Question 1

Find an equation of an ellipse satisfying the given conditions: Foci (-3, 0) and (3, 0) and length of major axis 6.

Accepted Answer

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

Question 2

Which of the following best describes the parametric curve x = sec(t), y =   (t),-  $\le$ t $\le$   ?

Accepted Answer

A)  part of the parabola y = x2 - 1 lying under the line y = 1 
B)  the parabola y = x2 - 1 
C)  part of the parabola y = x2 - 1 lying above the line y = 1 
D)  part of the parabola y = x2 - 1 lying above the line y = -1 
E)  part of the parabola y = x2 - 1 lying under the line y = 1 in the first quadrant 
A)  part of the parabola y = x2 - 1 lying under the line y = 1 
B)  the parabola y = x2 - 1 
C)  part of the parabola y = x2 - 1 lying above the line y = 1 
D)  part of the parabola y = x2 - 1 lying above the line y = -1 
E)  part of the parabola y = x2 - 1 lying under the line y = 1 in the first quadrant

Question 3

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

Accepted Answer

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

Question 4

Find the points on the hyperbola x2 - y2 = 1 nearest to the point (0, 1).

Accepted Answer

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

Question 5

At what values of t does the curve x = t - sin t, y = 1 - cos t have (a) a horizontal tangent, (b) a vertical tangent, and (c) no tangent?

Accepted Answer

A)  (a) t = 2k$\pi$, k is an integer; (b) t = (2k + 1)$\pi$; (c) nowhere 
B)  (a) t = k$\pi$, k is an integer; (b) nowhere; (c) nowhere 
C)  (a) t = (2k + 1)$\pi$, k is an integer; (b) t = 2k$\pi$, k is an integer; (c) nowhere 
D)  (a) t = (2k + 1)$\pi$, k is an integer; (b) nowhere; (c) t = 2k$\pi$, k is an integer 
E)  (a) t = k$\pi$, k is an integer; (b) t = 2k$\pi$, k is an integer; (c) nowhere 
A)  (a) t = 2k$\pi$, k is an integer; (b) t = (2k + 1)$\pi$; (c) nowhere 
B)  (a) t = k$\pi$, k is an integer; (b) nowhere; (c) nowhere 
C)  (a) t = (2k + 1)$\pi$, k is an integer; (b) t = 2k$\pi$, k is an integer; (c) nowhere 
D)  (a) t = (2k + 1)$\pi$, k is an integer; (b) nowhere; (c) t = 2k$\pi$, k is an integer 
E)  (a) t = k$\pi$, k is an integer; (b) t = 2k$\pi$, k is an integer; (c) nowhere

Question 6

Which of the following is an equation of the polar graph shown in the figure below?

Accepted Answer

A)  r = 3 - 3cos($\theta$) 
B)  r = 6 cos(2$\theta$) 
C)  r = 3 +3 sin($\theta$) 
D)  r = 3 - 3sin($\theta$) 
E)  r = 3 + 3cos($\theta$) 
A)  r = 3 - 3cos($\theta$) 
B)  r = 6 cos(2$\theta$) 
C)  r = 3 +3 sin($\theta$) 
D)  r = 3 - 3sin($\theta$) 
E)  r = 3 + 3cos($\theta$)

Question 7

Find the area of the region lying inside the circle r =   sin $\theta$ and outside the lemniscate   .

Accepted Answer

A)    +   square units 
B)    -   square units 
C)    +   square units 
D)    -   square units 
E)    +   square units 
A)    +   square units 
B)    -   square units 
C)    +   square units 
D)    -   square units 
E)    +   square units

Question 8

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

Accepted Answer

A)      square units 
B)      square units 
C)      square units 
D)      square units 
E)      square units 
A)      square units 
B)      square units 
C)      square units 
D)      square units 
E)      square units

Question 9

Which of the following sets of parametric equations constitute a parametrization of the whole parabola y = x2?

Accepted Answer

A)  (a), (c), and (e) only 
B)  (a) and (e) only 
C)  (a), (b), and (c) only 
D)  all of them 
E)  none of them 
A)  (a), (c), and (e) only 
B)  (a) and (e) only 
C)  (a), (b), and (c) only 
D)  all of them 
E)  none of them

Question 10

Find the slopes of two lines tangent to the parametric curve x =   +   - 6t + 1,y =   + t - 4 at the point   on the curve.

Accepted Answer

A)  2, -3 
B)  -   ,   
C)  -6, -   
D)    , -   
E)  -   ,   
A)  2, -3 
B)  -   ,   
C)  -6, -   
D)    , -   
E)  -   ,

Question 11

Describe the plane curve represented in polar coordinates (r , $\theta$) by the equation r = 4 sin($\theta$) , $\theta$   [0 , $\theta$].

Accepted Answer

A)  a straight line through the origin of slope 4 
B)  a circle centred at (x , y) = (2 , 0) and is of radius 2 
C)  a circle centred at (x , y) = (0 , 2) and is of radius 2 
D)  a circle centred at (x , y) = (0 , 2) and is of radius 4 
E)  a circle centred at (x , y) = (2 , 0) and is of radius 4 
A)  a straight line through the origin of slope 4 
B)  a circle centred at (x , y) = (2 , 0) and is of radius 2 
C)  a circle centred at (x , y) = (0 , 2) and is of radius 2 
D)  a circle centred at (x , y) = (0 , 2) and is of radius 4 
E)  a circle centred at (x , y) = (2 , 0) and is of radius 4

Question 12

Find all values of the constant real number k so that the second degree equation   represents a pair of lines.

Accepted Answer

A)  k = -1, k =   
B)  k = 1, -   
C)  k = -   ,   
D)  -$\infty$< k < $\infty$ 
E)  k $\neq$0 
A)  k = -1, k =   
B)  k = 1, -   
C)  k = -   ,   
D)  -$\infty$< k < $\infty$ 
E)  k $\neq$0

Question 13

What do the parametric equations x = 7 cos(t) and y = 3 sin(t) describe?

Accepted Answer

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

Question 14

For the hyperbola   -   = 8x - 2y - 13, find the centre, the vertices, the foci, and the asymptotes.

Accepted Answer

A)  Centre (4, 1), Vertices (4 ±   , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5 
B)  Centre (-4, -1), Vertices (-4 ±   , -1), Foci (-4 ± 2, -1), Asymptotes x - y = -3 and x + y = -5 
C)  Centre (4, 1), Vertices (4 ±2   , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5 
D)  Centre (-4, 1), Vertices (-4 ±2   , 1), Foci (-4 ± 2, 1), Asymptotes x - y = -3 and x + y = 5 
E)  Centre (4, -1), Vertices (4 ±   , -1), Foci (4 ± 2, -1), Asymptotes x + y = 3 and x - y = 5 
A)  Centre (4, 1), Vertices (4 ±   , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5 
B)  Centre (-4, -1), Vertices (-4 ±   , -1), Foci (-4 ± 2, -1), Asymptotes x - y = -3 and x + y = -5 
C)  Centre (4, 1), Vertices (4 ±2   , 1), Foci (4 ± 2, 1), Asymptotes x - y = 3 and x + y = 5 
D)  Centre (-4, 1), Vertices (-4 ±2   , 1), Foci (-4 ± 2, 1), Asymptotes x - y = -3 and x + y = 5 
E)  Centre (4, -1), Vertices (4 ±   , -1), Foci (4 ± 2, -1), Asymptotes x + y = 3 and x - y = 5

Question 15

Determine the length of the arc r =   , 1 $\le$ $\theta$ $\le$2.

Accepted Answer

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

Question 16

Find the Cartesian equation of the unique straight line that is tangent to the cardioid   at two different points.

Accepted Answer

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

Question 17

Find the slope of the curve x = 3 csc(t), y = 2 cot(t) at the point t =   .

Accepted Answer

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

Question 18

Find the rectangular coordinates of the point with polar coordinates   .

Accepted Answer

A)  (   , 1) 
B)  (1,   ) 
C)  (   , 2) 
D)  (2,   ) 
E)  (3, 1) 
A)  (   , 1) 
B)  (1,   ) 
C)  (   , 2) 
D)  (2,   ) 
E)  (3, 1)

Question 19

Where is the curve x = ln t, y = et concave upward?

Accepted Answer

A) at all points on the curve B) at all points corresponding to values of t satisfying t > 1 C) at all points corresponding to values of t satisfying 0 < t < 1 D) at all points corresponding to values of t satisfying 0 < t $\le$ 1 E) nowhere A) at all points on the curve B) at all points corresponding to values of t satisfying t > 1 C) at all points corresponding to values of t satisfying 0 < t < 1 D) at all points corresponding to values of t satisfying 0 < t $\le$ 1 E) nowhere

Question 20

Sketch the graph of the conic section given by the polar equation r =   forε = 0, 1, 0.7, and 2.Identify each of the conic sections.

Accepted Answer

@#IMG-DLM& 
For ε = 0 , the conic section is a cir

Find an equation of an ellipse satisfying the given conditions: Foci (-3, 0) and (3, 0) and length of major axis 6.

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

Find the points on the hyperbola x² - y² = 1 nearest to the point (0, 1).

At what values of t does the curve x = t - sin t, y = 1 - cos t have (a) a horizontal tangent, (b) a vertical tangent, and (c) no tangent?

Which of the following is an equation of the polar graph shown in the figure below?

Find the area of the region lying inside the circle r = $Find the area of the region lying inside the circle r = sin \theta and outside the lemniscate .$ sin $\theta$ and outside the lemniscate $Find the area of the region lying inside the circle r = sin \theta and outside the lemniscate .$ .

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

Which of the following sets of parametric equations constitute a parametrization of the whole parabola y = x²?

Find the slopes of two lines tangent to the parametric curve x = + - 6t + 1,y = + t - 4 at the point on the curve.

Describe the plane curve represented in polar coordinates (r , $\theta$ ) by the equation r = 4 sin( $\theta$ ) , $\theta$ $Describe the plane curve represented in polar coordinates (r , \theta ) by the equation r = 4 sin( \theta ) , \theta [0 , \theta ].$ [0 , $\theta$ ].

Find all values of the constant real number k so that the second degree equation represents a pair of lines.

What do the parametric equations x = 7 cos(t) and y = 3 sin(t) describe?

For the hyperbola - = 8x - 2y - 13, find the centre, the vertices, the foci, and the asymptotes.

Determine the length of the arc r = $Determine the length of the arc r = , 1 \le \theta \le 2.$ , 1 $\le$ $\theta$ $\le$ 2.

Find the Cartesian equation of the unique straight line that is tangent to the cardioid at two different points.

Find the slope of the curve x = 3 csc(t), y = 2 cot(t) at the point t = .

Find the rectangular coordinates of the point with polar coordinates .

Where is the curve x = ln t, y = e^t concave upward?

Sketch the graph of the conic section given by the polar equation r = forε = 0, 1, 0.7, and 2.Identify each of the conic sections.

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 an equation of an ellipse satisfying the given conditions: Foci (-3, 0) and (3, 0) and length of major axis 6.

Which of the following best describes the parametric curve x = sec(t), y = (t),- ≤\le≤ t ≤\le≤ ?

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

Find the points on the hyperbola x2 - y2 = 1 nearest to the point (0, 1).

At what values of t does the curve x = t - sin t, y = 1 - cos t have (a) a horizontal tangent, (b) a vertical tangent, and (c) no tangent?

Which of the following is an equation of the polar graph shown in the figure below?

Find the area of the region lying inside the circle r = sin θ\thetaθ and outside the lemniscate .

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

Which of the following sets of parametric equations constitute a parametrization of the whole parabola y = x2?

Find the slopes of two lines tangent to the parametric curve x = + - 6t + 1,y = + t - 4 at the point on the curve.

Describe the plane curve represented in polar coordinates (r , θ\thetaθ ) by the equation r = 4 sin( θ\thetaθ ) , θ\thetaθ [0 , θ\thetaθ ].

Find all values of the constant real number k so that the second degree equation represents a pair of lines.

What do the parametric equations x = 7 cos(t) and y = 3 sin(t) describe?

For the hyperbola - = 8x - 2y - 13, find the centre, the vertices, the foci, and the asymptotes.

Determine the length of the arc r = , 1 ≤\le≤ θ\thetaθ ≤\le≤ 2.

Find the Cartesian equation of the unique straight line that is tangent to the cardioid at two different points.

Find the slope of the curve x = 3 csc(t), y = 2 cot(t) at the point t = .

Find the rectangular coordinates of the point with polar coordinates .

Where is the curve x = ln t, y = et concave upward?

Sketch the graph of the conic section given by the polar equation r = forε = 0, 1, 0.7, and 2.Identify each of the conic sections.

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 the points on the hyperbola x² - y² = 1 nearest to the point (0, 1).

Find the area of the region lying inside the circle r = $Find the area of the region lying inside the circle r = sin \theta and outside the lemniscate .$ sin $\theta$ and outside the lemniscate $Find the area of the region lying inside the circle r = sin \theta and outside the lemniscate .$ .

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

Which of the following sets of parametric equations constitute a parametrization of the whole parabola y = x²?

Describe the plane curve represented in polar coordinates (r , $\theta$ ) by the equation r = 4 sin( $\theta$ ) , $\theta$ $Describe the plane curve represented in polar coordinates (r , \theta ) by the equation r = 4 sin( \theta ) , \theta [0 , \theta ].$ [0 , $\theta$ ].

Determine the length of the arc r = $Determine the length of the arc r = , 1 \le \theta \le 2.$ , 1 $\le$ $\theta$ $\le$ 2.

Where is the curve x = ln t, y = e^t concave upward?