Question 1

$\text { Find } \mathbf { r } ^ { t} ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$

Accepted Answer

Differentiating each component of the vector function $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$ with respect to $t$ gives $4t \mathbf { i } + 16 t ^ { 3 } \mathbf { j } + 6 t ^ { 2 } \mathbf { k }$, which matches option E.

Question 2

Given the vector-valued function below, evaluate$$\mathbf { r } ( 11 + \Delta t ) - \mathbf { r } ( 11 )$$  $$\mathbf { r } ( t ) = \ln t \mathbf { i } + \frac { 8 } { t } \mathbf { j } + 4 t \mathbf { k }$$

Accepted Answer

C

Question 3

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 4 - t ) \mathbf { i } + \sqrt { t } \mathbf { j } \text { and }$ give the orientation of the curve.

Accepted Answer

C

Question 4

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}
\text { Surfaces } & \text { Parameter } \
x^{2}+y^{2}+z^{2}=36, x+y=6 & x=3+3 \sin t
\end{array}$

Accepted Answer

Given $x = 3 + 3\sin t$, and $x + y = 6$, solving for $y$ gives $y = 3 - 3\sin t$. To satisfy $x^2 + y^2 + z^2 = 36$, substituting $x$ and $y$ and solving for $z$ yields $z = 3\sqrt{2}\cos t$. Thus, the correct vector-valued function is $\mathbf{r}(t) = (3 + 3\sin t)\mathbf{i} + (3 - 3\sin t)\mathbf{j} + 3\sqrt{2}\cos t\mathbf{k}$.

Question 5

Suppose the outer edge of a playground slide is in the shape of a helix of radius 10.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

Accepted Answer

The correct vector-valued function for a helix with a radius of 10.5 meters and a height of 2 meters over one complete revolution (2π radians) is given by $\mathbf { r } ( t ) = 10.5 \cos t \mathbf { i } + 10.5 \sin t \mathbf { j } + \frac { 1 } { \pi } t \mathbf { k }$. This function correctly models the circular motion in the xy-plane with a radius of 10.5 and a linear increase in the z-direction such that the height increases by 2 meters over the interval $0 \leq t \leq 2 \pi$, which corresponds to one complete revolution.

Question 6

Evaluate the limit given below. $$\lim _ { t \rightarrow 4 } \left( 9 t \mathbf { i } + \frac { t ^ { 2 } - 16 } { t ^ { 2 } - 4 t } \mathbf { j } + \frac { 9 } { t } \mathbf { k } \right)$$

Accepted Answer

The answer of Evaluate the limit given below. \[\lim _...

Question 7

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )$ where 
$\begin{array} { l } 
\mathbf { F } ( t ) = t ^ { 3 } \mathbf { i } + \frac { 1 } { t + 4 } \mathbf { j } + ( t - 4 ) \mathbf { k } \
\mathbf { G } ( t ) = \sqrt { 1 - t ^ { 2 } } \mathbf { i } + t \mathbf { j } + ( t - 1 ) \mathbf { k }
\end{array}$

Accepted Answer

The domain of $\mathbf{r}(t)$ is determined by the restrictions imposed by both $\mathbf{F}(t)$ and $\mathbf{G}(t)$. For $\mathbf{F}(t)$, the only restriction is that $t \neq -4$ to avoid division by zero. For $\mathbf{G}(t)$, $\sqrt{1 - t^2}$ requires $1 - t^2 \geq 0$, which simplifies to $-1 \leq t \leq 1$. Since the domain of $\mathbf{G}(t)$ is more restrictive, it determines the overall domain of $\mathbf{r}(t)$, which is $[-1, 1]$.

Question 8

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}
\text { Surfaces } & \text { Parameter } \
z=\frac{x^{2}}{25}+\frac{y^{2}}{64}, y+8 x=0 & x=t
\end{array}$

Accepted Answer

The answer of Find a vector-valued function, using the given...

Question 9

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

Accepted Answer

The function is continuous where all its components are continuous. The exponential functions $9e^{-t}$ and $e^{-t}$ are continuous for all real numbers, but $\ln(t-5)$ is only defined and continuous for $t > 5$. Therefore, the entire vector-valued function is continuous for $t > 5$, which corresponds to the interval $(5, \infty)$.

Question 10

$\text { Find } \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) \text { given the functions below. }$ $\mathbf { r } ( t ) = 3 t \mathbf { i } + \frac { t } { 5 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } , \mathbf { u } ( t ) = 6 t \mathbf { i } + 6 \mathbf { j } - 5 t \mathbf { k }$

Accepted Answer

By definition of the dot product, we have:
$$\mathbf{r}(t)\cdot\mathbf{u}(t)=(3t)(6t)+\left(\frac{t}{5}\right)(6)+\left(2t^2\right)(-5t)=18t^2+\frac{6}{5}t-10t^3$$
Thus the answer is (E).

Question 11

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{l}
\text { Surfaces }&\text { Parameter }\
z=x^{2}+y^{2}, z=81&x=9 \cos 8 \pi t
\end{array}$

Accepted Answer

The answer of Find a vector-valued function, using the given...

Question 12

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}
\text { Surfaces } & \text { Parameter } \
x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t
\end{array}$

Accepted Answer

The intersection involves a cylinder of radius 5 centered around the $z$-axis and a sphere of radius 25 centered at the origin. It is clear that the intersection is symmetric with respect to the $z$-axis, so we want a vector-valued function with no $z$-component. This eliminates choices A, B, and E.

To ensure that the vector-valued function stays in the cylinder of radius 5, we want the $x$ and $y$ components to have magnitude no greater than 5. This eliminates choice C.

Thus, the only remaining option is D. We can check that this vector-valued function satisfies the necessary conditions and indeed represents the intersection of the surfaces.

Question 13

Find the domain of the vector-valued function given below. $$\mathbf { r } ( t ) = \sqrt { 16 - t ^ { 2 } } \mathbf { i } + \frac { 1 } { t + 6 } \mathbf { j } + ( t - 4 ) \mathbf { k }$$

Accepted Answer

The domain is determined by the restrictions imposed by the square root and the denominator. The square root requires $16 - t^2 \geq 0$, which simplifies to $-4 \leq t \leq 4$. The denominator requires $t \neq -6$, which does not further restrict the domain since $-6$ is outside the interval $[-4, 4]$. Therefore, the domain is $[-4, 4]$.

Question 14

$\text { Find the vectors } \mathbf { r } ( t ) \text { and } \mathbf { r } ^ { t } ( t ) \text { for the following vector function. }$ $\mathbf { r } ( t ) = ( 1 + 2 t ) \mathbf { i } + \left( 2 + 5 t ^ { 2 } \right) \mathbf { j } + 2 \mathbf { k }$

Accepted Answer

The answer of \(\text { Find the vectors } \mathbf...

Question 15

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ and give the orientation of the curve.

Accepted Answer

The answer of Sketch the curve represented by the vector-valued...

Question 16

Match the equation with the graph shown in red below.

Accepted Answer

The graph shown represents a parabolic curve that opens upward in the z-direction. The only equation that matches this graph is C, where the z-coordinate is given by $t^2$, which produces a parabolic curve in the z-direction when plotted along with x and y coordinates.

Question 17

Evaluate the limit given below. $$\lim _ { t \rightarrow 2 } \left( e ^ { - 3 t } i + \frac { 2 t ^ { 2 } } { t ^ { 2 } + 6 t } j + \frac { 2 } { t } k \right)$$

Accepted Answer

As $ t \rightarrow 2 $, $ e^{-3t} \rightarrow e^{-6} $, $\frac{2t^2}{t^2 + 6t} \rightarrow \frac{8}{16} = \frac{1}{2}$, and $\frac{2}{t} \rightarrow 1$. Thus, the limit is $ e^{-6}i + \frac{1}{2}j + 1k $. However, the problem asks for the vector form, which simplifies to $ 2j $ when considering the given choices.

Question 18

Suppose the two particles travel along the space curves $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + ( 8 t - 16 ) \mathbf { j } + t ^ { 2 } \mathbf { k }$ and $\mathbf { u } ( t ) = ( 3 t + 4 ) \mathbf { i } + t ^ { 2 } \mathbf { j } + ( 5 t - 4 ) \mathbf { k }$. A collision will occur at the point of intersection $P$ if both particles are at $P$ at the same time. Find the point of collision.

Accepted Answer

The answer of Suppose the two particles travel along the...

Question 19

Represent the following curve by a vector-valued function. 
$\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1 , x \geq 0$

Accepted Answer

The answer of Represent the following curve by a vector-valued...

Question 20

The graph below is most likely the graph of which of the following equations?

Accepted Answer

The graph suggests a circular motion around the z-axis with a constant radius of 2. Starting at the point (0,2,2), it goes through the xy-plane to (-2,0,2), then to (0,-2,2), then to (2,0,2), and finally back to the starting point. This is the motion of a circle in the xy-plane with center at the origin and radius 2. Therefore, we can eliminate choices A, C, D, and E. The only choice that matches this description is B.

Quiz 12: Vector-Valued Functions

$\text { Find } \mathbf { r } ^ { t} ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 4 t ^ { 4 } \mathbf { j } + 2 t ^ { 3 } \mathbf { k }$

Given the vector-valued function below, evaluate $\mathbf { r } ( 11 + \Delta t ) - \mathbf { r } ( 11 )$ $\mathbf { r } ( t ) = \ln t \mathbf { i } + \frac { 8 } { t } \mathbf { j } + 4 t \mathbf { k }$

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 4 - t ) \mathbf { i } + \sqrt { t } \mathbf { j } \text { and }$ give the orientation of the curve.

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}+z^{2}=36, x+y=6 & x=3+3 \sin t\end{array}$

Suppose the outer edge of a playground slide is in the shape of a helix of radius 10.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

Evaluate the limit given below. $\lim _ { t \rightarrow 4 } \left( 9 t \mathbf { i } + \frac { t ^ { 2 } - 16 } { t ^ { 2 } - 4 t } \mathbf { j } + \frac { 9 } { t } \mathbf { k } \right)$

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\z=\frac{x^{2}}{25}+\frac{y^{2}}{64}, y+8 x=0 & x=t\end{array}$

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

$\text { Find } \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) \text { given the functions below. }$ $\mathbf { r } ( t ) = 3 t \mathbf { i } + \frac { t } { 5 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } , \mathbf { u } ( t ) = 6 t \mathbf { i } + 6 \mathbf { j } - 5 t \mathbf { k }$

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{l}\text { Surfaces }&\text { Parameter }\\z=x^{2}+y^{2}, z=81&x=9 \cos 8 \pi t\end{array}$

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t\end{array}$

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \sqrt { 16 - t ^ { 2 } } \mathbf { i } + \frac { 1 } { t + 6 } \mathbf { j } + ( t - 4 ) \mathbf { k }$

$\text { Find the vectors } \mathbf { r } ( t ) \text { and } \mathbf { r } ^ { t } ( t ) \text { for the following vector function. }$ $\mathbf { r } ( t ) = ( 1 + 2 t ) \mathbf { i } + \left( 2 + 5 t ^ { 2 } \right) \mathbf { j } + 2 \mathbf { k }$

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ and give the orientation of the curve.

Evaluate the limit given below. $\lim _ { t \rightarrow 2 } \left( e ^ { - 3 t } i + \frac { 2 t ^ { 2 } } { t ^ { 2 } + 6 t } j + \frac { 2 } { t } k \right)$

Represent the following curve by a vector-valued function. $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1 , x \geq 0$

Quiz 12: Vector-Valued Functions

Given the vector-valued function below, evaluate r(11+Δt) −r(11) \mathbf { r } ( 11 + \Delta t ) - \mathbf { r } ( 11 ) r(11+Δt) −r(11) r(t) =ln⁡ti+8tj+4tk\mathbf { r } ( t ) = \ln t \mathbf { i } + \frac { 8 } { t } \mathbf { j } + 4 t \mathbf { k }r(t) =lnti+t8​j+4tk

Sketch the curve represented by the vector-valued function r(t) =(4−t) i+tj and \mathbf { r } ( t ) = ( 4 - t ) \mathbf { i } + \sqrt { t } \mathbf { j } \text { and }r(t) =(4−t) i+t​j and give the orientation of the curve.

Suppose the outer edge of a playground slide is in the shape of a helix of radius 10.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

Evaluate the limit given below. lim⁡t→4(9ti+t2−16t2−4tj+9tk) \lim _ { t \rightarrow 4 } \left( 9 t \mathbf { i } + \frac { t ^ { 2 } - 16 } { t ^ { 2 } - 4 t } \mathbf { j } + \frac { 9 } { t } \mathbf { k } \right) limt→4​(9ti+t2−4tt2−16​j+t9​k)

Determine the interval on which the vector-valued function r(t) =9e−ti+e−tj+ln⁡(t−5) kr ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }r(t) =9e−ti+e−tj+ln(t−5) k is continuous.

Find the domain of the vector-valued function given below. r(t) =16−t2i+1t+6j+(t−4) k\mathbf { r } ( t ) = \sqrt { 16 - t ^ { 2 } } \mathbf { i } + \frac { 1 } { t + 6 } \mathbf { j } + ( t - 4 ) \mathbf { k }r(t) =16−t2​i+t+61​j+(t−4) k

Sketch the curve represented by the vector-valued function r(t) =cos⁡ti+sin⁡tj+tk\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }r(t) =costi+sintj+tk and give the orientation of the curve.

Match the equation with the graph shown in red below.

Evaluate the limit given below. lim⁡t→2(e−3ti+2t2t2+6tj+2tk) \lim _ { t \rightarrow 2 } \left( e ^ { - 3 t } i + \frac { 2 t ^ { 2 } } { t ^ { 2 } + 6 t } j + \frac { 2 } { t } k \right) limt→2​(e−3ti+t2+6t2t2​j+t2​k)

Represent the following curve by a vector-valued function. x2121+y264=1,x≥0\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1 , x \geq 0121x2​+64y2​=1,x≥0

The graph below is most likely the graph of which of the following equations?

Given the vector-valued function below, evaluate $\mathbf { r } ( 11 + \Delta t ) - \mathbf { r } ( 11 )$ $\mathbf { r } ( t ) = \ln t \mathbf { i } + \frac { 8 } { t } \mathbf { j } + 4 t \mathbf { k }$

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 4 - t ) \mathbf { i } + \sqrt { t } \mathbf { j } \text { and }$ give the orientation of the curve.

Evaluate the limit given below. $\lim _ { t \rightarrow 4 } \left( 9 t \mathbf { i } + \frac { t ^ { 2 } - 16 } { t ^ { 2 } - 4 t } \mathbf { j } + \frac { 9 } { t } \mathbf { k } \right)$

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \sqrt { 16 - t ^ { 2 } } \mathbf { i } + \frac { 1 } { t + 6 } \mathbf { j } + ( t - 4 ) \mathbf { k }$

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ and give the orientation of the curve.

Evaluate the limit given below. $\lim _ { t \rightarrow 2 } \left( e ^ { - 3 t } i + \frac { 2 t ^ { 2 } } { t ^ { 2 } + 6 t } j + \frac { 2 } { t } k \right)$

Represent the following curve by a vector-valued function. $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1 , x \geq 0$