Question 1

Find the arc length of the curve $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Accepted Answer

A) 287.453 
B) 191.635 
C) 193.606 
D) 66.480 
E) 99.721 
A) 287.453 
B) 191.635 
C) 193.606 
D) 66.480 
E) 99.721

Question 2

Find the arc length of the curve $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Accepted Answer

A) 287.453 
B) 191.635 
C) 193.606 
D) 66.480 
E) 99.721 
A) 287.453 
B) 191.635 
C) 193.606 
D) 66.480 
E) 99.721

Question 3

The path of a projectile is modeled by the parametric equations $x = \left( 85 \cos 60 ^ { \circ } \right) t \text { and }$ $y = \left( 85 \sin 60 ^ { \circ } \right) t - 16 t ^ { 2 }$ where $x$ and $y$ are measured in feet. Use a graphing utility to approximate the range of the projectile. Round your answer to two decimal places.

Accepted Answer

A)  $312.85 \mathrm { ft }$ 
B)  $521.42 \mathrm { ft }$ 
C)  $131.09 \mathrm { ft }$ 
D)  $391.06 \mathrm { ft }$ 
E)  $195.53 \mathrm { ft }$ 
A)  $312.85 \mathrm { ft }$ 
B)  $521.42 \mathrm { ft }$ 
C)  $131.09 \mathrm { ft }$ 
D)  $391.06 \mathrm { ft }$ 
E)  $195.53 \mathrm { ft }$

Question 4

Find the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$
(i) $x$-axis;(ii) $y$-axis

Accepted Answer

A) (i) $\frac { 27 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
B)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 54 } { 5 } \pi$ 
C)  (i) $\frac { 27 } { 10 } \pi$;(ii) $\frac { 27 } { 5 } \pi$ 
D)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
E)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 108 } { 5 } \pi$ 
A) (i) $\frac { 27 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
B)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 54 } { 5 } \pi$ 
C)  (i) $\frac { 27 } { 10 } \pi$;(ii) $\frac { 27 } { 5 } \pi$ 
D)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 27 } { 5 } \pi$ 
E)  (i) $\frac { 54 } { 5 } \pi$; (ii) $\frac { 108 } { 5 } \pi$

Question 5

$\text { Find an equation of the tangent line at a point } ( 0,0 ) \text { on the curve }$ $x = t ^ { 2 } - 4 , y = t ^ { 2 } - 2 t$

Accepted Answer

A)  $2 y + x = 0$ 
B)  $y - x = 0$ 
C)  $y - 2 x = 0$ 
D)  $y + 2 x = 0$ 
E)  $2 y - x = 0$ 
A)  $2 y + x = 0$ 
B)  $y - x = 0$ 
C)  $y - 2 x = 0$ 
D)  $y + 2 x = 0$ 
E)  $2 y - x = 0$

Question 6

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 7

$\text { Find } \frac { d y } { d x } \text { and } \frac { d ^ { 2 } y } { d t ^ { 2 } } \text { if possible, and find the slope and concavity (if possible) at the }$ 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 8

Suppose a cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 240 meters apart and 40 meters above the roadway as shown in the figure Given below. The cable touches the roadway midway between the towers. Find the length of the Parabolic supporting cable. Round your answer to two decimal places.

Accepted Answer

A)  $280.79$ meters 
B)  $327.58$ meters 
C)  $291.18$ meters 
D)  $245.71$ meters 
E)  $256.75$ meters 
A)  $280.79$ meters 
B)  $327.58$ meters 
C)  $291.18$ meters 
D)  $245.71$ meters 
E)  $256.75$ meters

Question 9

Find the eccentricity of the ellipse given by $$( x + 3 ) ^ { 2 } + \frac { ( y + 1 ) ^ { 2 } } { 1 / 16 } = 1$$

Accepted Answer

A)  $\frac { \sqrt { 15 } } { 4 }$ 
B)  $\frac { 17 } { 16 }$ 
C)  $\frac { 15 } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 16 }$ 
E)  $\frac { \sqrt { 17 } } { 16 }$ 
A)  $\frac { \sqrt { 15 } } { 4 }$ 
B)  $\frac { 17 } { 16 }$ 
C)  $\frac { 15 } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 16 }$ 
E)  $\frac { \sqrt { 17 } } { 16 }$

Question 10

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$downward or concave upward.

Accepted Answer

A)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < \infty$ 
B)  concave downward: $0 < t < \infty$; concave upward: $- \infty < t < 0$ 
C)  concave downward: $- 6 < t < 3$; concave upward: $0 < t < 3$ 
D)  concave downward: $- 6 < t < 0 ;$ concave upward: $0 < t < \infty$ 
E)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < 3$ 
A)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < \infty$ 
B)  concave downward: $0 < t < \infty$; concave upward: $- \infty < t < 0$ 
C)  concave downward: $- 6 < t < 3$; concave upward: $0 < t < 3$ 
D)  concave downward: $- 6 < t < 0 ;$ concave upward: $0 < t < \infty$ 
E)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < 3$

Question 11

Earth moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the maximum Distance (aphelion) of Earth from the sun. Round your answer to nearest kilometer.

Accepted Answer

A) 146,606,040 km 
B) 2,498,287 km 
C) 152,589,960 km 
D) 152,096,287 km 
E) 147,099,713 km 
A) 146,606,040 km 
B) 2,498,287 km 
C) 152,589,960 km 
D) 152,096,287 km 
E) 147,099,713 km

Question 12

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$y ^ { 2 } + 9 y + 2 x - 4 = 0$$

Accepted Answer

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

Question 13

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$ downward or concave upward.

Accepted Answer

A)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < \infty$ 
B)  concave downward: $0 < t < \infty$; concave upward: $- \infty < t < 0$ 
C)  concave downward: $- 6 < t < 3$; concave upward: $0 < t < 3$ 
D)  concave downward: $- 6 < t < 0$; concave upward: $0 < t < \infty$ 
E)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < 3$ 
A)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < \infty$ 
B)  concave downward: $0 < t < \infty$; concave upward: $- \infty < t < 0$ 
C)  concave downward: $- 6 < t < 3$; concave upward: $0 < t < 3$ 
D)  concave downward: $- 6 < t < 0$; concave upward: $0 < t < \infty$ 
E)  concave downward: $- \infty < t < 0$; concave upward: $0 < t < 3$

Question 14

Use the result, "the set of parametric equations for the ellipse is $x = h + a \cos \theta , y = k + b \sin \theta$ " to find a set of parametric equations for the ellipse with vertices $( 3,11 )$ and $( 3,1 )$ and with foci at $( 3,9 )$ and $( 3,3 )$.

Accepted Answer

A)  $x = 4 \sin \theta , y = 5 \cos \theta$ 
B)  $x = 6 + 5 \cos \theta , y = 3 + 4 \sin \theta$ 
C)  $x = 5 \cos \theta , y = 4 \sin \theta$ 
D)  $x = 3 - 5 \cos \theta , y = 6 - 4 \sin \theta$ 
E)  $x = 3 + 5 \cos \theta , y = 6 + 4 \sin \theta$ 
A)  $x = 4 \sin \theta , y = 5 \cos \theta$ 
B)  $x = 6 + 5 \cos \theta , y = 3 + 4 \sin \theta$ 
C)  $x = 5 \cos \theta , y = 4 \sin \theta$ 
D)  $x = 3 - 5 \cos \theta , y = 6 - 4 \sin \theta$ 
E)  $x = 3 + 5 \cos \theta , y = 6 + 4 \sin \theta$

Question 15

$\text { Find an equation of the hyperbola with vertices } ( 0 , - 5 ) , ( 0,5 ) \text { and asymptotes }$ $y = \pm \frac { 1 } { 6 } x$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 900 } = 1$ 
B)  $\frac { y ^ { 2 } } { 25 } + \frac { x ^ { 2 } } { 900 } = 1$ 
C)  $\frac { y ^ { 2 } } { 30 } - \frac { x ^ { 2 } } { 5 } = 1$ 
D)  $\frac { y ^ { 2 } } { 5 } + \frac { x ^ { 2 } } { 30 } = 1$ 
E)  $\frac { y ^ { 2 } } { 5 } - \frac { x ^ { 2 } } { 30 } = 1$ 
A)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 900 } = 1$ 
B)  $\frac { y ^ { 2 } } { 25 } + \frac { x ^ { 2 } } { 900 } = 1$ 
C)  $\frac { y ^ { 2 } } { 30 } - \frac { x ^ { 2 } } { 5 } = 1$ 
D)  $\frac { y ^ { 2 } } { 5 } + \frac { x ^ { 2 } } { 30 } = 1$ 
E)  $\frac { y ^ { 2 } } { 5 } - \frac { x ^ { 2 } } { 30 } = 1$

Question 16

Find an equation of the hyperbola with vertices $( - 7,0 ) , ( 7,0 ) \text { and asymptotes }$ $y = \pm 2 x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 14 } - \frac { y ^ { 2 } } { 7 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 196 } = 1$ 
C)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 196 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } - \frac { y ^ { 2 } } { 49 } = 1$ 
E)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 14 } = 1$ 
A)  $\frac { x ^ { 2 } } { 14 } - \frac { y ^ { 2 } } { 7 } = 1$ 
B)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 196 } = 1$ 
C)  $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 196 } = 1$ 
D)  $\frac { x ^ { 2 } } { 196 } - \frac { y ^ { 2 } } { 49 } = 1$ 
E)  $\frac { x ^ { 2 } } { 7 } - \frac { y ^ { 2 } } { 14 } = 1$

Question 17

Find the focus of the parabola given by $$( x + 4 ) + ( y - 2 ) ^ { 2 } = 0$$

Accepted Answer

A)  $( 4 , - 2 )$ 
B)  $\left( \frac { 17 } { 4 } , - 2 \right)$ 
C)  $( - 4,2 )$ 
D)  $\left( - \frac { 17 } { 4 } , 2 \right)$ 
E)  $\left( 2 , - \frac { 9 } { 4 } \right)$ 
A)  $( 4 , - 2 )$ 
B)  $\left( \frac { 17 } { 4 } , - 2 \right)$ 
C)  $( - 4,2 )$ 
D)  $\left( - \frac { 17 } { 4 } , 2 \right)$ 
E)  $\left( 2 , - \frac { 9 } { 4 } \right)$

Question 18

Find a polar equation for the hyperbola with its focus at the pole and vertices $( 20,0 ) , ( 100,0 )$

Accepted Answer

A)  $r = \frac { 100 } { 2 + 3 \cos \theta }$ 
B)  $r = \frac { 60 } { 3 + 5 \cos \theta }$ 
C)  $r = \frac { 50 } { 3 + 5 \cos \theta }$ 
D)  $r = \frac { 100 } { 3 + 5 \cos \theta }$ 
E)  $r = \frac { 50 } { 2 + 3 \cos \theta }$ 
A)  $r = \frac { 100 } { 2 + 3 \cos \theta }$ 
B)  $r = \frac { 60 } { 3 + 5 \cos \theta }$ 
C)  $r = \frac { 50 } { 3 + 5 \cos \theta }$ 
D)  $r = \frac { 100 } { 3 + 5 \cos \theta }$ 
E)  $r = \frac { 50 } { 2 + 3 \cos \theta }$

Question 19

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)  $\frac { \ln ( 864 + 24 \sqrt { 327 } ) - \ln 2 + 12 \sqrt { 327 } } { 24 }$

Question 20

Find the foci of the ellipse given by $$\frac { ( x - 4 ) ^ { 2 } } { 144 } + \frac { ( y - 8 ) ^ { 2 } } { 400 } = 1$$

Accepted Answer

A)  $( 20,8 ) , ( - 12,8 )$ 
B)  $( 20,28 ) , ( 20 , - 4 )$ 
C)  $( 36,12 ) , ( 4,12 )$ 
D)  $( 4,8 ) , ( 12,20 )$ 
E)  $( 4,24 ) , ( 4 , - 8 )$ 
A)  $( 20,8 ) , ( - 12,8 )$ 
B)  $( 20,28 ) , ( 20 , - 4 )$ 
C)  $( 36,12 ) , ( 4,12 )$ 
D)  $( 4,8 ) , ( 12,20 )$ 
E)  $( 4,24 ) , ( 4 , - 8 )$

Find the arc length of the curve $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Find the arc length of the curve $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Find the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$ (i) $x$ -axis;(ii) $y$ -axis

$\text { Find an equation of the tangent line at a point } ( 0,0 ) \text { on the curve }$ $x = t ^ { 2 } - 4 , y = t ^ { 2 } - 2 t$

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

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

Find the eccentricity of the ellipse given by $( x + 3 ) ^ { 2 } + \frac { ( y + 1 ) ^ { 2 } } { 1 / 16 } = 1$

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$ downward or concave upward.

Earth moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the maximum Distance (aphelion) of Earth from the sun. Round your answer to nearest kilometer.

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 9 y + 2 x - 4 = 0$

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$ downward or concave upward.

Use the result, "the set of parametric equations for the ellipse is $x = h + a \cos \theta , y = k + b \sin \theta$ " to find a set of parametric equations for the ellipse with vertices $( 3,11 )$ and $( 3,1 )$ and with foci at $( 3,9 )$ and $( 3,3 )$ .

$\text { Find an equation of the hyperbola with vertices } ( 0 , - 5 ) , ( 0,5 ) \text { and asymptotes }$ $y = \pm \frac { 1 } { 6 } x$

Find an equation of the hyperbola with vertices $( - 7,0 ) , ( 7,0 ) \text { and asymptotes }$ $y = \pm 2 x$

Find the focus of the parabola given by $( x + 4 ) + ( y - 2 ) ^ { 2 } = 0$

Find a polar equation for the hyperbola with its focus at the pole and vertices $( 20,0 ) , ( 100,0 )$

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

Find the foci of the ellipse given by $\frac { ( x - 4 ) ^ { 2 } } { 144 } + \frac { ( y - 8 ) ^ { 2 } } { 400 } = 1$

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 x=2t2,y=3t3 on the interval 1≤t≤4. Round x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }x=2t2,y=3t3 on the interval 1≤t≤4. Round your answer to three decimal places.

Find the arc length of the curve x=2t2,y=3t3 on the interval 1≤t≤4. Round x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }x=2t2,y=3t3 on the interval 1≤t≤4. Round your answer to three decimal places.

Find the area of the surface generated by revolving the curve about the given axis. x=3cos⁡3θ,y=3sin⁡3θ,0≤θ≤π/2x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2x=3cos3θ,y=3sin3θ,0≤θ≤π/2 (i) xxx -axis;(ii) yyy -axis

Find an equation of the tangent line at a point (0,0) on the curve \text { Find an equation of the tangent line at a point } ( 0,0 ) \text { on the curve } Find an equation of the tangent line at a point (0,0) on the curve x=t2−4,y=t2−2tx = t ^ { 2 } - 4 , y = t ^ { 2 } - 2 tx=t2−4,y=t2−2t

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

Find the eccentricity of the ellipse given by (x+3)2+(y+1)21/16=1( x + 3 ) ^ { 2 } + \frac { ( y + 1 ) ^ { 2 } } { 1 / 16 } = 1(x+3)2+1/16(y+1)2​=1

Determine the t intervals on which the curve x=6t2,y=t3−3t is concave x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }x=6t2,y=t3−3t is concave downward or concave upward.

Earth moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the maximum Distance (aphelion) of Earth from the sun. Round your answer to nearest kilometer.

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. y2+9y+2x−4=0y ^ { 2 } + 9 y + 2 x - 4 = 0y2+9y+2x−4=0

Determine the t intervals on which the curve x=6t2,y=t3−3t is concave x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }x=6t2,y=t3−3t is concave downward or concave upward.

Find an equation of the hyperbola with vertices (−7,0),(7,0) and asymptotes ( - 7,0 ) , ( 7,0 ) \text { and asymptotes }(−7,0),(7,0) and asymptotes y=±2xy = \pm 2 xy=±2x

Find the focus of the parabola given by (x+4)+(y−2)2=0( x + 4 ) + ( y - 2 ) ^ { 2 } = 0(x+4)+(y−2)2=0

Find a polar equation for the hyperbola with its focus at the pole and vertices (20,0),(100,0)( 20,0 ) , ( 100,0 )(20,0),(100,0)

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

Find the foci of the ellipse given by (x−4)2144+(y−8)2400=1\frac { ( x - 4 ) ^ { 2 } } { 144 } + \frac { ( y - 8 ) ^ { 2 } } { 400 } = 1144(x−4)2​+400(y−8)2​=1

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 $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Find the arc length of the curve $x = 2 t ^ { 2 } , y = 3 t ^ { 3 } \text { on the interval } 1 \leq t \leq 4 \text {. Round }$ your answer to three decimal places.

Find the area of the surface generated by revolving the curve about the given axis. $x = 3 \cos ^ { 3 } \theta , y = 3 \sin ^ { 3 } \theta , 0 \leq \theta \leq \pi / 2$ (i) $x$ -axis;(ii) $y$ -axis

$\text { Find an equation of the tangent line at a point } ( 0,0 ) \text { on the curve }$ $x = t ^ { 2 } - 4 , y = t ^ { 2 } - 2 t$

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

Find the eccentricity of the ellipse given by $( x + 3 ) ^ { 2 } + \frac { ( y + 1 ) ^ { 2 } } { 1 / 16 } = 1$

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$ downward or concave upward.

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 9 y + 2 x - 4 = 0$

Determine the t intervals on which the curve $x = 6 t ^ { 2 } , y = t ^ { 3 } - 3 t \text { is concave }$ downward or concave upward.

Find an equation of the hyperbola with vertices $( - 7,0 ) , ( 7,0 ) \text { and asymptotes }$ $y = \pm 2 x$

Find the focus of the parabola given by $( x + 4 ) + ( y - 2 ) ^ { 2 } = 0$

Find a polar equation for the hyperbola with its focus at the pole and vertices $( 20,0 ) , ( 100,0 )$

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

Find the foci of the ellipse given by $\frac { ( x - 4 ) ^ { 2 } } { 144 } + \frac { ( y - 8 ) ^ { 2 } } { 400 } = 1$