8.2: Calculus of Vector-Valued Functions
- Page ID
- 144337
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- 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.