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Mathematics LibreTexts

12.1E: Exercises for Section 12.1

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    Introduction to Vector-Valued Functions

    1) Give the component functions \(x=f(t)\) and \(y=g(t)\) for the vector-valued function \(\vecs r(t)=3 \sec t \, \hat{\mathbf{i}}+2 \tan t \,\hat{\mathbf{j}}\).

    Answer:
    Here we can say that \(f(t)=3 \sec t, \quad g(t)=2 \tan t\)

    so we have \(x(t)=3 \sec t, \quad y(t)=2 \tan t\).

    2) Given \(\vecs r(t)=3 \sec t \hat{\mathbf{i}}+2 \tan t \hat{\mathbf{j}}\), find the following values (if possible).

    1. \(\vecs r(\frac{\pi}{4})\)
    2. \(\vecs r(\pi)\)
    3. \(\vecs r(\frac{\pi}{2})\)

    3) Sketch the curve of the vector-valued function \( \vecs r(t)=3 \sec t \hat{\mathbf{i}}+2 \tan t \hat{\mathbf{j}}\) and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

    Answer:
    Hyperbolic path along a horizontally oriented hyperbola.

    Limits of Vector-Valued Functions

    4) Evaluate \(\lim \limits_{t \to 0}\left(e^t \hat{\mathbf{i}}+\frac{\sin t}{t} \hat{\mathbf{j}}+e^{−t} \hat{\mathbf{k}}\right)\)

    5) Given the vector-valued function \(\vecs r(t)=⟨\cos t,\sin t⟩\) find the following values:

    1. \(\lim \limits_{t \to \frac{\pi}{4}} \vecs r(t)\)
    2. \(\vecs r(\frac{\pi}{3})\)
    3. Is \(\vecs r(t)\) continuous at \(t=\frac{\pi}{3}\)?
    4. Graph \(\vecs r(t)\).
    Answer:

    a. \(⟨\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}⟩\),
    b. \(⟨\frac{1}{2},\frac{\sqrt{3}}{2}⟩\),
    c. Yes, the limit as t approaches \(\mathrm{\frac{\pi}{3}}\) is equal to \(\mathrm{r(\frac{\pi}{3})}\),
    d.

    Counterclockwise oriented path on the unit circle.

    6) Given the vector-valued function \(\vecs r(t)=⟨t,t^2+1⟩\), find the following values:

    1. \(\lim \limits_{t \to -3} \vecs r(t)\)
    2. \(\vecs r(−3)\)
    3. Is \(\vecs r(t)\) continuous at \(x=−3\)?
    4. \(\vecs r(t+2)−\vecs r(t)\)

    7) Let \(\vecs r(t)=e^t \hat{\mathbf{i}}+\sin t \hat{\mathbf{j}}+\ln t \hat{\mathbf{k}}\). Find the following values:

    1. \(\vecs r(\frac{\pi}{4})\)
    2. \(\lim \limits_{t \to \frac{\pi}{4} } \vecs r(t)\)
    3. Is \(\vecs r(t)\) continuous at \(t=\frac{\pi}{4}\)?
    Answer:
    a. ⟨\(e^{\frac{\pi}{4}},\frac{\sqrt{2}}{2},\ln (\frac{\pi}{4})\)⟩;
    b. ⟨\(e^{\frac{\pi}{4}},\frac{\sqrt{2}}{2},\ln (\frac{\pi}{4})\)⟩;
    c. Yes

    For exercises 8 - 13, find the limit of the following vector-valued functions at the indicated value of \(t\).

    8) \(\lim \limits_{t \to 4}⟨\sqrt{t−3},\frac{\sqrt{t}−2}{t−4},\tan(\frac{\pi}{t})⟩\)

    9) \(\lim \limits_{t \to \frac{\pi}{2}} \vecs r(t)\) for \(\vecs r(t)=e^t \hat{\mathbf{i}}+\sin t \hat{\mathbf{j}}+\ln t \hat{\mathbf{k}}\)

    Answer:
    \(⟨e^{\frac{\pi}{2}},1,\ln(\frac{\pi}{2})⟩\)

    10) \(\lim \limits_{t \to \infty}⟨e^{−2t},\frac{2t+3}{3t−1},\arctan(2t)⟩\)

    11) \(\lim \limits_{t \to e^2}⟨t \ln (t),\frac{\ln t}{t^2},\sqrt{\ln(t^2)⟩}\)

    Answer:
    \(2e^2 \hat{\mathbf{i}}+\frac{2}{e^4}\hat{\mathbf{j}}+2\hat{\mathbf{k}}\)

    12) \(\lim \limits_{t \to \frac{\pi}{6}}⟨\cos 2t,\sin 2t,1⟩\)

    13) \(\lim \limits_{t \to \infty} \vecs r(t)\) for \(\vecs r(t)=2e^{−t} \mathbf{ i}+e^{−t} \hat{\mathbf{j}}+\ln(t−1) \hat{\mathbf{k}}\)

    Answer:
    The limit does not exist because the limit of \(\ln(t−1)\) as \(t\) approaches infinity does not exist.


    Domain of a Vector-Valued Function

    For problems 14 - 17, find the domain of the vector-valued functions.

    14) Domain: \(\vecs r(t)=⟨t^2,t,\sin t⟩\)

    15) Domain: \(\vecs r(t)=⟨t^2,\tan t,\ln t⟩\)

    Answer:
    \(\text{D}_{\vecs r} = \left \{ t \,|\, t>0,t≠(2k+1)\frac{\pi}{2}, \, \text{where} \, k \,\text{is any integer} \right \}\)

    16) Domain: \(\vecs r(t)=⟨t^2,\sqrt{t−3},\frac{3}{2t+1}⟩\)

    17) Domain: \(\vecs r(t)=⟨\csc(t),\frac{1}{\sqrt{t−3}}, \ln(t−2)⟩\)

    Answer:
    \(\text{D}_{\vecs r} = \left \{ t \,|\, t>3,t≠n\pi, \, \text{where} \, n \,\text{is any integer} \right \}\)

    18) a. Find the domain of \(\vecs r(t)=2e^{-t} \hat{\mathbf{i}}+e^{−t}\hat{\mathbf{j}}+\ln(t−1)\hat{\mathbf{k}}\).

    b. For what values of \(t\) is \(\vecs r(t)=2e^{-t} \hat{\mathbf{i}}+e^{−t}\hat{\mathbf{j}}+\ln(t−1)\hat{\mathbf{k}}\) continuous?

    Answer:
    a. \(\text{D}_{\vecs r}: ( 1, \infty )\)
    b. All \(t\) such that \(t∈(1,\infty)\)

    19) Domain: \(\vecs r(t)=(\arccos t) \, \hat{\mathbf{i}} + \sqrt{2t−1} \, \hat{\mathbf{j}}+\ln(t) \, \hat{\mathbf{k}}\)

    Answer:
    \(\text{D}_{\vecs r}: \big[ \frac{1}{2}, 1 \big]\)
     

    Visualizing Vector-Valued Functions

    20) Describe the curve defined by the vector-valued function \(\vecs r(t)=(1+t)\hat{\mathbf{i}}+(2+5t)\hat{\mathbf{j}}+(−1+6t)\hat{\mathbf{k}}\).

    21) Let \(\vecs r(t)=⟨\cos t,t,\sin t⟩\) and use it to answer the following questions.

    1. For what values of \(t\) is \(\vecs r(t)\) continuous?
    2. Sketch the graph of \(\vecs r(t)\).
    Answer:
    a. \(\vecs r\) is continuous for all real numbers, i.e., for \(t \in \mathbb{R}\).
    b. Note that there should be a \(z\) on the vertical axis in the cross-section in image (a) below instead of the \(y\).

    Top image shows counterclockwise oriented path on the unit circle.   Bottom image shows corkscrew path with z-coordinate varying as the circular motion continues as in the image above.

    22) Produce a careful sketch of the graph of \(\vecs r(t) = t^2 \, \hat{\mathbf{i}} + t \, \hat{\mathbf{j}}\).

    In questions 23 - 25, use a graphing utility to sketch each of the vector-valued functions:

    23) [T] \(\vecs r(t)=2 \cos^2 t \hat{\mathbf{i}}+(2−\sqrt{t})\hat{\mathbf{j}}\)

    Answer:

    24) [T] \(\vecs r(t)=⟨e^{\cos (3t)},e^{−\sin(t)}⟩\)

    25) [T] \(\vecs r(t)=⟨2−\sin (2t),3+2 \cos t⟩\)

    Answer:
    A figure eight oriented path.

    Finding Equations in \(x\) and \(y\) for the Path Traced out by Vector-Valued Functions

    For questions 26-33, eliminate the parameter \(t\), write the equation in Cartesian coordinates, then sketch the graph of the vector-valued functions.

    26) \(\vecs r(t)=2t\hat{\mathbf{i}}+t^2 \hat{\mathbf{j}}\)
    (Hint: Let \(x=2t\) and \(y=t^2\). Solve the first equation for \(t\) in terms of \(x\) and substitute this result into the second equation.)

    27) \(\vecs r(t)=t^3 \hat{\mathbf{i}}+2t \hat{\mathbf{j}}\)

    Answer:

    \(y=2\sqrt[3]{x}\), a variation of the cube-root function

    Oriented path along the graph of y equals 2 times the cube-root of x.  Motion along the path is oriented from left-to-right.

    28) \(\vecs r(t)=\sin t\,\hat{\mathbf{i}}+\cos t\,\hat{\mathbf{j}}\)

    29) \(\vecs r(t)=3\cos t\,\hat{\mathbf{i}}+3\sin t\,\hat{\mathbf{j}}\)

    Answer:

    \(x^2+y^2=9\), a circle centered at \((0,0)\) with radius 3, and a counterclockwise orientation

    Counterclockwise motion along the circle of radius 3, centered at the origin.

    30) \(\vecs r(t)=⟨ \sin t,4 \cos t⟩\)

    31) \(\vecs r(t)=2\sin t\,\hat{\mathbf{i}}-3\cos t\,\hat{\mathbf{j}}\)

    Answer:

    \(\frac{x^2}{4}+\frac{y^2}{9}=1\), an ellipse centered at \((0,0)\) with intercepts at \(x = \pm2\) and \(y =\pm3\), and a clockwise orientation

    Ellipse with clockwise orientation passing thru (-2,0), (0, 3), (2, 0), (0, -3)

    32) \(\vecs r(t)=\tan t\,\hat{\mathbf{i}}-2\sec t\,\hat{\mathbf{j}}\)

    33) \(\vecs r(t)=3\sec t\,\hat{\mathbf{i}}+4\tan t\,\hat{\mathbf{j}}\)

    Answer:

    \(\frac{x^2}{9}-\frac{y^2}{16}=1\), a hyperbola centered at \((0,0)\) with \(x\)-intercepts \((3, 0)\) and \((-3, 0)\), with orientation shown

    Oriented Hyperbola

    Finding a Vector-Valued Function to Trace out the Graph of an Equation in \(x\) and \(y\)

    For questions 34 - 40, find a vector-valued function that traces out the given curve in the indicated direction.

    34) \(4x^2+9y^2=36\); clockwise and counterclockwise

    35) \(y=x^2\); from left to right

    Answer:
    \(\vecs r(t)=⟨t,t^2⟩\), where \(t\) increases

    36) The line through \(P\) and \(Q\) where \(P\) is \((1,4,−2)\) and \(Q\) is \((3,9,6)\)

    37) The circle, \(x^2 + y^2 = 36\), oriented clockwise, with position \((-6, 0)\) at time \(t = 0\).

    Answer:
    \(\vecs r(t)=-6\cos t\,\hat{\mathbf{i}}+6\sin t\,\hat{\mathbf{j}}\)

    38) The ellipse, \(x^2 + \dfrac{y^2}{36} = 1\), oriented counterclockwise

    Answer:
    \(\vecs r(t)=\cos t\,\hat{\mathbf{i}}+6\sin t\,\hat{\mathbf{j}}\)

    39) The hyperbola, \(\dfrac{y^2}{36} - x^2 = 1\), top piece is oriented from left-to-right

    Answer:
    \(\vecs r(t)=\tan t\,\hat{\mathbf{i}}+6\sec t\,\hat{\mathbf{j}}\)

    40) The hyperbola, \(\dfrac{x^2}{49} - \dfrac{y^2}{64} = 1\), right piece is oriented from bottom-to-top

    Answer:
    \(\vecs r(t)=7\sec t\,\hat{\mathbf{i}}+8\tan t\,\hat{\mathbf{j}}\)

    Parameterizing a Piecewise Path

    For questions 41 - 44, provide a parameterization for each piecewise path. Try to write a parameterization that starts with \(t = 0\) and progresses on through values of \(t\) as you move from one piece to another.

    41)

    Counterclockwise-oriented boundary of a closed region formed by y = x^4 and y equals the cube root of x. Clockwise-oriented boundary of a closed region formed by y = x^4 and y equals the cube root of x.

    Answer:
    a. \(\vecs r_1(t)= t\,\hat{\mathbf{i}} + t^4 \,\hat{\mathbf{j}}\) for \(0 \le t \le 1\)
    \(\vecs r_2(t)= -t\,\hat{\mathbf{i}} + \sqrt[3]{-t} \,\hat{\mathbf{j}}\) for \(-1 \le t \le 0\)

    So a piecewise parameterization of this path is:
    \(\vecs r(t) = \begin{cases}
    t\,\hat{\mathbf{i}} + t^4 \,\hat{\mathbf{j}}, & 0 \le t \le 1 \\
    \left(2-t\right)\,\hat{\mathbf{i}} + \sqrt[3]{2-t} \,\hat{\mathbf{j}}, & 1 \lt t\le 2
    \end{cases}\)

    b. \(\vecs r_1(t)= t\,\hat{\mathbf{i}} + \sqrt[3]{t} \,\hat{\mathbf{j}}\) for \(0 \le t \le 1\)
    \(\vecs r_2(t)= -t\,\hat{\mathbf{i}} + (-t)^4 \,\hat{\mathbf{j}}\) for \(-1 \le t \le 0\)

    So a piecewise parameterization of this path is:
    \(\vecs r(t) = \begin{cases}
    t\,\hat{\mathbf{i}} + \sqrt[3]{t} \,\hat{\mathbf{j}}, & 0 \le t \le 1 \\
    \left(2-t\right)\,\hat{\mathbf{i}} + \left(2-t\right)^4 \,\hat{\mathbf{j}}, & 1 \lt t\le 2
    \end{cases}\)

    42)

    Counterclockwise-oriented boundary of a closed region formed by y = x^3 and y = 4x. Clockwise-oriented boundary of a closed region formed by y = x^3 and y = 4x.

    43)

    Counterclockwise-oriented boundary of a closed region formed by y = x^3 and y = 2 - x and the x-axis. Clockwise-oriented boundary of a closed region formed by y = x^3 and y = 2 - x and the x-axis.

    Answer:
    a. \(\vecs r_1(t)= t\,\hat{\mathbf{i}} +0 \,\hat{\mathbf{j}}\) for \(0 \le t \le 2\)
    \(\vecs r_2(t)= -t\,\hat{\mathbf{i}} + \left(2 + t\right) \,\hat{\mathbf{j}}\) for \(-2 \le t \le -1\)
    \(\vecs r_3(t)= -t\,\hat{\mathbf{i}} + \left(-t\right)^3 \,\hat{\mathbf{j}}\) for \(-1 \le t \le 0\)

    So a piecewise parameterization of this path is:
    \(\vecs r(t) = \begin{cases}
    t\,\hat{\mathbf{i}}, & 0 \le t \le 2 \\
    \left(4-t\right)\,\hat{\mathbf{i}} + \left(t-2\right) \,\hat{\mathbf{j}}, & 2 \lt t\le 3 \\
    \left(4-t\right) \, \hat{\mathbf{i}} + \left(4-t\right)^3 \,\hat{\mathbf{j}}, & 3 \lt t\le 4
    \end{cases}\)

    b. \(\vecs r_1(t)= t\,\hat{\mathbf{i}} + t^3 \,\hat{\mathbf{j}}\) for \(0 \le t \le 1\)
    \(\vecs r_2(t)= t\,\hat{\mathbf{i}} + \left(2 - t\right) \,\hat{\mathbf{j}}\) for \(1 \le t \le 2\)
    \(\vecs r_3(t)= -t\,\hat{\mathbf{i}} + 0 \,\hat{\mathbf{j}}\) for \(-2 \le t \le 0\)

    So a piecewise parameterization of this path is:
    \(\vecs r(t) = \begin{cases}
    t\,\hat{\mathbf{i}} + t^3 \,\hat{\mathbf{j}}, & 0 \le t \le 1 \\
    t\,\hat{\mathbf{i}} + \left(2 - t\right) \,\hat{\mathbf{j}}, & 1 \lt t\le 2 \\
    \left(4-t\right) \, \hat{\mathbf{i}}, & 2 \lt t\le 4
    \end{cases}\)

    44)

    Counterclockwise-oriented boundary of a closed region formed by y = 1-x/2 and y = 3x/2 - 3 and y = 1 plus the square root of x. Clockwise-oriented boundary of a closed region formed by y = 1-x/2 and y = 3x/2 - 3 and y = 1 plus the square root of x.

    Additional Vector-Valued Function Questions

    For questions 45 - 48, consider the curve described by the vector-valued function \(\vecs r(t)=(50e^{−t}\cos t)\hat{\mathbf{i}}+(50e^{−t}\sin t)\hat{\mathbf{j}}+(5−5e^{−t})\hat{\mathbf{k}}\).

    45) What is the initial point of the path corresponding to \(\vecs r(0)\)?

    Answer:
    \((50,0,0)\)

    46) What is \(\lim \limits_{t \to \infty} \vecs r(t) \)?

    47) [T] Use technology to sketch the curve.

    Answer:
    Partial path for r(t)=(50e^(−t) cos t)i+(50e^(−t) sin t)j+(5−5e^(−t))k.

    48) Eliminate the parameter t to show that \(z=5−\dfrac{r}{10}\) where \(r^2=x^2+y^2\).

    49) [T] Let \(\vecs r(t)=\cos t \hat{\mathbf{i}}+\sin t\hat{\mathbf{j}}+0.3 \sin (2t)\hat{\mathbf{k}}\). Use technology to graph the curve (called the roller-coaster curve) over the interval \([0,2\pi)\). Choose at least two views to determine the peaks and valleys.

    Answer:
    Two views of the path traced out by r(t)=(cos t)i + (sin t)j + (0.3 sin 2t)k.

    50) [T] Use the result of the preceding problem to construct an equation of a roller coaster with a steep drop from the peak and steep incline from the “valley.” Then, use technology to graph the equation.

    51) Use the results of the preceding two problems to construct an equation of a path of a roller coaster with more than two turning points (peaks and valleys).

    Answer:

    One possibility is \(\vecs r(t)=\cos t \hat{\mathbf{i}}+\sin t\hat{\mathbf{j}}+\sin (4t)\hat{\mathbf{k}}\). By increasing the coefficient of \(t\) in the third component, the number of turning points will increase.

    Path traced out by r(t)=(cos t)i + (sin t)j + (sin 4t)k.

    52) Complete the following investigation.

    1. Graph the curve \(\vecs r(t)=(4+\cos(18t))\cos(t)\hat{\mathbf{i}}+(4+\cos (18t)\sin (t))\hat{\mathbf{j}}+0.3 \sin(18t)\hat{\mathbf{k}}\) using two viewing angles of your choice to see the overall shape of the curve.
    2. Does the curve resemble a “slinky”?
    3. What changes to the equation should be made to increase the number of coils of the slinky?

     

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

    Paul Seeburger (Monroe Community College) created problems 12, 14, 19, 22, 30-33, 37- 44.