Question 1

If u and v are differentiable functions of t, then

Accepted Answer

The answer of If u and v are differentiable functions...

Question 2

If r(t) , v(t) and a(t) are the position, velocity, and acceleration of a moving particle at any time t, respectively, and if the speed at any time t is a constant , then the velocity and acceleration vectors are perpendicular.

Accepted Answer

When the speed of a particle is constant, the direction of the velocity may change but not its magnitude. The acceleration is responsible for this change in direction, implying it acts perpendicular to the velocity to avoid changing the speed.

Question 3

A particle moves around the elliptical cylinder  +  = 1 in such a way that its position at time t is  Find the maximum and minimum values of both its speed  and its magnitude of its acceleration  .
A) max  = 3, min  = 2; max  = 3, min  = 2
B) max  = 5, min  = 2  ; max  = 3, min  = 2
C) max  = 5, min  = 2  ; max  = 5, min  = 2 
D) max  = 3, min  = 2; max  = 3, min  = 0
E) max  = 5, min  = 2  ; max  = 3, min  = 0

Accepted Answer

The answer of A particle moves around the elliptical cylinder...

Question 4

A ball of ice having mass 100 g at time t = 0 s is melting, and therefore losing mass, at a steady rate of 1 g/s. The ball has velocity i + 2 j at time t = 0 and is subject to a constant force F = 3 i thereafter. What is its velocity after 1 minute?
A) 10 i + 5 j
B) 10 i + 10 j
C) 4 i + 5 j
D) 4 i + 8 j
E) 10 i + 8 j

Accepted Answer

The force only affects the i-component of velocity, adding 1 m/s each second due to F=ma and mass reducing by 1g/s. The j-component remains unchanged. After 60s, velocity is 10 i + 2 j.

Question 5

Solve the initial-value problem  = k × r with initial condition r(0) = i + 2j + 3k. Describe the solution curve.
A) r(t) =  i +  j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.
B) r(t) =  i -  j + 3 k,a circle with centre (0, 0, 3) and radius  in the plane z = 3.
C) r(t) =  i -  j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.
D) r(t) =  i +  j + 3 k,a circle with centre (0, 0, 3) and radius  in the plane z = 3.
E) r(t) =  i +  j + 3 k,a circle with centre (0, 0, 3) and radius 5 in the plane z = 3.

Accepted Answer

The answer of Solve the initial-value problem  = k...

Question 6

You are at the origin in the xy-plane. At time t = 0 an incoming missile is at position (1000, 500) and has velocity -30 i + 3 j. (All distances are in metres and time is measured in seconds.) At that instant you fire an anti-missile missile to intercept the incoming missile. If your missile has an initial speed of 100 m/s, at what angle of elevation above the horizontal should you fire your missile to ensure that it intercepts the incoming missile? Assume that gravity is the only force acting on the projectiles.
A) 0.42535 radians
B) 0.52535 radians
C) 0.62535 radians
D) 0.72535 radians
E) 0.56535 radians

Accepted Answer

The relative velocity and position equations can be used to determine the angle. Solving these equations, the angle of elevation required is approximately 0.62535 radians.

Question 7

A rocket is fired vertically in a vacuum (free space where the  gravitational field is negligible).During the burning process, the exhaust gases are ejected at a constant rate of 1000 kg/s and at a constant velocity with magnitude 400 m/s relative to the rocket.Let M be the total initial mass of the rocket, and assume the rocket starts motion from rest.In order to accelerate to the speed of 800 m/s, the rocket has to burn P % of the total initial mass M as a fuel. Assuming there is sufficient amount of fuel on board, the value of P is equal to:
A) 1 - 
B) 100 
C) 100 
D) 
E) 100 -

Accepted Answer

The answer of A rocket is fired vertically in a...

Question 8

A rocket with mass 60,000 kilogram (kg), which includes 30,000 kg of fuel mixture, is fired vertically in a vacuum (free space where the gravitational field is negligible).During the burning process, the exhaust gases are ejected at a constant rate of 1000 kg/s and at a constant velocity with magnitude 400 m/s relative to the rocket.If the rocket was initially at rest, find its speed after 50 seconds.
A) 400 ln(3) m/s
B) 400 ln  m/s
C) 400 m/s
D) 400 ln (2) m/s
E) 400 ln (6) m/s

Accepted Answer

The answer of A rocket with mass 60,000 kilogram (kg),...

Question 9

Which of the following parametrize the circle  +  - 4y = 0 with clockwise orientation?
(a) r = (2 cos $\theta$) i + (2 + 2 sin $\theta$) j
 (b) r = (2 cos $\theta$) i - (2 + 2 sin $\theta$) j
(c) r = (2 cos $\theta$) i + (2 - 2 sin $\theta$) j 
(d) r = (2 sin $\theta$) i + (2 + 2 cos $\theta$) j
A) only (a)
B) only (b)
C) only (c)
D) only (d)
E) both (a) and (b)
F) both (c) and (d)
G) none of the above

Accepted Answer

The answer of Which of the following parametrize the circle...

Question 10

Find a parametrization of the ellipse in which the plane z = 3y intersects the cylinder  +  = 4, using the polar angle $\theta$ in the xy-plane as the parameter, [0, 2$\pi$] as parameter interval, and ensuring that the ellipse is oriented counterclockwise as viewed from high on the z-axis.
A) r = 2 cos($\theta$) i + 2 sin($\theta$) j + 6 sin($\theta$) k
B) r = 2 cos($\theta$) i - 2 sin($\theta$) j + 6 sin($\theta$) k
C) r = 2 cos($\theta$) i + 2 sin($\theta$) j - 6 sin($\theta$) k
D) r = 2 cos($\theta$) i - 2 sin($\theta$) j - 6 sin($\theta$) k
E) r = 2 cos($\theta$) i + 2 sin($\theta$) j + 3 sin($\theta$) k

Accepted Answer

The answer of Find a parametrization of the ellipse in...

Question 11

Parametrize the parabola in which the plane z = 1 + y intersects the cone  +  =  using x as the parameter.
A) r = x i +  j +  k, -$\infty$ < x < $\infty$
B) r = x i +  j +  k, -$\infty$ < x < $\infty$
C) r = x i +  j +  k, -$\infty$ < x < $\infty$
D) r = x i +  j +  k, -$\infty$ < x < $\infty$
E) r = x i +  j +  k, -$\infty$ < x < $\infty$

Accepted Answer

The answer of Parametrize the parabola in which the plane...

Question 12

Find the length of the arc r = u i +  j, 0 $\le$ u $\le$  .
A)  units
B)  units
C)  units
D)  units
E)  units

Accepted Answer

The answer of Find the length of the arc r...

Question 13

Find the perimeter of the astroid curve x = 2  $\theta$, y = 2  $\theta$.
A) 16 units
B) 6$\pi$ units
C) 3$\pi$ units
D) 12 units
E) 9$\pi$ units

Accepted Answer

The answer of Find the perimeter of the astroid curve...

Question 14

Find the arc length of the arc r = (ln (cos $\theta$) ) i + $\theta$ j, -  $\le$  $\theta$ $\le$ 0.
A)  units
B) ln (2 +  ) units
C) 2 units
D)  units
E)  units

Accepted Answer

The answer of Find the arc length of the arc...

Question 15

Find the length of the curve r(t) = (  sin t) i + (  cos t) j + 2t k for 0 $\le$ t $\le$ 2$\pi$.
A)  + 2$\pi$ units
B)  + 4$\pi$ units
C)  + 4$\pi$ units
D)  + 2$\pi$ units
E)  + 2$\pi$ units

Accepted Answer

The answer of Find the length of the curve r(t)...

Question 16

Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 -  ) k, 1 $\le$ t $\le$2.
A) 1 + 2ln(4) units
B)  unit
C) 2 units
D) 1 unit
E) 3 units

Accepted Answer

The answer of Find the arc length of the space...

Question 17

Parametrize the curve of intersection of the cylinder y = 1 -  and the plane z = - x using as parameter the slope m =  of the curve of intersection of the cylinder with the xy-plane.
A) x = -  i +  j + m k
B) x = - m i +  j + m k
C) x = -  i +  j +  k
D) x = -  i +  j -  k
E) x =  i +  j -  k

Accepted Answer

The answer of Parametrize the curve of intersection of the...

Question 18

Find a parametric representation of the curve of intersection of the paraboloid z = x² + y² and the plane 8x - 4y - z - 11 = 0. A) r = (3 - 4cos(t)) i + (3 + 2sin(t)) j + (1 - 32cos(t) - 8sin(t)) k, 0 $\le$ t $\le$ 2$\pi$ B) r = (3 + 4cos(t)) i + (3 - 2sin(t)) j + (1 + 32cos(t) + 8sin(t)) k, 0 $\le$ t $\le$ 2$\pi$ C) r = (4 + 3cos(t)) i + (-2 + 3sin(t)) j + (24cos(t) - 12sin(t) - 29) k, 0 $\le$ t $\le$ 2$\pi$ D) r = (-4 + 3cos(t)) i + (2 + 3sin(t)) j + (24cos(t) - 12sin(t) - 51) k, 0 $\le$ t $\le$2$\pi$ E) r = (-3 + 4cos(t)) i + (-3 - 2sin(t)) j + (32cos(t) + 8sin(t) - 23) k, 0 $\le$ t $\le$ 2$\pi$

Accepted Answer

The answer of Find a parametric representation of the curve...

Question 19

Reparametrize the curve r =  cos t) i + (  sin t) j +  k in terms of arc length s measured from the point where t = 0.
A) r =  cos  i +  sin  j +  k
B) r =  cos  i +  sin  j +  k
C) r =  cos  i +  sin  j +  k
D) r =  cos  i +  sin  j +  k
E) r =  i +  j +  k

Accepted Answer

The answer of Reparametrize the curve r =  cos...

Question 20

A parametric representation of the curve of intersection of the two surfaces 4x² + y² + z² = 8 and is given by which of the following vector equations? A) r = 2cos(t) i + sin(t) j - 2 k, 0 $\le$ t $\le$ 2$\pi$ B) r = 2cos(t) i + sin(t) j + 2 k, 0 $\le$ t $\le$ 2$\pi$ C) r = cos(t) i + 2sin(t) j +2 k, 0 $\le$ t $\le$ $\pi$ D) r = cos(t) i + 2sin(t) j - 2 k, 0 $\le$ t $\le$ 2$\pi$ E) r = 2cos(t) i + sin(t) j - 2 k, 0 $\le$ t $\le$ $\pi$

Accepted Answer

The answer of A parametric representation of the curve of...

Quiz 12: Vector Functions and Curves

If u and v are differentiable functions of t, then

If r(t) , v(t) and a(t) are the position, velocity, and acceleration of a moving particle at any time t, respectively, and if the speed at any time t is a constant , then the velocity and acceleration vectors are perpendicular.

A particle moves around the elliptical cylinder + = 1 in such a way that its position at time t is Find the maximum and minimum values of both its speed and its magnitude of its acceleration .

A ball of ice having mass 100 g at time t = 0 s is melting, and therefore losing mass, at a steady rate of 1 g/s. The ball has velocity i + 2 j at time t = 0 and is subject to a constant force F = 3 i thereafter. What is its velocity after 1 minute?

Solve the initial-value problem = k × r with initial condition r(0) = i + 2j + 3k. Describe the solution curve.

Find the length of the arc r = u i + $Find the length of the arc r = u i + j, 0 \le u \le . A) units B) units C) units D) units E) units$ j, 0 $\le$ u $\le$ $Find the length of the arc r = u i + j, 0 \le u \le . A) units B) units C) units D) units E) units$ .

Find the arc length of the arc r = (ln (cos $\theta$ ) ) i + $\theta$ j, - $Find the arc length of the arc r = (ln (cos \theta ) ) i + \theta j, - \le \theta \le 0. A) units B) ln (2 + ) units C) 2 units D) units E) units$ $\le$ $\theta$ $\le$ 0.

Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - $Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - ) k, 1 \le t \le 2. A) 1 + 2ln(4) units B) unit C) 2 units D) 1 unit E) 3 units$ ) k, 1 $\le$ t $\le$ 2.

Parametrize the curve of intersection of the cylinder y = 1 - and the plane z = - x using as parameter the slope m = of the curve of intersection of the cylinder with the xy-plane.

Find a parametric representation of the curve of intersection of the paraboloid z = x² + y² and the plane 8x - 4y - z - 11 = 0.

Reparametrize the curve r = cos t) i + ( sin t) j + k in terms of arc length s measured from the point where t = 0.

Quiz 12: Vector Functions and Curves

If u and v are differentiable functions of t, then

If r(t) , v(t) and a(t) are the position, velocity, and acceleration of a moving particle at any time t, respectively, and if the speed at any time t is a constant , then the velocity and acceleration vectors are perpendicular.

A particle moves around the elliptical cylinder + = 1 in such a way that its position at time t is Find the maximum and minimum values of both its speed and its magnitude of its acceleration .

A ball of ice having mass 100 g at time t = 0 s is melting, and therefore losing mass, at a steady rate of 1 g/s. The ball has velocity i + 2 j at time t = 0 and is subject to a constant force F = 3 i thereafter. What is its velocity after 1 minute?

Solve the initial-value problem = k × r with initial condition r(0) = i + 2j + 3k. Describe the solution curve.

Find a parametrization of the ellipse in which the plane z = 3y intersects the cylinder + = 4, using the polar angle θ\thetaθ in the xy-plane as the parameter, [0, 2 π\piπ ] as parameter interval, and ensuring that the ellipse is oriented counterclockwise as viewed from high on the z-axis.

Parametrize the parabola in which the plane z = 1 + y intersects the cone + = using x as the parameter.

Find the length of the arc r = u i + j, 0 ≤\le≤ u ≤\le≤ .

Find the perimeter of the astroid curve x = 2 θ\thetaθ , y = 2 θ\thetaθ .

Find the arc length of the arc r = (ln (cos θ\thetaθ ) ) i + θ\thetaθ j, - ≤\le≤ θ\thetaθ ≤\le≤ 0.

Find the length of the curve r(t) = ( sin t) i + ( cos t) j + 2t k for 0 ≤\le≤ t ≤\le≤ 2 π\piπ .

Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - ) k, 1 ≤\le≤ t ≤\le≤ 2.

Parametrize the curve of intersection of the cylinder y = 1 - and the plane z = - x using as parameter the slope m = of the curve of intersection of the cylinder with the xy-plane.

Find a parametric representation of the curve of intersection of the paraboloid z = x2 + y2 and the plane 8x - 4y - z - 11 = 0.

Reparametrize the curve r = cos t) i + ( sin t) j + k in terms of arc length s measured from the point where t = 0.

A parametric representation of the curve of intersection of the two surfaces 4x2 + y2 + z2 = 8 and is given by which of the following vector equations?

Find the length of the arc r = u i + $Find the length of the arc r = u i + j, 0 \le u \le . A) units B) units C) units D) units E) units$ j, 0 $\le$ u $\le$ $Find the length of the arc r = u i + j, 0 \le u \le . A) units B) units C) units D) units E) units$ .

Find the arc length of the arc r = (ln (cos $\theta$ ) ) i + $\theta$ j, - $Find the arc length of the arc r = (ln (cos \theta ) ) i + \theta j, - \le \theta \le 0. A) units B) ln (2 + ) units C) 2 units D) units E) units$ $\le$ $\theta$ $\le$ 0.

Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - $Find the arc length of the space curve given by the vector equation r = t i + 2ln(t) j +(1 - ) k, 1 \le t \le 2. A) 1 + 2ln(4) units B) unit C) 2 units D) 1 unit E) 3 units$ ) k, 1 $\le$ t $\le$ 2.

Find a parametric representation of the curve of intersection of the paraboloid z = x² + y² and the plane 8x - 4y - z - 11 = 0.