8.2: Calculus of Vector-Valued Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
- 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.