Question 1

Find the points of intersection of the graphs of the equations. 
$\begin{array} { l } 
r = \frac { \theta } { 1.6 } \
r = 1.6
\end{array}$

Accepted Answer

A)  $( \sqrt { 1.6 } , - 1.6 ) , ( \sqrt { 1.6 } , 1.6 )$ 
B)  $( \sqrt { 1.6 } , 1.6 )$ 
C)  $( 1.6,2.56 ) , ( - 1.6 , - 2.56 )$ 
D)  $( 1.6,1.6 )$ 
E)  $( - 1.6,2.56 )$ 
A)  $( \sqrt { 1.6 } , - 1.6 ) , ( \sqrt { 1.6 } , 1.6 )$ 
B)  $( \sqrt { 1.6 } , 1.6 )$ 
C)  $( 1.6,2.56 ) , ( - 1.6 , - 2.56 )$ 
D)  $( 1.6,1.6 )$ 
E)  $( - 1.6,2.56 )$

Question 2

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

Accepted Answer

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

Question 3

Find the vertices of the ellipse given by 
$16 x ^ { 2 } + 9 y ^ { 2 } + 160 x - 18 y + 265 = 0 .$

Accepted Answer

A)  $( 5,3 ) , ( 5 , - 5 )$ 
B)  $( - 2,1 ) , ( - 2,1 )$ 
C)  $( 5,4 ) , ( 5 , - 2 )$ 
D)  $( - 1,1 ) , ( - 1,1 )$ 
E)  $( - 5,5 ) , ( - 5 , - 3 )$ 
A)  $( 5,3 ) , ( 5 , - 5 )$ 
B)  $( - 2,1 ) , ( - 2,1 )$ 
C)  $( 5,4 ) , ( 5 , - 2 )$ 
D)  $( - 1,1 ) , ( - 1,1 )$ 
E)  $( - 5,5 ) , ( - 5 , - 3 )$

Question 4

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$

Accepted Answer

A)  horizontal tangents: $( 9 , - 54 ) , ( 3,54 )$, vertical tangent: none 
B)  horizontal tangent: none, vertical tangents: $( 9 , - 54 ) , ( 3,54 )$ 
C)  horizontal tangents: $( 9,54 ) , ( 3,54 )$, vertical tangent: none 
D)  horizontal tangent: none, vertical tangents: $( 9,54 ) , ( 3,54 )$ 
E)  horizontal tangent: none, vertical tangent: none 
A)  horizontal tangents: $( 9 , - 54 ) , ( 3,54 )$, vertical tangent: none 
B)  horizontal tangent: none, vertical tangents: $( 9 , - 54 ) , ( 3,54 )$ 
C)  horizontal tangents: $( 9,54 ) , ( 3,54 )$, vertical tangent: none 
D)  horizontal tangent: none, vertical tangents: $( 9,54 ) , ( 3,54 )$ 
E)  horizontal tangent: none, vertical tangent: none

Question 5

Find a polar equation for the hyperbola with its focus at the pole, eccentricity $e = \frac { 7 } { 6 }$ and directrix $y = 7$.

Accepted Answer

A)  $r = \frac { 49 } { 6 + 7 \sin \theta }$ 
B)  $r = \frac { - 49 } { 6 + 7 \sin \theta }$ 
C)  $r = \frac { 49 } { 6 - 7 \cos \theta }$ 
D)  $r = \frac { - 49 } { 6 - 7 \cos \theta }$ 
E)  $r = \frac { - 49 } { 6 - 7 \sin \theta }$ 
A)  $r = \frac { 49 } { 6 + 7 \sin \theta }$ 
B)  $r = \frac { - 49 } { 6 + 7 \sin \theta }$ 
C)  $r = \frac { 49 } { 6 - 7 \cos \theta }$ 
D)  $r = \frac { - 49 } { 6 - 7 \cos \theta }$ 
E)  $r = \frac { - 49 } { 6 - 7 \sin \theta }$

Question 6

Match the equation with its graph. $$( x + 3 ) ^ { 2 } = - 2 ( y - 3 )$$

Accepted Answer

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

Question 7

Find the eccentricity of the polar equation $$r ( 14 + \sin \theta ) = 28$$

Accepted Answer

A)  28 
B)  14 
C)  $\frac { 1 } { 28 }$ 
D)  $\frac { 1 } { 14 }$ 
E)  7 
A)  28 
B)  14 
C)  $\frac { 1 } { 28 }$ 
D)  $\frac { 1 } { 14 }$ 
E)  7

Question 8

Write an integral that represents the area of the shaded region for $r = \cos 2 \theta \text { as }$ shown in the figure. Do not evaluate the integral.

Accepted Answer

A) $\int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
B) $\int _ { - 5 \pi / 2 } ^ { - 3 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
C)  $\frac { 1 } { 2 } \int _ { 3 \pi / 2 } ^ { 5 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
D)  $\frac { 1 } { 2 } \int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
E)  $\frac { 1 } { 2 } \int _ { 3 \pi / 4 } ^ { 5 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
A) $\int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
B) $\int _ { - 5 \pi / 2 } ^ { - 3 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
C)  $\frac { 1 } { 2 } \int _ { 3 \pi / 2 } ^ { 5 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
D)  $\frac { 1 } { 2 } \int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
E)  $\frac { 1 } { 2 } \int _ { 3 \pi / 4 } ^ { 5 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$

Question 9

$\text { Find the second derivative } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text { of the parametric equations } x = 4 t , y = 8 t - 3 \text {. }$ 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 10

Find the length of the curve over the given interval. $$r = 9 + 9 \sin \theta , 0 \leq \theta \leq 2 \pi$$

Accepted Answer

A)  72 
B)  54 
C)  9 
D)  36 
E)  18 
A)  72 
B)  54 
C)  9 
D)  36 
E)  18

Question 11

Match the equation with its graph. $$y ^ { 2 } = 6 x$$

Accepted Answer

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

Question 12

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. $\begin{array} { l } 
x = 16 + 8 \cos \theta \
y = - 4 + 4 \sin \theta
\end{array}$

Accepted Answer

A)  $y = x ^ { 4 } + 1$
  
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 64 } = 1$
  
C)  $y = \frac { 5 } { 4 } ( x + 1 ) + 1$
  
D)  $y = x ^ { 2 } - 4 , x \geq 0$
  
E)  none of the above 
A)  $y = x ^ { 4 } + 1$
  
B)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 64 } = 1$
  
C)  $y = \frac { 5 } { 4 } ( x + 1 ) + 1$
  
D)  $y = x ^ { 2 } - 4 , x \geq 0$
  
E)  none of the above

Question 13

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

Question 14

Find all points (if any) of horizontal and vertical tangency to the curve $x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$

Accepted Answer

A)  horizontal tangents: $( 7 , - 2 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
B)  horizontal tangents: $( 10 , - 3 ) , ( 4 , - 3 )$, vertical tangents: $( 7 , - 2 ) , ( 7 , - 4 )$ 
C)  horizontal tangent: $( 7 , - 2 )$, vertical tangent: $( 10 , - 3 )$ 
D)  horizontal tangents: $( 7 , - 4 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
E)  horizontal tangent: $( 10 , - 3 )$, vertical tangent: $( 7 , - 2 )$ 
A)  horizontal tangents: $( 7 , - 2 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
B)  horizontal tangents: $( 10 , - 3 ) , ( 4 , - 3 )$, vertical tangents: $( 7 , - 2 ) , ( 7 , - 4 )$ 
C)  horizontal tangent: $( 7 , - 2 )$, vertical tangent: $( 10 , - 3 )$ 
D)  horizontal tangents: $( 7 , - 4 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
E)  horizontal tangent: $( 10 , - 3 )$, vertical tangent: $( 7 , - 2 )$

Question 15

Find the area of the region lying between the loops of $r = 2 - 4 \cos \theta$

Accepted Answer

A)  $4 ( 2 \sqrt { 3 } - 3 \pi )$ 
B)  $4 ( \sqrt { 3 } + 2 \pi )$ 
C)  $8 \sqrt { 3 } - \pi$ 
D)  $4 ( 3 \sqrt { 3 } + \pi )$ 
E)  $12 \sqrt { 3 } + \pi$ 
A)  $4 ( 2 \sqrt { 3 } - 3 \pi )$ 
B)  $4 ( \sqrt { 3 } + 2 \pi )$ 
C)  $8 \sqrt { 3 } - \pi$ 
D)  $4 ( 3 \sqrt { 3 } + \pi )$ 
E)  $12 \sqrt { 3 } + \pi$

Question 16

Find the area of the surface formed by revolving about the polar axis the following curve over the given interval. $r = 6 \cos \theta , 0 \leq \theta \leq \frac { \pi } { 2 }$

Accepted Answer

A)  $36 \pi$ 
B)  $13 \pi$ 
C)  $6 \pi$ 
D)  $12 \pi$ 
E)  $14 \pi$ 
A)  $36 \pi$ 
B)  $13 \pi$ 
C)  $6 \pi$ 
D)  $12 \pi$ 
E)  $14 \pi$

Question 17

Uranus moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 2,876,769,540 kilometers, and the eccentricity is 0.0444. Find the minimum Distance (perihelion) of Uranus from the sun. Round your answer to nearest kilometer.

Accepted Answer

A) 3,004,498,108 km 
B) 2,749,040,972 km 
C) 2,819,234,149 km 
D) 127,728,568 km 
E) 2,934,304,931 km 
A) 3,004,498,108 km 
B) 2,749,040,972 km 
C) 2,819,234,149 km 
D) 127,728,568 km 
E) 2,934,304,931 km

Question 18

Find all points (if any) of horizontal and vertical tangency to the curve $$x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$$

Accepted Answer

A)  horizontal tangents: $( 7 , - 2 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
B)  horizontal tangents: $( 10 , - 3 ) , ( 4 , - 3 )$, vertical tangents: $( 7 , - 2 ) , ( 7 , - 4 )$ 
C)  horizontal tangent: $( 7 , - 2 )$, vertical tangent: $( 10 , - 3 )$ 
D)  horizontal tangents: $( 7 , - 4 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
E)  horizontal tangent: $( 10 , - 3 )$, vertical tangent: $( 7 , - 2 )$ 
A)  horizontal tangents: $( 7 , - 2 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
B)  horizontal tangents: $( 10 , - 3 ) , ( 4 , - 3 )$, vertical tangents: $( 7 , - 2 ) , ( 7 , - 4 )$ 
C)  horizontal tangent: $( 7 , - 2 )$, vertical tangent: $( 10 , - 3 )$ 
D)  horizontal tangents: $( 7 , - 4 ) , ( 7 , - 4 )$, vertical tangents: $( 10 , - 3 ) , ( 4 , - 3 )$ 
E)  horizontal tangent: $( 10 , - 3 )$, vertical tangent: $( 7 , - 2 )$

Question 19

The parametric equations for the path of a projectile launched at a height h feet above the ground, at an angle $\theta$ with the horizontal and having an initial velocity of $v _ { 0 }$ feet per second is given by $x = \left( v _ { 0 } \cos \theta \right) t$ and $y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$. The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 2 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour as shown in the figure. Write a set of parametric equations for the path of the ball.

Accepted Answer

A) $x = \left( \frac { 440 } { 3 } \cos \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t$ 
B)  $x = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t , y = \left( \frac { 440 } { 3 } \cos \theta \right) t$ 
C) $x = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t - 16 t ^ { 2 } , y = \left( \frac { 440 } { 3 } \cos \theta \right) t$ 
D)  $x = \left( \frac { 440 } { 3 } \sin \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \cos \theta \right) t - 16 t ^ { 2 }$ 
E) $x = \left( \frac { 440 } { 3 } \cos \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t - 16 t ^ { 2 }$ 
A) $x = \left( \frac { 440 } { 3 } \cos \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t$ 
B)  $x = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t , y = \left( \frac { 440 } { 3 } \cos \theta \right) t$ 
C) $x = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t - 16 t ^ { 2 } , y = \left( \frac { 440 } { 3 } \cos \theta \right) t$ 
D)  $x = \left( \frac { 440 } { 3 } \sin \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \cos \theta \right) t - 16 t ^ { 2 }$ 
E) $x = \left( \frac { 440 } { 3 } \cos \theta \right) t , y = 2 + \left( \frac { 440 } { 3 } \sin \theta \right) t - 16 t ^ { 2 }$

Question 20

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$

Accepted Answer

A)  horizontal tangents: $( 9 , - 54 ) , ( 3,54 )$, vertical tangent: none 
B)  horizontal tangent: none, vertical tangents: $( 9 , - 54 ) , ( 3,54 )$ 
C)  horizontal tangents: $( 9,54 ) , ( 3,54 )$, vertical tangent: none 
D)  horizontal tangent: none, vertical tangents: $( 9,54 ) , ( 3,54 )$ 
E)  horizontal tangent: none, vertical tangent: none 
A)  horizontal tangents: $( 9 , - 54 ) , ( 3,54 )$, vertical tangent: none 
B)  horizontal tangent: none, vertical tangents: $( 9 , - 54 ) , ( 3,54 )$ 
C)  horizontal tangents: $( 9,54 ) , ( 3,54 )$, vertical tangent: none 
D)  horizontal tangent: none, vertical tangents: $( 9,54 ) , ( 3,54 )$ 
E)  horizontal tangent: none, vertical tangent: none

Find the points of intersection of the graphs of the equations. r= r=1.6

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

Find the vertices of the ellipse given by $16 x ^ { 2 } + 9 y ^ { 2 } + 160 x - 18 y + 265 = 0 .$

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$

Find a polar equation for the hyperbola with its focus at the pole, eccentricity $e = \frac { 7 } { 6 }$ and directrix $y = 7$ .

Match the equation with its graph. $( x + 3 ) ^ { 2 } = - 2 ( y - 3 )$

Find the eccentricity of the polar equation $r ( 14 + \sin \theta ) = 28$

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

Find the length of the curve over the given interval. $r = 9 + 9 \sin \theta , 0 \leq \theta \leq 2 \pi$

Match the equation with its graph. $y ^ { 2 } = 6 x$

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. x=16+8\theta y=-4+4\theta

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

Find all points (if any) of horizontal and vertical tangency to the curve $x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$

Find the area of the region lying between the loops of $r = 2 - 4 \cos \theta$

Find the area of the surface formed by revolving about the polar axis the following curve over the given interval. $r = 6 \cos \theta , 0 \leq \theta \leq \frac { \pi } { 2 }$

Uranus moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 2,876,769,540 kilometers, and the eccentricity is 0.0444. Find the minimum Distance (perihelion) of Uranus from the sun. Round your answer to nearest kilometer.

Find all points (if any) of horizontal and vertical tangency to the curve $x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$

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 points of intersection of the graphs of the equations. r= r=1.6

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 5x2−7y2+2x+2y−7=05 x ^ { 2 } - 7 y ^ { 2 } + 2 x + 2 y - 7 = 05x2−7y2+2x+2y−7=0

Find the vertices of the ellipse given by 16x2+9y2+160x−18y+265=0.16 x ^ { 2 } + 9 y ^ { 2 } + 160 x - 18 y + 265 = 0 .16x2+9y2+160x−18y+265=0.

Find all points (if any) of horizontal and vertical tangency to the curve x=t+6,y=t3−27tx = t + 6 , y = t ^ { 3 } - 27 tx=t+6,y=t3−27t

Find a polar equation for the hyperbola with its focus at the pole, eccentricity e=76e = \frac { 7 } { 6 }e=67​ and directrix y=7y = 7y=7 .

Match the equation with its graph. (x+3)2=−2(y−3)( x + 3 ) ^ { 2 } = - 2 ( y - 3 )(x+3)2=−2(y−3)

Find the eccentricity of the polar equation r(14+sin⁡θ)=28r ( 14 + \sin \theta ) = 28r(14+sinθ)=28

Write an integral that represents the area of the shaded region for r=cos⁡2θ as r = \cos 2 \theta \text { as }r=cos2θ as shown in the figure. Do not evaluate the integral.

Find the length of the curve over the given interval. r=9+9sin⁡θ,0≤θ≤2πr = 9 + 9 \sin \theta , 0 \leq \theta \leq 2 \pir=9+9sinθ,0≤θ≤2π

Match the equation with its graph. y2=6xy ^ { 2 } = 6 xy2=6x

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter. x=16+8\theta y=-4+4\theta

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

Find all points (if any) of horizontal and vertical tangency to the curve x=7+3cos⁡θ,y=−3+sin⁡θx = 7 + 3 \cos \theta , y = - 3 + \sin \thetax=7+3cosθ,y=−3+sinθ

Find the area of the region lying between the loops of r=2−4cos⁡θr = 2 - 4 \cos \thetar=2−4cosθ

Find the area of the surface formed by revolving about the polar axis the following curve over the given interval. r=6cos⁡θ,0≤θ≤π2r = 6 \cos \theta , 0 \leq \theta \leq \frac { \pi } { 2 }r=6cosθ,0≤θ≤2π​

Uranus moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 2,876,769,540 kilometers, and the eccentricity is 0.0444. Find the minimum Distance (perihelion) of Uranus from the sun. Round your answer to nearest kilometer.

Find all points (if any) of horizontal and vertical tangency to the curve x=7+3cos⁡θ,y=−3+sin⁡θx = 7 + 3 \cos \theta , y = - 3 + \sin \thetax=7+3cosθ,y=−3+sinθ

Find all points (if any) of horizontal and vertical tangency to the curve x=t+6,y=t3−27tx = t + 6 , y = t ^ { 3 } - 27 tx=t+6,y=t3−27t

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

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

Find the vertices of the ellipse given by $16 x ^ { 2 } + 9 y ^ { 2 } + 160 x - 18 y + 265 = 0 .$

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$

Find a polar equation for the hyperbola with its focus at the pole, eccentricity $e = \frac { 7 } { 6 }$ and directrix $y = 7$ .

Match the equation with its graph. $( x + 3 ) ^ { 2 } = - 2 ( y - 3 )$

Find the eccentricity of the polar equation $r ( 14 + \sin \theta ) = 28$

Find the length of the curve over the given interval. $r = 9 + 9 \sin \theta , 0 \leq \theta \leq 2 \pi$

Match the equation with its graph. $y ^ { 2 } = 6 x$

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

Find all points (if any) of horizontal and vertical tangency to the curve $x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$

Find the area of the region lying between the loops of $r = 2 - 4 \cos \theta$

Find the area of the surface formed by revolving about the polar axis the following curve over the given interval. $r = 6 \cos \theta , 0 \leq \theta \leq \frac { \pi } { 2 }$

Find all points (if any) of horizontal and vertical tangency to the curve $x = 7 + 3 \cos \theta , y = - 3 + \sin \theta$

Find all points (if any) of horizontal and vertical tangency to the curve $x = t + 6 , y = t ^ { 3 } - 27 t$