Timedora
Free Practice Quiz Question List

Vector-Valued Functions Online Quiz Questions

Use this free practice quiz with 20 questions to review Vector-Valued Functions, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

What is lim⁡t→0⟨t2+1,sin⁡tt,et⟩\lim_{t\to 0}\left\langle t^2+1,\frac{\sin t}{t},e^t\right\rangle?

  1. A

    The limit does not exist because the second component contains a quotient

  2. B

    ⟨1,1,1⟩\langle 1,1,1\rangle

  3. C

    ⟨1,0,1⟩\langle 1,0,1\rangle

  4. D

    ⟨0,1,0⟩\langle 0,1,0\rangle

02
Choose one
1 point

A particle has position r(t)=⟨t2,3t−t3⟩\mathbf r(t)=\langle t^2,3t-t^3\rangle. Which expression gives its velocity vector?

  1. A

    ⟨t,3−t2⟩\langle t,3-t^2\rangle

  2. B

    ⟨2t,3t2⟩\langle 2t,3t^2\rangle

  3. C

    ⟨2t,3−3t2⟩\langle 2t,3-3t^2\rangle

  4. D

    ⟨2,3−6t⟩\langle 2,3-6t\rangle

03
Choose all
1 point

Select all identities that correctly describe differentiation of vector-valued functions.

  1. A

    ddt[u+v]=u′×v′\frac{d}{dt}[\mathbf u+\mathbf v]=\mathbf u'\times\mathbf v'

  2. B

    ddt[u+v]=u′+v′\frac{d}{dt}[\mathbf u+\mathbf v]=\mathbf u'+\mathbf v'

  3. C

    ddt[fu]=f′u+fu′\frac{d}{dt}[f\mathbf u]=f'\mathbf u+f\mathbf u'

  4. D

    ddt[fu]=f′u′\frac{d}{dt}[f\mathbf u]=f'\mathbf u'

04
Fill in the blank
1 point

Complete the derivative rule for r(t)=⟨f(t),g(t),h(t)⟩\mathbf r(t)=\langle f(t),g(t),h(t)\rangle: r′(t)=\mathbf r'(t)=.

05
Choose one
1 point

For r(t)=⟨2cos⁡t,2sin⁡t⟩\mathbf r(t)=\langle 2\cos t,2\sin t\rangle, which is the unit tangent vector at t=π4t=\frac{\pi}{4}?

  1. A

    ⟨−2,2⟩\langle-\sqrt2,\sqrt2\rangle

  2. B

    ⟨−12,12⟩\left\langle-\frac{1}{\sqrt2},\frac{1}{\sqrt2}\right\rangle

  3. C

    ⟨−1,1⟩\langle-1,1\rangle

  4. D

    ⟨12,−12⟩\left\langle\frac{1}{\sqrt2},-\frac{1}{\sqrt2}\right\rangle

06
Fill in the blank
1 point

In vector-valued motion, the derivative of position is , and the derivative of velocity is .

07
Choose all
1 point

Select all valid statements about recovering position and velocity by integration.

  1. A

    r(t)=r(t0)+∫t0tv(u) du\mathbf r(t)=\mathbf r(t_0)+\int_{t_0}^{t}\mathbf v(u)\,du

  2. B

    Integrating acceleration once directly gives position without any additional information

  3. C

    v(t)=v(t0)+∫t0ta(u) du\mathbf v(t)=\mathbf v(t_0)+\int_{t_0}^{t}\mathbf a(u)\,du

  4. D

    The components of a vector integral cannot be integrated separately

08
Written response
1 point

For r(t)=⟨t2,3t−t3⟩\mathbf r(t)=\langle t^2,3t-t^3\rangle, what is the slope dydx\frac{dy}{dx} at t=1t=1? Enter the value as a number.

09
Choose one
1 point

A particle has r(0)=⟨3,4⟩\mathbf r(0)=\langle3,4\rangle and r′(0)=⟨1,−2⟩\mathbf r'(0)=\langle1,-2\rangle. Which equation represents the tangent line at t=0t=0?

  1. A

    L(s)=⟨3,4⟩+s⟨1,−2⟩\mathbf L(s)=\langle3,4\rangle+s\langle1,-2\rangle

  2. B

    L(s)=⟨1,−2⟩+s⟨3,4⟩\mathbf L(s)=\langle1,-2\rangle+s\langle3,4\rangle

  3. C

    L(s)=⟨3,4⟩+s⟨2,6⟩\mathbf L(s)=\langle3,4\rangle+s\langle2,6\rangle

  4. D

    L(s)=⟨0,0⟩+s⟨1,−2⟩\mathbf L(s)=\langle0,0\rangle+s\langle1,-2\rangle

10
Choose one
1 point

Suppose a particle remains at a constant distance from the origin, so r(t)⋅r(t)=c\mathbf r(t)\cdot\mathbf r(t)=c. Which conclusion follows?

  1. A

    r(t)⋅r′(t)=0\mathbf r(t)\cdot\mathbf r'(t)=0

  2. B

    r(t)×r′(t)=0\mathbf r(t)\times\mathbf r'(t)=0

  3. C

    r(t)+r′(t)=0\mathbf r(t)+\mathbf r'(t)=\mathbf 0

  4. D

    ∥r′(t)∥=0\lVert\mathbf r'(t)\rVert=0

11
Open ended
1 point

Given a(t)=⟨2,6t⟩\mathbf a(t)=\langle2,6t\rangle, v(0)=⟨1,−2⟩\mathbf v(0)=\langle1,-2\rangle, and r(0)=⟨3,4⟩\mathbf r(0)=\langle3,4\rangle, derive the position function r(t)\mathbf r(t). Show how the initial conditions determine the constants of integration.

12
Choose one
1 point

Which condition is equivalent to continuity of a vector-valued function r(t)=⟨f(t),g(t),h(t)⟩\mathbf r(t)=\langle f(t),g(t),h(t)\rangle at t=at=a?

  1. A

    A vector-valued function is continuous whenever at least one component is continuous.

  2. B

    A vector-valued function is continuous exactly when every component is continuous.

  3. C

    A vector-valued function is continuous only when all components are constant.

  4. D

    A vector-valued function is continuous only when its derivative is zero.

13
Choose one
1 point

Evaluate lim⁡t→0⟨t2+1,sin⁡tt,et⟩\displaystyle\lim_{t\to 0}\left\langle t^2+1,\frac{\sin t}{t},e^t\right\rangle.

  1. A

    ⟨0,0,0⟩\langle 0,0,0\rangle

  2. B

    ⟨1,0,1⟩\langle 1,0,1\rangle

  3. C

    ⟨1,1,1⟩\langle 1,1,1\rangle

  4. D

    The limit does not exist because the second component is undefined at t=0t=0.

14
True or false
1 point

True or false: If r(t)=⟨f(t),g(t),h(t)⟩\mathbf r(t)=\langle f(t),g(t),h(t)\rangle, then r′(t)=⟨f′(t),g′(t),h′(t)⟩\mathbf r'(t)=\langle f'(t),g'(t),h'(t)\rangle.

  1. A

    True

  2. B

    False

15
Choose one
1 point

For r(t)=⟨t2,3t2−4t,cos⁡t⟩\mathbf r(t)=\langle t^2,3t^2-4t,\cos t\rangle, what is the velocity vector v(1)\mathbf v(1)?

  1. A

    ⟨1,2,−sin⁡1⟩\langle 1,2,-\sin 1\rangle

  2. B

    ⟨2,2,−sin⁡1⟩\langle 2,2,-\sin 1\rangle

  3. C

    ⟨2,6,−cos⁡1⟩\langle 2,6,-\cos 1\rangle

  4. D

    ⟨2,2,sin⁡1⟩\langle 2,2,\sin 1\rangle

16
Written response
1 point

A particle has velocity v(t)=⟨−3,4⟩\mathbf v(t)=\langle -3,4\rangle at a particular instant. What is its speed?

17
Written response
1 point

For r(t)=⟨3cos⁡t,3sin⁡t⟩\mathbf r(t)=\langle 3\cos t,3\sin t\rangle, enter the x-component of the unit tangent vector at t=π6t=\frac{\pi}{6}. Give your answer as a simplified fraction.

18
True or false
1 point

True or false: A particle can have constant speed and nonzero acceleration.

  1. A

    True

  2. B

    False

19
True or false
1 point

True or false: Two vector-valued functions can trace the same geometric curve but differ in the speed or direction with which the curve is traversed.

  1. A

    True

  2. B

    False

20
Written response
1 point

A particle has position r(t)=⟨t2,3t−t3⟩\mathbf r(t)=\langle t^2,3t-t^3\rangle. What is the y-component of its acceleration at t=2t=2? Enter the value as a number.