Question 1

Accepted Answer

+   =

Question 2

Find   at the highest point on the cycloid x = a$\theta$
 - a sin $\theta$
, y = a - a cos $\theta$
.

Accepted Answer

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

Question 3

Determine the area above the x-axis and under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).

Accepted Answer

A)  2$\pi$   square units 
B)  3$\pi$   square units 
C)  4$\pi$  square units 
D)  6$\pi$  square units 
E)  5$\pi$   square units 
A)  2$\pi$   square units 
B)  3$\pi$   square units 
C)  4$\pi$  square units 
D)  6$\pi$  square units 
E)  5$\pi$   square units B

Question 4

Find a rectangular equation equivalent to the polar equation r = tan$\theta$
.

Accepted Answer

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

Question 5

Find parametric equations of the plane curve C given by 4x2 + 9y2 - 8x -32 = 0.

Accepted Answer

A)  x(t) = 1 + 2cos(t), y(t) = 3sin(t), t    [0 , 2$\pi$] 
B)  x(t) = - 1 + 3cos(t), y(t) = 2sin(t),    [0 , 2$\pi$] 
C)  x(t) = - 1 + 2cos(t), y(t) = 3sin(t),    [0 , 2$\pi$] 
D)  x(t) = 1 + 3cos(t), y(t) = 2sin(t), t    [0 , 2$\pi$] 
E)    
A)  x(t) = 1 + 2cos(t), y(t) = 3sin(t), t    [0 , 2$\pi$] 
B)  x(t) = - 1 + 3cos(t), y(t) = 2sin(t),    [0 , 2$\pi$] 
C)  x(t) = - 1 + 2cos(t), y(t) = 3sin(t),    [0 , 2$\pi$] 
D)  x(t) = 1 + 3cos(t), y(t) = 2sin(t), t    [0 , 2$\pi$] 
E)

Question 6

Find the arc length of r = sec($\theta$), 0 $\le$ $\theta$ $\le$   .

Accepted Answer

A)    
B)    
C)  2 
D)  1 
E)  2   
A)    
B)    
C)  2 
D)  1 
E)  2

Question 7

Find the length of x = ln sin $\theta$, y = $\theta$,   $\le$ $\theta$ $\le$  .

Accepted Answer

A)  ln (4 -   ) units 
B)  ln (3 +   ) units 
C)  ln (2 +   ) units 
D)  ln (1 + 2   ) units 
E)  ln (1 - 3   ) units 
A)  ln (4 -   ) units 
B)  ln (3 +   ) units 
C)  ln (2 +   ) units 
D)  ln (1 + 2   ) units 
E)  ln (1 - 3   ) units

Question 8

Where does the curve x = 2   - 5, y =   + t have a tangent line that is perpendicular to the line   ?

Accepted Answer

A)    and (-3, 2) 
B)    and (3, -2) 
C)    and (3, -2) 
D)    and (-3, 2) 
E)    and (-3, -2) 
A)    and (-3, 2) 
B)    and (3, -2) 
C)    and (3, -2) 
D)    and (-3, 2) 
E)    and (-3, -2)

Question 9

Parametrize the curve y =   + 3x using its slope m as the parameter.

Accepted Answer

A)  x =   , y =   
B)  x =   , y =   
C)  x =   , y =   
D)  x =   , y =   
E)  x =   , y =   
A)  x =   , y =   
B)  x =   , y =   
C)  x =   , y =   
D)  x =   , y =   
E)  x =   , y =

Question 10

Find a rectangular equation equivalent to the polar equation $\theta$
 =   .

Accepted Answer

A)  y = x 
B)  y =   x 
C)  y =   x 
D)  y = 2x 
E)  y = 3x 
A)  y = x 
B)  y =   x 
C)  y =   x 
D)  y = 2x 
E)  y = 3x

Question 11

Find the slope of the curve x =   sin 2t, y =   cos 3t at t = 0.

Accepted Answer

A)  -   
B)  -   
C)  0 
D)    
E)    
A)  -   
B)  -   
C)  0 
D)    
E)

Question 12

Find the length of the curve x =   t, y =   , 0 $\le$ t $\le$ 1.

Accepted Answer

A)  1 +   ln(   + 1) units 
B)    +   ln(   + 1) units 
C)    -   ln(   + 1) units 
D)  1 -   ln(   + 1) units 
E)  1 +   ln(   ) units 
A)  1 +   ln(   + 1) units 
B)    +   ln(   + 1) units 
C)    -   ln(   + 1) units 
D)  1 -   ln(   + 1) units 
E)  1 +   ln(   ) units

Question 13

Sketch the polar curve r2 = 9 sin(3θ).

Accepted Answer

Question 14

Convert x2 + y2 = xy to polar coordinates.

Accepted Answer

A)  r2 =   
B)  r =   
C)  r =   
D)  r2 =   
E)  r2 =   
A)  r2 =   
B)  r =   
C)  r =   
D)  r2 =   
E)  r2 =

Question 15

Find the area of the surface generated by rotating x =     , y =   , 0$\le$ t $\le$1 about the y-axis.

Accepted Answer

A)  4 $\pi$(5   - 8) square units 
B)  4 $\pi$(5   + 8) square units 
C)  2$\pi$(5   - 8) square units 
D)  2 $\pi$(5   + 8) square units 
E)   $\pi$(5   - 8) square units 
A)  4 $\pi$(5   - 8) square units 
B)  4 $\pi$(5   + 8) square units 
C)  2$\pi$(5   - 8) square units 
D)  2 $\pi$(5   + 8) square units 
E)   $\pi$(5   - 8) square units

Question 16

A plane curve C is given parametrically by the functions x (t) = cosh(t) - 2, y(t) = sinh(t), t    R.
A Cartesian equation of the curve C is given by:

Accepted Answer

A)    -   = 1 
B)    +   = 1 
C)    -   =1 
D)    +   = 5 
E)    -   = 5 
A)    -   = 1 
B)    +   = 1 
C)    -   =1 
D)    +   = 5 
E)    -   = 5

Question 17

Find an equation of a parabola satisfying the given conditions Focus (4, 1) and directrixx = -2.

Accepted Answer

A)    = 12x 
B)    = 12(x - 1) 
C)    = 12(x + 1) 
D)    = -12(x - 1) 
E)    = 12x + 1 
A)    = 12x 
B)    = 12(x - 1) 
C)    = 12(x + 1) 
D)    = -12(x - 1) 
E)    = 12x + 1

Question 18

Find an equation of a hyperbola with vertices (-1, 3) and (-1, 7) and   = 4.

Accepted Answer

A)    -   = 1 
B)    -   = 1 
C)    -   = 1 
D)    -   = 1 
E)    -   = 1 
A)    -   = 1 
B)    -   = 1 
C)    -   = 1 
D)    -   = 1 
E)    -   = 1

Question 19

A circle passes through both foci of an ellipse and is tangent to the ellipse at two points. Find the eccentricity of the ellipse.

Accepted Answer

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

Question 20

Find the angle at which the parabolas y2 = 4x + 4 and y2 = -6x + 9 intersect at each of their intersection points.

Accepted Answer

A)  90º at each intersection point 
B)  120º at each intersection point 
C)  60º at each intersection point 
D)  75º at each intersection point 
E)  50º at each intersection point 
A)  90º at each intersection point 
B)  120º at each intersection point 
C)  60º at each intersection point 
D)  75º at each intersection point 
E)  50º at each intersection point

Find $Find at the highest point on the cycloid x = a \theta - a sin \theta , y = a - a cos \theta .$ at the highest point on the cycloid x = a $\theta$ - a sin $\theta$ , y = a - a cos $\theta$ .

Determine the area above the x-axis and under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).

Find a rectangular equation equivalent to the polar equation r = tan $\theta$ .

Find parametric equations of the plane curve C given by 4x² + 9y² - 8x -32 = 0.

Find the arc length of r = sec( $\theta$ ), 0 $\le$ $\theta$ $\le$ $Find the arc length of r = sec( \theta ), 0 \le \theta \le .$ .

Find the length of x = ln sin $\theta$ , y = $\theta$ , $Find the length of x = ln sin \theta , y = \theta , \le \theta \le .$ $\le$ $\theta$ $\le$ $Find the length of x = ln sin \theta , y = \theta , \le \theta \le .$ .

Where does the curve x = 2 - 5, y = + t have a tangent line that is perpendicular to the line ?

Parametrize the curve y = + 3x using its slope m as the parameter.

Find a rectangular equation equivalent to the polar equation $\theta$ = $Find a rectangular equation equivalent to the polar equation \theta = .$ .

Find the slope of the curve x = sin 2t, y = cos 3t at t = 0.

Find the length of the curve x = $Find the length of the curve x = t, y = , 0 \le t \le 1.$ t, y = $Find the length of the curve x = t, y = , 0 \le t \le 1.$ , 0 $\le$ t $\le$ 1.

Sketch the polar curve r² = 9 sin(3θ).

Convert x² + y² = xy to polar coordinates.

A plane curve C is given parametrically by the functions x (t) = cosh(t) - 2, y(t) = sinh(t), t R. A Cartesian equation of the curve C is given by:

Find an equation of a parabola satisfying the given conditions Focus (4, 1) and directrixx = -2.

Find an equation of a hyperbola with vertices (-1, 3) and (-1, 7) and = 4.

A circle passes through both foci of an ellipse and is tangent to the ellipse at two points. Find the eccentricity of the ellipse.

Find the angle at which the parabolas y² = 4x + 4 and y² = -6x + 9 intersect at each of their intersection points.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 9: Conics, Parametric Curves, and Polar Curves

Find at the highest point on the cycloid x = a θ\thetaθ - a sin θ\thetaθ , y = a - a cos θ\thetaθ .

Determine the area above the x-axis and under one arch of the cycloid x = a(t - sin t), y = a(1 - cos t).

Find a rectangular equation equivalent to the polar equation r = tan θ\thetaθ .

Find parametric equations of the plane curve C given by 4x2 + 9y2 - 8x -32 = 0.

Find the arc length of r = sec( θ\thetaθ ), 0 ≤\le≤ θ\thetaθ ≤\le≤ .

Find the length of x = ln sin θ\thetaθ , y = θ\thetaθ , ≤\le≤ θ\thetaθ ≤\le≤ .

Where does the curve x = 2 - 5, y = + t have a tangent line that is perpendicular to the line ?

Parametrize the curve y = + 3x using its slope m as the parameter.

Find a rectangular equation equivalent to the polar equation θ\thetaθ = .

Find the slope of the curve x = sin 2t, y = cos 3t at t = 0.

Find the length of the curve x = t, y = , 0 ≤\le≤ t ≤\le≤ 1.

Sketch the polar curve r2 = 9 sin(3θ).

Convert x2 + y2 = xy to polar coordinates.

Find the area of the surface generated by rotating x = , y = , 0 ≤\le≤ t ≤\le≤ 1 about the y-axis.

A plane curve C is given parametrically by the functions x (t) = cosh(t) - 2, y(t) = sinh(t), t R. A Cartesian equation of the curve C is given by:

Find an equation of a parabola satisfying the given conditions Focus (4, 1) and directrixx = -2.

Find an equation of a hyperbola with vertices (-1, 3) and (-1, 7) and = 4.

A circle passes through both foci of an ellipse and is tangent to the ellipse at two points. Find the eccentricity of the ellipse.

Find the angle at which the parabolas y2 = 4x + 4 and y2 = -6x + 9 intersect at each of their intersection points.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find $Find at the highest point on the cycloid x = a \theta - a sin \theta , y = a - a cos \theta .$ at the highest point on the cycloid x = a $\theta$ - a sin $\theta$ , y = a - a cos $\theta$ .

Find a rectangular equation equivalent to the polar equation r = tan $\theta$ .

Find parametric equations of the plane curve C given by 4x² + 9y² - 8x -32 = 0.

Find the arc length of r = sec( $\theta$ ), 0 $\le$ $\theta$ $\le$ $Find the arc length of r = sec( \theta ), 0 \le \theta \le .$ .

Find the length of x = ln sin $\theta$ , y = $\theta$ , $Find the length of x = ln sin \theta , y = \theta , \le \theta \le .$ $\le$ $\theta$ $\le$ $Find the length of x = ln sin \theta , y = \theta , \le \theta \le .$ .

Find a rectangular equation equivalent to the polar equation $\theta$ = $Find a rectangular equation equivalent to the polar equation \theta = .$ .

Find the length of the curve x = $Find the length of the curve x = t, y = , 0 \le t \le 1.$ t, y = $Find the length of the curve x = t, y = , 0 \le t \le 1.$ , 0 $\le$ t $\le$ 1.

Sketch the polar curve r² = 9 sin(3θ).

Convert x² + y² = xy to polar coordinates.

Find the angle at which the parabolas y² = 4x + 4 and y² = -6x + 9 intersect at each of their intersection points.