Question 1

Find the equation of the tangent line to the curve at the given t.
X = cos 3t, y = 3 sin 5t at t =   .

Accepted Answer

A)  x + y = 2 
B)  y = -   
C)  x = -1 
D)  x + y + 1 +   = 0 
E)  x = 1 
A)  x + y = 2 
B)  y = -   
C)  x = -1 
D)  x + y + 1 +   = 0 
E)  x = 1

Question 2

Accepted Answer

A)  10 units 
B)  15 units 
C)  20 units 
D)  25 units 
E)  16 units 
A)  10 units 
B)  15 units 
C)  20 units 
D)  25 units 
E)  16 units

Question 3

Find the centre, the foci, and the asymptotes of the hyperbola 4   - 9   -16x - 54y = 101.

Accepted Answer

A)  Centre (2, -3), Foci (2 ±   , -3), Asymptotes   = ±   
B)  Centre (-2, 3), Foci (-2 ±   , 3), Asymptotes   = ±   
C)  Centre (2, -3), Foci (2 ±   , -3), Asymptotes   = ±   
D)  Centre (2, 3), Foci (2 ±   , 3), Asymptotes   = ±   
E)  Centre (-2, 3), Foci (-2 ±   , 3), Asymptotes   = ±   
A)  Centre (2, -3), Foci (2 ±   , -3), Asymptotes   = ±   
B)  Centre (-2, 3), Foci (-2 ±   , 3), Asymptotes   = ±   
C)  Centre (2, -3), Foci (2 ±   , -3), Asymptotes   = ±   
D)  Centre (2, 3), Foci (2 ±   , 3), Asymptotes   = ±   
E)  Centre (-2, 3), Foci (-2 ±   , 3), Asymptotes   = ±

Question 4

Convert to Cartesian coordinates the polar equation r2 =   .

Accepted Answer

A)    -   = 1 
B)    -   = 1 
C)    -   = 36 
D)    +   = 36 
E)    +   = 36 
A)    -   = 1 
B)    -   = 1 
C)    -   = 36 
D)    +   = 36 
E)    +   = 36

Question 5

Determine the coordinates of the points where the curve x =   + 2t, y = 2   + 7 has (a) a horizontal tangent and (b) a vertical tangent.

Accepted Answer

(a) (0, -9

Question 6

Find the slope of the cardioid r = 1 - cos $\theta$ at $\theta$ =   .

Accepted Answer

A)  2 
B)  1 
C)  0 
D)  -1 
E)  -2 
A)  2 
B)  1 
C)  0 
D)  -1 
E)  -2

Question 7

To eliminate the xy-term from the general equation of a conic section,   ,   , we rotate the coordinate axes about the origin through an   , where cot(2$\theta$) =   .

Accepted Answer

A) True 
 B)False

Question 8

Sketch the polar curve r = 4 + 4 sin θ.

Accepted Answer

The answer of Sketch the polar curve r = 4...

Question 9

Find the arc length of the curve x =   ln (1 +   ), y =   t, from t = 0 to t = 1.

Accepted Answer

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

Question 10

Describe the curve x = 3 - cos(t), y = -2 + 2 sin(t).

Accepted Answer

A)  ellipse, centre (3, -2) with major axis along the line x = 3 
B)  ellipse, centre (3, -2) with major axis along the line y = -2 
C)  hyperbola, centre (3, -2) with transverse axis along the line x = -3 
D)  hyperbola, centre (3, -2) with transverse axis along the line x = 3 
E)  ellipse, centre (-3, 2) with major axis along the line x = -3 
A)  ellipse, centre (3, -2) with major axis along the line x = 3 
B)  ellipse, centre (3, -2) with major axis along the line y = -2 
C)  hyperbola, centre (3, -2) with transverse axis along the line x = -3 
D)  hyperbola, centre (3, -2) with transverse axis along the line x = 3 
E)  ellipse, centre (-3, 2) with major axis along the line x = -3

Question 11

Find the area of the region bounded by the cardioid r = 3 - 2 sin $\theta$.

Accepted Answer

A)  5$\pi$ square units 
B)  7$\pi$ square units 
C)  9$\pi$ square units 
D)  11$\pi$ square units 
E)  3$\pi$ square units 
A)  5$\pi$ square units 
B)  7$\pi$ square units 
C)  9$\pi$ square units 
D)  11$\pi$ square units 
E)  3$\pi$ square units

Question 12

Find an equation of an ellipse containing the point (-   ,   ) and with vertices (0, -3) and (0, 3).

Accepted Answer

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

Question 13

Find the tangent line(s) to the parametric curve given by x =   - 4   , y=   at (0, 4).

Accepted Answer

A)  y = 4 ±   x 
B)  y = 4 ±   x 
C)  y = 8 ±   x 
D)  y = 8 ±   x 
E)  y = 2 ±   x 
A)  y = 4 ±   x 
B)  y = 4 ±   x 
C)  y = 8 ±   x 
D)  y = 8 ±   x 
E)  y = 2 ±   x

Question 14

Find the angles of intersection of the curves r = 3 cos $\theta$ and r = 1 + cos $\theta$ at their intersection points.

Accepted Answer

A)    at the points   ,   at (0, 0) 
B)    at the points   , 0 at (0, 0) 
C)    at the points   ,   at (0, 0) 
D)    at the points   , 0 at (0, 0) 
E)    at the points   , 0 at (0, 0) 
A)    at the points   ,   at (0, 0) 
B)    at the points   , 0 at (0, 0) 
C)    at the points   ,   at (0, 0) 
D)    at the points   , 0 at (0, 0) 
E)    at the points   , 0 at (0, 0)

Question 15

Find the arc length of the curve x = et sin t, y = et cos t, from t = -   to t =   .

Accepted Answer

A)      - 1) units 
B)  2     + 1) units 
C)  2     - 1) units 
D)  2   sinh   units 
E)  2   cosh   units 
A)      - 1) units 
B)  2     + 1) units 
C)  2     - 1) units 
D)  2   sinh   units 
E)  2   cosh   units

Question 16

Find the coordinates of the highest point of the curve x = 6t, y = 6t -   .

Accepted Answer

A)  (18, 9) 
B)  (0, 0) 
C)  (12, 6) 
D)  (6, 5) 
E)  (24, 8) 
A)  (18, 9) 
B)  (0, 0) 
C)  (12, 6) 
D)  (6, 5) 
E)  (24, 8)

Question 17

Find the Cartesian coordinates of points of intersection of the plane parametric curves   ,   and x =   , y = -u - 1.

Accepted Answer

There are

Question 18

Sketch the polar graph of r = 2 + 3cos(θ), 0 ≤ θ ≤ 2π.

Accepted Answer

The answer of Sketch the polar graph of r =...

Question 19

Find the surface area generated by revolving the arc of r = a(1 - cos$\theta$), 0$\le$ $\theta$ $\le$ $\pi$ about the line $\theta$ = 0.

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 20

Find the slope of the curve x = 5 cos t, y = 3 sin t at t =   .

Accepted Answer

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

Find the equation of the tangent line to the curve at the given t. X = cos 3t, y = 3 sin 5t at t = .

Find the centre, the foci, and the asymptotes of the hyperbola 4 - 9 -16x - 54y = 101.

Convert to Cartesian coordinates the polar equation r² = .

Determine the coordinates of the points where the curve x = + 2t, y = 2 + 7 has (a) a horizontal tangent and (b) a vertical tangent.

Find the slope of the cardioid r = 1 - cos $\theta$ at $\theta$ = $Find the slope of the cardioid r = 1 - cos \theta at \theta = .$ .

Sketch the polar curve r = 4 + 4 sin θ.

Find the arc length of the curve x = ln (1 + ), y = t, from t = 0 to t = 1.

Describe the curve x = 3 - cos(t), y = -2 + 2 sin(t).

Find the area of the region bounded by the cardioid r = 3 - 2 sin $\theta$ .

Find an equation of an ellipse containing the point (- , ) and with vertices (0, -3) and (0, 3).

Find the tangent line(s) to the parametric curve given by x = - 4 , y= at (0, 4).

Find the angles of intersection of the curves r = 3 cos $\theta$ and r = 1 + cos $\theta$ at their intersection points.

Find the arc length of the curve x = e^t sin t, y = e^t cos t, from t = - to t = .

Find the coordinates of the highest point of the curve x = 6t, y = 6t - .

Find the Cartesian coordinates of points of intersection of the plane parametric curves , and x = , y = -u - 1.

Sketch the polar graph of r = 2 + 3cos(θ), 0 ≤ θ ≤ 2π.

Find the surface area generated by revolving the arc of r = a(1 - cos $\theta$ ), 0 $\le$ $\theta$ $\le$ $\pi$ about the line $\theta$ = 0.

Find the slope of the curve x = 5 cos t, y = 3 sin t at t = .

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 the equation of the tangent line to the curve at the given t. X = cos 3t, y = 3 sin 5t at t = .

Find the centre, the foci, and the asymptotes of the hyperbola 4 - 9 -16x - 54y = 101.

Convert to Cartesian coordinates the polar equation r2 = .

Determine the coordinates of the points where the curve x = + 2t, y = 2 + 7 has (a) a horizontal tangent and (b) a vertical tangent.

Find the slope of the cardioid r = 1 - cos θ\thetaθ at θ\thetaθ = .

To eliminate the xy-term from the general equation of a conic section, , , we rotate the coordinate axes about the origin through an , where cot(2 θ\thetaθ ) = .

Sketch the polar curve r = 4 + 4 sin θ.

Find the arc length of the curve x = ln (1 + ), y = t, from t = 0 to t = 1.

Describe the curve x = 3 - cos(t), y = -2 + 2 sin(t).

Find the area of the region bounded by the cardioid r = 3 - 2 sin θ\thetaθ .

Find an equation of an ellipse containing the point (- , ) and with vertices (0, -3) and (0, 3).

Find the tangent line(s) to the parametric curve given by x = - 4 , y= at (0, 4).

Find the angles of intersection of the curves r = 3 cos θ\thetaθ and r = 1 + cos θ\thetaθ at their intersection points.

Find the arc length of the curve x = et sin t, y = et cos t, from t = - to t = .

Find the coordinates of the highest point of the curve x = 6t, y = 6t - .

Find the Cartesian coordinates of points of intersection of the plane parametric curves , and x = , y = -u - 1.

Sketch the polar graph of r = 2 + 3cos(θ), 0 ≤ θ ≤ 2π.

Find the surface area generated by revolving the arc of r = a(1 - cos θ\thetaθ ), 0 ≤\le≤ θ\thetaθ ≤\le≤ π\piπ about the line θ\thetaθ = 0.

Find the slope of the curve x = 5 cos t, y = 3 sin t at t = .

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

Convert to Cartesian coordinates the polar equation r² = .

Find the slope of the cardioid r = 1 - cos $\theta$ at $\theta$ = $Find the slope of the cardioid r = 1 - cos \theta at \theta = .$ .

Find the area of the region bounded by the cardioid r = 3 - 2 sin $\theta$ .

Find the angles of intersection of the curves r = 3 cos $\theta$ and r = 1 + cos $\theta$ at their intersection points.

Find the arc length of the curve x = e^t sin t, y = e^t cos t, from t = - to t = .

Find the surface area generated by revolving the arc of r = a(1 - cos $\theta$ ), 0 $\le$ $\theta$ $\le$ $\pi$ about the line $\theta$ = 0.