8.2: Calculus of Vector-Valued Functions

1. Compute the derivative of $\mathbf{r}(t) = 2t^2\,\mathbf{i} - \frac{1}{t}\,\mathbf{j} + \sqrt{t}\,\mathbf{k}$.
    
2. Compute the derivative of $\mathbf{r}(t) = (\cos t, \sin t, t)$.
    
3. Compute the derivative of $\mathbf{r}(t) = t^2e^t\,\mathbf{i} + \ln(\sec t)\,\mathbf{j}$.
    
4. Compute the derivative of $\mathbf{r}(t) = \dfrac{\tan t}{t^2 + t + 1}\,\mathbf{i} - \mathbf{j} + \arctan(t^2)\,\mathbf{k}$.
    
5. Find the tangent and the unit tangent vectors to the curve $\mathbf{r}(t) = (\ln(2t), t^2 + \dfrac{3}{t})$ at $t = 1$.
    
6. Find the tangent and unit tangent vectors to the curve $\mathbf{r}(t) = (\cos(2t), 2t, \sin(3t))$ at $t = \dfrac{\pi}{2}$.
    
7. 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 $t = \ln(2)$.
    
8. Find the equation of the tangent line to the curve $\mathbf{r}(t) = (\sin t, \cos t)$ at $t = \dfrac{\pi}{4}$.
    
9. Find the equation of the tangent line to the curve $\mathbf{r}(t) = t^2\,\mathbf{i} + t\,\mathbf{j} + t\,\mathbf{k}$ at $t = -1$.
    
10. 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 $t = \dfrac{\pi}{4}$.
     
11. 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 $t = 1$.
     
12. 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 $\dfrac{d}{dt}\Bigl(f(t)\mathbf{r}(t)\Bigr)\Bigg|_{t = 0}$.
     
13. 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 $\dfrac{d}{dt}\Bigl(\mathbf{r}(t) \cdot \mathbf{s}(t)\Bigr)\Bigg|_{t = 5}$.
     
14. 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 $\dfrac{d}{dt}\Bigl(\mathbf{r}(t) \times \mathbf{s}(t)\Bigr)\Bigg|_{t = 1}$.
     
15. Evaluate the integral $\displaystyle \int_1^e \frac{1}{t}\,\mathbf{i} + t^2\,\mathbf{j}\ dt$.
     
16. Evaluate the integral $\displaystyle \int_0^1 (\sqrt[3]{t}, \dfrac{1}{t + 1}, e^{-t})\ dt$.
     
17. 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.