8.2: Calculus of Vector-Valued Functions
- Page ID
- 144337
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Compute the derivative of \(\mathbf{r}(t) = 2t^2\,\mathbf{i} - \frac{1}{t}\,\mathbf{j} + \sqrt{t}\,\mathbf{k}\).
- Compute the derivative of \(\mathbf{r}(t) = (\cos t, \sin t, t)\).
- Compute the derivative of \(\mathbf{r}(t) = t^2e^t\,\mathbf{i} + \ln(\sec t)\,\mathbf{j}\).
- Compute the derivative of \(\mathbf{r}(t) = \dfrac{\tan t}{t^2 + t + 1}\,\mathbf{i} - \mathbf{j} + \arctan(t^2)\,\mathbf{k}\).
- Find the tangent and the unit tangent vectors to the curve \(\mathbf{r}(t) = (\ln(2t), t^2 + \dfrac{3}{t})\) at \(t = 1\).
- Find the tangent and unit tangent vectors to the curve \(\mathbf{r}(t) = (\cos(2t), 2t, \sin(3t))\) at \(t = \dfrac{\pi}{2}\).
- Find the tangent and the unit tangent vectors to the curve \(\mathbf{r}(t) = e^t\,\mathbf{i} - \dfrac{e^{3t}}{2}\,\mathbf{j} + 4e^{-2t}\,\mathbf{k}\) at \(t = \ln(2)\).
- Find the equation of the tangent line to the curve \(\mathbf{r}(t) = (\sin t, \cos t)\) at \(t = \dfrac{\pi}{4}\).
- Find the equation of the tangent line to the curve \(\mathbf{r}(t) = t^2\,\mathbf{i} + t\,\mathbf{j} + t\,\mathbf{k}\) at \(t = -1\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = (\cos t, \sin 3t)\). Find the velocity, speed, and acceleration of the particle at \(t = \dfrac{\pi}{4}\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = t\,\mathbf{i} - t^2\,\mathbf{j} + t^3\,\mathbf{k}\). Find the velocity, speed, and acceleration of the particle at \(t = 1\).
- Consider a scalar function \(f(t)\) and a vector-valued function \(\mathbf{r}(t)\). Suppose \(f(0) = 4\), \(f'(0) = -2\), \(\mathbf{r}(0) = (5, 3, 4)\), and \(\mathbf{r}'(0) = (0, -2, -4)\). Find \(\dfrac{d}{dt}\Bigl(f(t)\mathbf{r}(t)\Bigr)\Bigg|_{t = 0}\).
- Consider vector-valued functions \(\mathbf{r}(t)\) and \(\mathbf{s}(t)\). Suppose \(\mathbf{r}(5) = \mathbf{i} - 3\,\mathbf{j}\), \(\mathbf{r}'(5) = -\mathbf{i} + 4\,\mathbf{j} + 5\,\mathbf{k}\), \(\mathbf{s}(5) = -4\,\mathbf{i} + 3\,\mathbf{j} + \mathbf{k}\), and \(\mathbf{s}'(5) = -5\,\mathbf{i} + 2\,\mathbf{j} - 2\,\mathbf{k}\). Find \(\dfrac{d}{dt}\Bigl(\mathbf{r}(t) \cdot \mathbf{s}(t)\Bigr)\Bigg|_{t = 5}\).
- Consider vector-valued functions \(\mathbf{r}(t)\) and \(\mathbf{s}(t)\). Suppose \(\mathbf{r}(1) = (-2, 2, 4)\), \(\mathbf{r}'(1) = (4, 0, 4)\), \(\mathbf{s}(1) = (1, 2, -2)\), and \(\mathbf{s}'(1) = (4, 0, 3)\). Find \(\dfrac{d}{dt}\Bigl(\mathbf{r}(t) \times \mathbf{s}(t)\Bigr)\Bigg|_{t = 1}\).
- Evaluate the integral \(\displaystyle \int_1^e \frac{1}{t}\,\mathbf{i} + t^2\,\mathbf{j}\ dt\).
- Evaluate the integral \(\displaystyle \int_0^1 (\sqrt[3]{t}, \dfrac{1}{t + 1}, e^{-t})\ dt\).
- The acceleration function of a particle at time \(t\) is given by \(\mathbf{a}(t) = -5\cos t\,\mathbf{i} - 5\sin t\,\mathbf{j}\). If the particle has an initial velocity of \(\mathbf{v}(0) = 9\,\mathbf{i} + 2\,\mathbf{j}\) and starts at position \(\mathbf{r}(0) = 5\,\mathbf{i}\), find the velocity function \(\mathbf{v}(t)\) and position function \(\mathbf{r}(t)\) of the particle.