  parabola 
B)  circle 
C)  ellipse 
D)  hyperbola

Question 10

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 \cos \theta , y = 4 \sin \theta \text {. }$

Accepted Answer

A)  $- \frac { \tan ^ { 3 } \theta } { 4 }$ 
B)  $- \frac { \csc ^ { 3 } \theta } { 4 }$ 
C)  $\frac { \csc ^ { 3 } \theta } { 4 }$ 
D)  $- \frac { \sec ^ { 3 } \theta } { 4 }$ 
E)  $\frac { \sec ^ { 3 } \theta } { 4 }$ 
A)  $- \frac { \tan ^ { 3 } \theta } { 4 }$ 
B)  $- \frac { \csc ^ { 3 } \theta } { 4 }$ 
C)  $\frac { \csc ^ { 3 } \theta } { 4 }$ 
D)  $- \frac { \sec ^ { 3 } \theta } { 4 }$ 
E)  $\frac { \sec ^ { 3 } \theta } { 4 }$

Question 11

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$
(i) $x$-axis; (ii) $y$-axis

Accepted Answer

A)  (i) $486 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
B)  (i) $324 \sqrt { 82 } \pi$; (ii) $36 \sqrt { 82 } \pi$ 
C)  (i) $486 \sqrt { 37 } \pi$; (ii) $36 \sqrt { 37 } \pi$ 
D)  (i) $81 \sqrt { 37 } \pi$; (ii) $486 \sqrt { 37 } \pi$ 
E)  (i) $324 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
A)  (i) $486 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
B)  (i) $324 \sqrt { 82 } \pi$; (ii) $36 \sqrt { 82 } \pi$ 
C)  (i) $486 \sqrt { 37 } \pi$; (ii) $36 \sqrt { 37 } \pi$ 
D)  (i) $81 \sqrt { 37 } \pi$; (ii) $486 \sqrt { 37 } \pi$ 
E)  (i) $324 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$

Question 12

Find the center, foci, and vertices of the hyperbola. $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 2 ) ^ { 2 } } { 1 } = 1$

Accepted Answer

A)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( 5 , - 2 )$; foci: $( 1 - \sqrt { 17 } , - 2 ) , ( 1 + \sqrt { 17 } , - 2 )$. 
B)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( 5 , - 2 )$; foci: $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
C)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( - 5 , - 2 )$; foci: $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
D)  center: $( 1 , - 2 )$; vertices: $( 1 , - 3 ) , ( 1 , - 1 )$; foci $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
E)  center: $( 1 , - 2 )$; vertices: $( 1 , - 3 ) , ( 1 , - 1 )$; foci: $( 1 - \sqrt { 17 } , - 2 ) , ( 1 + \sqrt { 17 } , - 2 )$. 
A)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( 5 , - 2 )$; foci: $( 1 - \sqrt { 17 } , - 2 ) , ( 1 + \sqrt { 17 } , - 2 )$. 
B)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( 5 , - 2 )$; foci: $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
C)  center: $( 1 , - 2 )$; vertices: $( - 3 , - 2 ) , ( - 5 , - 2 )$; foci: $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
D)  center: $( 1 , - 2 )$; vertices: $( 1 , - 3 ) , ( 1 , - 1 )$; foci $( 1 , - 2 - \sqrt { 17 } ) , ( 1 , - 2 + \sqrt { 17 } )$. 
E)  center: $( 1 , - 2 )$; vertices: $( 1 , - 3 ) , ( 1 , - 1 )$; foci: $( 1 - \sqrt { 17 } , - 2 ) , ( 1 + \sqrt { 17 } , - 2 )$.

Question 13

Find the vertex of the parabola given by  $y ^ { 2 } = - 8 x$

Accepted Answer

A)  $( 0 , - 2 )$ 
B)  $( 0,0 )$ 
C)  $( 2,0 )$ 
D)  $( 0,2 )$ 
E)  $( - 2,0 )$ 
A)  $( 0 , - 2 )$ 
B)  $( 0,0 )$ 
C)  $( 2,0 )$ 
D)  $( 0,2 )$ 
E)  $( - 2,0 )$

Question 14

Find the length of the curve $r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }$ answer to two decimal places.

Accepted Answer

A) 0.58 
B) 3.46 
C) 2.00 
D) 1.15 
E) 1.73 
A) 0.58 
B) 3.46 
C) 2.00 
D) 1.15 
E) 1.73

Question 15

Find  the distance from the pole to the directrix for the conic $$r = - \frac { 30 } { 15 + 31 \sin \theta }$$

Accepted Answer

A)  $\frac { 30 } { 31 }$ 
B)  $\frac { 31 } { 30 }$ 
C)  30 
D)  $\frac { 31 } { 15 }$ 
E)  62 
A)  $\frac { 30 } { 31 }$ 
B)  $\frac { 31 } { 30 }$ 
C)  30 
D)  $\frac { 31 } { 15 }$ 
E)  62

Question 16

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$y ^ { 2 } + 6 y + 4 x + 1 = 0$$

Accepted Answer

A)  vertex: $( - 2,3 )$; focus: $( - 3 , - 3 )$; directrix $x = - 1$  
  
B)  vertex: $( 2 , - 3 )$; focus: $( 3 , - 3 )$;  directrix $x = - 1$
  
C)  vertex: $( 2 , - 3 )$; focus: $( 1 , - 3 )$; directrix $x = 3$
  
D)  vertex: $( 2 , - 3 )$; focus: $( 1 , - 3 )$; directrix $x = 3$

E)  vertex: $( - 2,3 ) ;$ focus: $( - 3 , - 3 )$;directrix $x = 1$
  
A)  vertex: $( - 2,3 )$; focus: $( - 3 , - 3 )$; directrix $x = - 1$  
  
B)  vertex: $( 2 , - 3 )$; focus: $( 3 , - 3 )$;  directrix $x = - 1$
  
C)  vertex: $( 2 , - 3 )$; focus: $( 1 , - 3 )$; directrix $x = 3$
  
D)  vertex: $( 2 , - 3 )$; focus: $( 1 , - 3 )$; directrix $x = 3$

E)  vertex: $( - 2,3 ) ;$ focus: $( - 3 , - 3 )$;directrix $x = 1$

Question 17

Find all points of intersection of the graphs of the equations. $\begin{array} { l } 
r = 1 + \sin \theta \
r = 3 \sin \theta
\end{array}$

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 } { 2 } , - \frac { \pi } { 6 } \right) , ( 0,0 )$ 
B)  $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 \sqrt { 3 } } { 2 } , - \frac { \pi } { 6 } \right) , ( 0,0 )$ 
C)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right) , \left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right) , ( 0,0 )$ 
D)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 } { 2 } , \frac { 5 \pi } { 6 } \right) , ( 0,0 )$ 
E)  $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 3 } \right) , \left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 3 } \right) , ( 0,0 )$ 
A)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 } { 2 } , - \frac { \pi } { 6 } \right) , ( 0,0 )$ 
B)  $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 \sqrt { 3 } } { 2 } , - \frac { \pi } { 6 } \right) , ( 0,0 )$ 
C)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right) , \left( \frac { 3 } { 2 } , \frac { \pi } { 3 } \right) , ( 0,0 )$ 
D)  $\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right) , \left( \frac { 3 } { 2 } , \frac { 5 \pi } { 6 } \right) , ( 0,0 )$ 
E)  $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 3 } \right) , \left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { \pi } { 3 } \right) , ( 0,0 )$

Question 18

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$
(i) $x$-axis;(ii) $y$-axis

Accepted Answer

A)  (i) $486 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
B)  (i) $324 \sqrt { 82 } \pi$; (ii) $36 \sqrt { 82 } \pi$ 
C)  (i) $486 \sqrt { 37 } \pi$; (ii) $36 \sqrt { 37 } \pi$ 
D)  (i) $81 \sqrt { 37 } \pi$; (ii) $486 \sqrt { 37 } \pi$ 
E)  (i) $324 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
A)  (i) $486 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$ 
B)  (i) $324 \sqrt { 82 } \pi$; (ii) $36 \sqrt { 82 } \pi$ 
C)  (i) $486 \sqrt { 37 } \pi$; (ii) $36 \sqrt { 37 } \pi$ 
D)  (i) $81 \sqrt { 37 } \pi$; (ii) $486 \sqrt { 37 } \pi$ 
E)  (i) $324 \sqrt { 37 } \pi$; (ii) $81 \sqrt { 37 } \pi$

Question 19

$\text { Find } \frac { d y } { d x } \text {. }$
$x = \sqrt [ 12 ] { t }$
$y = 6 - t$

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

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

Accepted Answer

A)  $\frac { \ln ( 864 + 24 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 327 } } { 24 }$ 
B)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) - \ln 2 + 6 \sqrt { 327 } } { 24 }$ 
C)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 165 } } { 48 }$ 
D)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 165 } } { 24 }$ 
E)  $\frac { \ln ( 864 + 24 \sqrt { 327 } ) - \ln 2 + 12 \sqrt { 327 } } { 24 }$ 
A)  $\frac { \ln ( 864 + 24 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 327 } } { 24 }$ 
B)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) - \ln 2 + 6 \sqrt { 327 } } { 24 }$ 
C)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 165 } } { 48 }$ 
D)  $\frac { \ln ( 864 + 12 \sqrt { 327 } ) + \ln 2 + 12 \sqrt { 165 } } { 24 }$ 
E)  $\frac { \ln ( 864 + 24 \sqrt { 327 } ) - \ln 2 + 12 \sqrt { 327 } } { 24 }$

Find the arc length of the curve on the given interval. $x = t ^ { 2 } , y = 8 t , 0 \leq t \leq 4$

Find the center of the ellipse given by $16 x ^ { 2 } + y ^ { 2 } = 16$

Find the distance from the pole to the directrix for the conic $r = \frac { 42 } { 14 + \cos \theta }$

Sketch the curve represented by the parametric equations $x = e ^ { t } , y = e ^ { 3 t } + 2$

Find the area of the interior of $r = 10 + 2 \sin \theta$

Find the area of inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$ equations using the graphing utility.

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

Sketch the curve represented by the parametric equations $x = \sqrt { t } , y = t - 5$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $4 x ^ { 2 } + 4 y ^ { 2 } + 3 x + 7 y - 6 = 0$

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 \cos \theta , y = 4 \sin \theta \text {. }$

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$ (i) $x$ -axis; (ii) $y$ -axis

Find the center, foci, and vertices of the hyperbola. $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 2 ) ^ { 2 } } { 1 } = 1$

Find the vertex of the parabola given by $y ^ { 2 } = - 8 x$

Find the length of the curve $r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }$ answer to two decimal places.

Find the distance from the pole to the directrix for the conic $r = - \frac { 30 } { 15 + 31 \sin \theta }$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $y ^ { 2 } + 6 y + 4 x + 1 = 0$

Find all points of intersection of the graphs of the equations. r=1+\theta r=3\theta

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$ (i) $x$ -axis;(ii) $y$ -axis

$\text { Find } \frac { d y } { d x } \text {. }$ $x = \sqrt [ 12 ] { t }$ $y = 6 - t$

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

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

Find the arc length of the curve on the given interval. x=t2,y=8t,0≤t≤4x = t ^ { 2 } , y = 8 t , 0 \leq t \leq 4x=t2,y=8t,0≤t≤4

Find the center of the ellipse given by 16x2+y2=1616 x ^ { 2 } + y ^ { 2 } = 1616x2+y2=16

Find the distance from the pole to the directrix for the conic r=4214+cos⁡θr = \frac { 42 } { 14 + \cos \theta }r=14+cosθ42​

Sketch the curve represented by the parametric equations x=et,y=e3t+2x = e ^ { t } , y = e ^ { 3 t } + 2x=et,y=e3t+2

Find the area of the interior of r=10+2sin⁡θr = 10 + 2 \sin \thetar=10+2sinθ

Find the area of inside r=88cos⁡θ and outside r=44 by sketching the graph of the r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }r=88cosθ and outside r=44 by sketching the graph of the equations using the graphing utility.

Sketch the curve represented by the parametric equations x=t,y=t−5x = \sqrt { t } , y = t - 5x=t​,y=t−5

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 4x2+4y2+3x+7y−6=04 x ^ { 2 } + 4 y ^ { 2 } + 3 x + 7 y - 6 = 04x2+4y2+3x+7y−6=0

Find the area of the surface generated by revolving the curve about the given axis. x=t,y=6t,0≤t≤9x = t , y = 6 t , 0 \leq t \leq 9x=t,y=6t,0≤t≤9 (i) xxx -axis; (ii) yyy -axis

Find the center, foci, and vertices of the hyperbola. (x−1)216−(y+2)21=1\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 2 ) ^ { 2 } } { 1 } = 116(x−1)2​−1(y+2)2​=1

Find the vertex of the parabola given by y2=−8xy ^ { 2 } = - 8 xy2=−8x

Find the length of the curve r=2sec⁡θ over the interval 0≤θ≤π3. Round your r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }r=2secθ over the interval 0≤θ≤3π​. Round your answer to two decimal places.

Find the distance from the pole to the directrix for the conic r=−3015+31sin⁡θr = - \frac { 30 } { 15 + 31 \sin \theta }r=−15+31sinθ30​

Find the vertex, focus, and directrix of the parabola and sketch its graph. y2+6y+4x+1=0y ^ { 2 } + 6 y + 4 x + 1 = 0y2+6y+4x+1=0

Find all points of intersection of the graphs of the equations. r=1+\theta r=3\theta

Find the area of the surface generated by revolving the curve about the given axis. x=t,y=6t,0≤t≤9x = t , y = 6 t , 0 \leq t \leq 9x=t,y=6t,0≤t≤9 (i) xxx -axis;(ii) yyy -axis

Find dydx. \text { Find } \frac { d y } { d x } \text {. } Find dxdy​. x=t12x = \sqrt [ 12 ] { t }x=12t​ y=6−ty = 6 - ty=6−t

Find the arc length of the curve on the given interval. x=t,y=3t−2,0≤t≤3x = \sqrt { t } , y = 3 t - 2,0 \leq t \leq 3x=t​,y=3t−2,0≤t≤3

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

Find the arc length of the curve on the given interval. $x = t ^ { 2 } , y = 8 t , 0 \leq t \leq 4$

Find the center of the ellipse given by $16 x ^ { 2 } + y ^ { 2 } = 16$

Find the distance from the pole to the directrix for the conic $r = \frac { 42 } { 14 + \cos \theta }$

Sketch the curve represented by the parametric equations $x = e ^ { t } , y = e ^ { 3 t } + 2$

Find the area of the interior of $r = 10 + 2 \sin \theta$

Find the area of inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$ equations using the graphing utility.

Sketch the curve represented by the parametric equations $x = \sqrt { t } , y = t - 5$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $4 x ^ { 2 } + 4 y ^ { 2 } + 3 x + 7 y - 6 = 0$

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$ (i) $x$ -axis; (ii) $y$ -axis

Find the center, foci, and vertices of the hyperbola. $\frac { ( x - 1 ) ^ { 2 } } { 16 } - \frac { ( y + 2 ) ^ { 2 } } { 1 } = 1$

Find the vertex of the parabola given by $y ^ { 2 } = - 8 x$

Find the length of the curve $r = 2 \sec \theta \text { over the interval } 0 \leq \theta \leq \frac { \pi } { 3 } \text {. Round your }$ answer to two decimal places.

Find the distance from the pole to the directrix for the conic $r = - \frac { 30 } { 15 + 31 \sin \theta }$

Find the vertex, focus, and directrix of the parabola and sketch its graph. $y ^ { 2 } + 6 y + 4 x + 1 = 0$

Find the area of the surface generated by revolving the curve about the given axis. $x = t , y = 6 t , 0 \leq t \leq 9$ (i) $x$ -axis;(ii) $y$ -axis

$\text { Find } \frac { d y } { d x } \text {. }$ $x = \sqrt [ 12 ] { t }$ $y = 6 - t$

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