8.1: Vector-Valued Functions and Space Curves
- Page ID
- 144336
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Given the vector-valued function \(\mathbf{r}(t) = (2t^2 + 3, \sqrt{3t - 2})\), find \(\mathbf{r}(2)\).
- Given the vector-valued function \(\mathbf{r}(t) = \sec t\mathbf{i} + \tan t\mathbf{j}\), find \(\mathbf{r}\left(\frac{\pi}{4}\right)\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = t\mathbf{i} + (1 - t^2)\mathbf{j}\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = (t, t^2, t)\).
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = (\cos t, \sin t)\). Be sure to indicate the orientation of the curve.
- Sketch the curve of the vector-valued function \(\mathbf{r}(t) = \sin t\mathbf{i} + \cos t\mathbf{j} + t\mathbf{k}\). Be sure to indicate the orientation of the curve.
- Find the domain of the vector-valued function \(\mathbf{r}(t) = \left(\dfrac{1}{t-3}, \sqrt{t^2 - 1}\right)\).
- Find the domain of the vector-valued function \(\mathbf{r}(t) = 2e^t\mathbf{i} + \ln t\mathbf{j} + \dfrac{1}{t^2 + 1}\mathbf{k}\).
- Evaluate \(\displaystyle \lim_{t \to 0} (2t + 1, \cos t)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to 2} \left(\frac{t - 2}{t^2 - 4}, e^t\right)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to \infty} \left(e^{-t}\mathbf{i} + \frac{3t^2 + 2t - 4}{2 - 5t^2}\mathbf{j} + \arctan(t)\mathbf{k}\right)\), if possible.
- Evaluate \(\displaystyle \lim_{t \to 0^+} \left( \frac{\sin t}{t}, \ln t, 4\right)\), if possible.
- Express the vector-valued function \(\mathbf{r}(t) = 2t\mathbf{i} - (3 + t)\mathbf{j}\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = (t^3, t^2)\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = \dfrac{1}{t}(\mathbf{i} + \mathbf{j})\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = (2\sin t, -3\cos t)\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Express the vector-valued function \(\mathbf{r}(t) = 3\sec^2t\mathbf{i} + \tan t\mathbf{j}\) in Cartesian coordinates by eliminating the parameter \(t\), then sketch the graph of the vector-valued function. Be sure to indicate the orientation of the curve.
- Find a vector-valued function that traces the line from point \(P(1, 0, 1)\) to point \(Q(0, 1, 0)\).
- Find a vector-valued function that traces the parabola \(y = x^2\) from left-to-right.
- Find a vector-valued function that traces the ellipse \(4x^2 + 9y^2 = 36\) in the counterclockwise direction.
- Find a vector-valued function that traces the hyperbola \(\dfrac{x^2}{16} - y^2 = 1\), where the right branch is oriented top-to-bottom.