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)  $16 ( \sqrt { 2 } + \ln \sqrt { 2 } + \ln ( 2 + \sqrt { 2 } ) )$ D

Question 2

Determine the t intervals on which the curve $x = 7 t ^ { 2 } , y = \ln t \text { is concave downward }$ 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: $- \infty < t < \infty$ 
D)  concave upward: $0 < t < \infty$ 
E)  concave downward: $0 < t < \infty$ 
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: $- \infty < t < \infty$ 
D)  concave upward: $0 < t < \infty$ 
E)  concave downward: $0 < t < \infty$ E

Question 3

Identify the graph for the polar equation $r = \frac { 5 } { 2 + \cos \theta }$

Accepted Answer

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

Question 4

Use the result, "the set of parametric equations for the line passing through $\left( x _ { 1 } , y _ { 1 } \right)$  and $\left( x _ { 2 } , y _ { 2 } \right)$ is $x = x _ { 1 } + t \left( x _ { 2 } - x _ { 1 } \right) , y = y _ { 1 } + t \left( y _ { 2 } - y _ { 1 } \right) ^ { n }$ to find a set of parametric equations for the line passing through $( 0,0 )$ and $( 9 , - 8 )$.

Accepted Answer

A)  $x = - \frac { 9 } { 8 } t + 9 , y = - \frac { 8 } { 9 } t + 8$ 
B) $x = - \frac { 9 } { 8 } t + 9 , y = - \frac { 8 } { 9 } t - 8$ 
C)  $x = 9 + t , y = - 8 + t$ 
D)  $x = 9 t , y = - 8 t$ 
E)  $x = 9 - t , y = - 8 - t$ 
A)  $x = - \frac { 9 } { 8 } t + 9 , y = - \frac { 8 } { 9 } t + 8$ 
B) $x = - \frac { 9 } { 8 } t + 9 , y = - \frac { 8 } { 9 } t - 8$ 
C)  $x = 9 + t , y = - 8 + t$ 
D)  $x = 9 t , y = - 8 t$ 
E)  $x = 9 - t , y = - 8 - t$

Question 5

$\text { Find } \frac { d y } { d x } \text {. }$ 
$x = 7 e ^ { \theta }$
$y = e ^ { - \frac { \theta } { 7 } }$

Accepted Answer

A) $\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { d } { 7 } }$ 
B)  $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$ 
C) $\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$ 
D)  $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$ 
E) $\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$ 
A) $\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { d } { 7 } }$ 
B)  $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$ 
C) $\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$ 
D)  $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$ 
E) $\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$

Question 6

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$7 x ^ { 2 } + 10 y ^ { 2 } + 7 x + 4 y - 5 = 0$$

Accepted Answer

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

Question 7

Find the area of the inner loop of $r = 3 + 6 \sin \theta$

Accepted Answer

A) 
$27 \pi + \frac { 27 \sqrt { 2 } } { 2 }$ 
B) 
$9 \pi - \frac { 27 \sqrt { 3 } } { 4 }$ 
C) 
$27 \pi - \frac { 27 \sqrt { 3 } } { 4 }$ 
D) 
$9 \pi - \frac { 27 \sqrt { 3 } } { 2 }$ 
E) 
$9 \pi + \frac { 27 \sqrt { 2 } } { 2 }$ 
A) 
$27 \pi + \frac { 27 \sqrt { 2 } } { 2 }$ 
B) 
$9 \pi - \frac { 27 \sqrt { 3 } } { 4 }$ 
C) 
$27 \pi - \frac { 27 \sqrt { 3 } } { 4 }$ 
D) 
$9 \pi - \frac { 27 \sqrt { 3 } } { 2 }$ 
E) 
$9 \pi + \frac { 27 \sqrt { 2 } } { 2 }$

Question 8

Find the directrix of the parabola given by $$y ^ { 2 } - 44 y - 484 x = 0$$

Accepted Answer

A)  $x = - 123$ 
B)  $x = - 122$ 
C)  $x = 120$ 
D)  $x = 121$ 
E)  $x = 122$ 
A)  $x = - 123$ 
B)  $x = - 122$ 
C)  $x = 120$ 
D)  $x = 121$ 
E)  $x = 122$

Question 9

$\text { Find } \frac { d y } { d x }$ $x = 7 e ^ { \theta }$
$y = e ^ { - \frac { \theta } { 7 } }$

Accepted Answer

A) $\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { \theta } { 7 } }$ 
B)  $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$ 
C) $\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$ 
D)  $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$ 
E)  $\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$ 
A) $\frac { d y } { d x } = \frac { 1 } { 49 } e ^ { - \frac { \theta } { 7 } }$ 
B)  $\frac { d y } { d x } = 49 e ^ { - 7 \theta }$ 
C) $\frac { d y } { d x } = - \frac { 1 } { 7 } e ^ { - \frac { 8 } { 7 } \theta }$ 
D)  $\frac { d y } { d x } = 7 e ^ { - 7 \theta }$ 
E)  $\frac { d y } { d x } = - \frac { 1 } { 49 } e ^ { - \frac { 8 } { 7 } \theta }$

Question 10

$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$

Accepted Answer

A)  $\frac { 13 \pi } { 12 }$ 
B)  $\frac { 2197 \pi } { 12 }$ 
C)  $\frac { 169 \pi } { 12 }$ 
D)  $\frac { 169 \pi } { 24 }$ 
E)  $\frac { 13 \pi } { 24 }$ 
A)  $\frac { 13 \pi } { 12 }$ 
B)  $\frac { 2197 \pi } { 12 }$ 
C)  $\frac { 169 \pi } { 12 }$ 
D)  $\frac { 169 \pi } { 24 }$ 
E)  $\frac { 13 \pi } { 24 }$

Question 11

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 3 + \cos \theta }$

Accepted Answer

A)  eccentricity: $\frac { 1 } { 3 }$
distance from pole to directrix: 2
The graph is an ellipse.
  
B)  eccentricity: 3
distance from pole to directrix: $\frac { 1 } { 2 } \quad$
The graph is an ellipse.
  
C)  eccentricity: $\frac { 1 } { 3 }$
distance from pole to directrix: $\frac { 1 } { 2 }$
The graph is an ellipse.
  
D)  eccentricity: 3
distance from pole to directrix: 2
The graph is a hyperbola.
  
E)  eccentricity: 3
 distance from pole to directrix: $\frac { 1 } { 2 }$
The graph is a hyperbola.
  
A)  eccentricity: $\frac { 1 } { 3 }$
distance from pole to directrix: 2
The graph is an ellipse.
  
B)  eccentricity: 3
distance from pole to directrix: $\frac { 1 } { 2 } \quad$
The graph is an ellipse.
  
C)  eccentricity: $\frac { 1 } { 3 }$
distance from pole to directrix: $\frac { 1 } { 2 }$
The graph is an ellipse.
  
D)  eccentricity: 3
distance from pole to directrix: 2
The graph is a hyperbola.
  
E)  eccentricity: 3
 distance from pole to directrix: $\frac { 1 } { 2 }$
The graph is a hyperbola.

Question 12

$\text { Identify the conic for the polar equation } r = \frac { 2 e } { 1 + e \cos \theta } \text { when } e = 1.8 \text {. }$

Accepted Answer

A) hyperbola 
B) ellipse 
C) parabola 
A) hyperbola 
B) ellipse 
C) parabola

Question 13

Find a set of parametric equations for the rectangular equation $$y = 3 x - 5$$

Accepted Answer

A)  $x = t - 1 , y = 3 t + 2$ 
B)  $x = t , y = 3 t - 5$ 
C)  $x = t , y = 4 t ^ { 2 } - t - 5$ 
D)  $x = t + 1 , y = 3 t - 2$ 
E)  $x = t - 1 , y = 4 t ^ { 2 } - 9 t - 6$ 
A)  $x = t - 1 , y = 3 t + 2$ 
B)  $x = t , y = 3 t - 5$ 
C)  $x = t , y = 4 t ^ { 2 } - t - 5$ 
D)  $x = t + 1 , y = 3 t - 2$ 
E)  $x = t - 1 , y = 4 t ^ { 2 } - 9 t - 6$

Question 14

$\text { Identify the graph for the polar equation } r = \frac { 6 } { 1 - \cos \theta } \text {. }$

Accepted Answer

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

Question 15

$\text { Identify any points at which the cycloid } x = \theta + \sin \theta , y = 1 - \cos \theta \text { is not smooth. }$

Accepted Answer

A)  not smooth when $\theta = ( 2 n + 1 ) \pi$ 
B)  smooth everywhere 
C)  not smooth when $\theta = ( 2 n ) \pi$ 
D)  not smooth when $\theta = \frac { 1 } { 2 } ( 2 n + 1 ) \pi$ 
E)  not smooth when $\theta = n \pi$ 
A)  not smooth when $\theta = ( 2 n + 1 ) \pi$ 
B)  smooth everywhere 
C)  not smooth when $\theta = ( 2 n ) \pi$ 
D)  not smooth when $\theta = \frac { 1 } { 2 } ( 2 n + 1 ) \pi$ 
E)  not smooth when $\theta = n \pi$

Question 16

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 17

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. $x = t$
$y = \ln t$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 25 } = 1$
  
B)  $y = e ^ { x }$
  
C)  $y = x ^ { 2 } - 1 , x \geq 0$
  
D)  $y = \ln x$
  
E)  $y = x + 1$
  
A)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 25 } = 1$
  
B)  $y = e ^ { x }$
  
C)  $y = x ^ { 2 } - 1 , x \geq 0$
  
D)  $y = \ln x$
  
E)  $y = x + 1$

Question 18

Find a polar equation for the ellipse with its focus at the pole, eccentricity $e = \frac { 5 } { 6 } , \text { and }$ directrix $y = - 10$.

Accepted Answer

A)  $r = \frac { - 50 } { 6 + 5 \sin \theta }$ 
B)  $r = \frac { - 50 } { 6 - 5 \cos \theta }$ 
C)  $r = \frac { 50 } { 6 - 5 \sin \theta }$ 
D)  $r = \frac { - 50 } { 6 + 5 \cos \theta }$ 
E)  $r = \frac { 50 } { 6 - 5 \cos \theta }$ 
A)  $r = \frac { - 50 } { 6 + 5 \sin \theta }$ 
B)  $r = \frac { - 50 } { 6 - 5 \cos \theta }$ 
C)  $r = \frac { 50 } { 6 - 5 \sin \theta }$ 
D)  $r = \frac { - 50 } { 6 + 5 \cos \theta }$ 
E)  $r = \frac { 50 } { 6 - 5 \cos \theta }$

Question 19

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 7 + 5 \sin \theta }$

Accepted Answer

A)  eccentricity: $\frac { 7 } { 5 }$
distance from pole to directrix: $\frac { 5 } { 7 } \quad$ 
The graph is an ellipse. 
  
B)  eccentricity: $\frac { 5 } { 7 }$
distance from pole to directrix: $\frac { 7 } { 5 } \quad$
The graph is a hyperbola.
  
C)  eccentricity: $\frac { 7 } { 5 }$
distance from pole to directrix: $\frac { 5 } { 7 }$
The graph is a hyperbola.
  
D)  eccentricity: $\frac { 5 } { 7 }$
distance from pole to directrix: $\frac { 7 } { 5 }$
$\quad$ The graph is an ellipse.
  
E)  eccentricity: $\frac { 7 } { 5 }$
 distance from pole to directrix: $\frac { 7 } { 5 }$
 $\quad$ The graph is an ellipse.
  
A)  eccentricity: $\frac { 7 } { 5 }$
distance from pole to directrix: $\frac { 5 } { 7 } \quad$ 
The graph is an ellipse. 
  
B)  eccentricity: $\frac { 5 } { 7 }$
distance from pole to directrix: $\frac { 7 } { 5 } \quad$
The graph is a hyperbola.
  
C)  eccentricity: $\frac { 7 } { 5 }$
distance from pole to directrix: $\frac { 5 } { 7 }$
The graph is a hyperbola.
  
D)  eccentricity: $\frac { 5 } { 7 }$
distance from pole to directrix: $\frac { 7 } { 5 }$
$\quad$ The graph is an ellipse.
  
E)  eccentricity: $\frac { 7 } { 5 }$
 distance from pole to directrix: $\frac { 7 } { 5 }$
 $\quad$ The graph is an ellipse.

Question 20

$\text { Find } \frac { d y } { d x } \text {. }$ 
$\begin{array} { l } 
x = t ^ { 2 } \
y = 10 - 10 t
\end{array}$

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)  $\frac { d y } { d x } = - \frac { t } { 10 }$

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

Determine the t intervals on which the curve $x = 7 t ^ { 2 } , y = \ln t \text { is concave downward }$ or concave upward.

Identify the graph for the polar equation $r = \frac { 5 } { 2 + \cos \theta }$

$\text { Find } \frac { d y } { d x } \text {. }$ $x = 7 e ^ { \theta }$ $y = e ^ { - \frac { \theta } { 7 } }$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $7 x ^ { 2 } + 10 y ^ { 2 } + 7 x + 4 y - 5 = 0$

Find the area of the inner loop of $r = 3 + 6 \sin \theta$

Find the directrix of the parabola given by $y ^ { 2 } - 44 y - 484 x = 0$

$\text { Find } \frac { d y } { d x }$ $x = 7 e ^ { \theta }$ $y = e ^ { - \frac { \theta } { 7 } }$

$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 3 + \cos \theta }$

$\text { Identify the conic for the polar equation } r = \frac { 2 e } { 1 + e \cos \theta } \text { when } e = 1.8 \text {. }$

Find a set of parametric equations for the rectangular equation $y = 3 x - 5$

$\text { Identify the graph for the polar equation } r = \frac { 6 } { 1 - \cos \theta } \text {. }$

$\text { Identify any points at which the cycloid } x = \theta + \sin \theta , y = 1 - \cos \theta \text { is not smooth. }$

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

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. $x = t$ $y = \ln t$

Find a polar equation for the ellipse with its focus at the pole, eccentricity $e = \frac { 5 } { 6 } , \text { and }$ directrix $y = - 10$ .

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 7 + 5 \sin \theta }$

$\text { Find } \frac { d y } { d x } \text {. }$ x= y=10-10t

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

Determine the t intervals on which the curve x=7t2,y=ln⁡t is concave downward x = 7 t ^ { 2 } , y = \ln t \text { is concave downward }x=7t2,y=lnt is concave downward or concave upward.

Identify the graph for the polar equation r=52+cos⁡θr = \frac { 5 } { 2 + \cos \theta }r=2+cosθ5​

Find dydx. \text { Find } \frac { d y } { d x } \text {. } Find dxdy​. x=7eθx = 7 e ^ { \theta }x=7eθ y=e−θ7y = e ^ { - \frac { \theta } { 7 } }y=e−7θ​

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 7x2+10y2+7x+4y−5=07 x ^ { 2 } + 10 y ^ { 2 } + 7 x + 4 y - 5 = 07x2+10y2+7x+4y−5=0

Find the area of the inner loop of r=3+6sin⁡θr = 3 + 6 \sin \thetar=3+6sinθ

Find the directrix of the parabola given by y2−44y−484x=0y ^ { 2 } - 44 y - 484 x = 0y2−44y−484x=0

Find dydx\text { Find } \frac { d y } { d x } Find dxdy​ x=7eθx = 7 e ^ { \theta }x=7eθ y=e−θ7y = e ^ { - \frac { \theta } { 7 } }y=e−7θ​

Find the area of one petal of r=13cos⁡3θ. \text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. } Find the area of one petal of r=13cos3θ.

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. r=23+cos⁡θr = \frac { 2 } { 3 + \cos \theta }r=3+cosθ2​

Identify the conic for the polar equation r=2e1+ecos⁡θ when e=1.8. \text { Identify the conic for the polar equation } r = \frac { 2 e } { 1 + e \cos \theta } \text { when } e = 1.8 \text {. } Identify the conic for the polar equation r=1+ecosθ2e​ when e=1.8.

Find a set of parametric equations for the rectangular equation y=3x−5y = 3 x - 5y=3x−5

Identify the graph for the polar equation r=61−cos⁡θ. \text { Identify the graph for the polar equation } r = \frac { 6 } { 1 - \cos \theta } \text {. } Identify the graph for the polar equation r=1−cosθ6​.

Identify any points at which the cycloid x=θ+sin⁡θ,y=1−cos⁡θ is not smooth. \text { Identify any points at which the cycloid } x = \theta + \sin \theta , y = 1 - \cos \theta \text { is not smooth. } Identify any points at which the cycloid x=θ+sinθ,y=1−cosθ is not smooth.

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

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. x=tx = tx=t y=ln⁡ty = \ln ty=lnt

Find a polar equation for the ellipse with its focus at the pole, eccentricity e=56, and e = \frac { 5 } { 6 } , \text { and }e=65​, and directrix y=−10y = - 10y=−10 .

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. r=27+5sin⁡θr = \frac { 2 } { 7 + 5 \sin \theta }r=7+5sinθ2​

Find dydx. \text { Find } \frac { d y } { d x } \text {. } Find dxdy​. x= y=10-10t

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. $x = t ^ { 2 } , y = 8 t , 0 \leq t \leq 4$

Determine the t intervals on which the curve $x = 7 t ^ { 2 } , y = \ln t \text { is concave downward }$ or concave upward.

Identify the graph for the polar equation $r = \frac { 5 } { 2 + \cos \theta }$

$\text { Find } \frac { d y } { d x } \text {. }$ $x = 7 e ^ { \theta }$ $y = e ^ { - \frac { \theta } { 7 } }$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $7 x ^ { 2 } + 10 y ^ { 2 } + 7 x + 4 y - 5 = 0$

Find the area of the inner loop of $r = 3 + 6 \sin \theta$

Find the directrix of the parabola given by $y ^ { 2 } - 44 y - 484 x = 0$

$\text { Find } \frac { d y } { d x }$ $x = 7 e ^ { \theta }$ $y = e ^ { - \frac { \theta } { 7 } }$

$\text { Find the area of one petal of } r = 13 \cos 3 \theta \text {. }$

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 3 + \cos \theta }$

$\text { Identify the conic for the polar equation } r = \frac { 2 e } { 1 + e \cos \theta } \text { when } e = 1.8 \text {. }$

Find a set of parametric equations for the rectangular equation $y = 3 x - 5$

$\text { Identify the graph for the polar equation } r = \frac { 6 } { 1 - \cos \theta } \text {. }$

$\text { Identify any points at which the cycloid } x = \theta + \sin \theta , y = 1 - \cos \theta \text { is not smooth. }$

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

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. $x = t$ $y = \ln t$

Find a polar equation for the ellipse with its focus at the pole, eccentricity $e = \frac { 5 } { 6 } , \text { and }$ directrix $y = - 10$ .

Find the eccentricity and distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $r = \frac { 2 } { 7 + 5 \sin \theta }$

$\text { Find } \frac { d y } { d x } \text {. }$ x= y=10-10t