91Ó°ÊÓ

Given the vector-valued function \(\mathbf{r}(t)=e^{-t / 10} \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k}\), we can identify the individual coordinate functions as: $$x(t) = e^{-t / 10}$$ $$y(t) = 3\cos t$$ $$z(t) = 3\sin t$$

To analyze the possible extreme points in our graph, we will look at the VERTICAL points (maxima/minima) of \(y(t)\) and \(z(t)\). Let's find their critical points by setting their first derivatives equal to zero: $$\frac{dy(t)}{dt} = -3\sin t \quad \Rightarrow \quad t = \pi k,\,\, k \in \mathbb{Z}$$ $$\frac{dz(t)}{dt} = 3\cos t \quad \Rightarrow \quad t = \frac{\pi}{2} + \pi k,\,\, k \in \mathbb{Z}$$ Therefore, extreme points in the \(y\)-axis occur at integer multiples of \(\pi\), and the extreme points in the \(z\)-axis occur at odd integer multiples of \(\pi/2\).

To determine the positive orientation of the curve, we'll look at the direction of the tangent vector \(\mathbf{r'}(t)\) at some point \(t_0\) in the domain. Let's choose \(t_0=0\): $$\mathbf{r'}(t) = \left(-\frac{1}{10} e^{-t / 10} \mathbf{i}\right) + \left(-3 \sin t \mathbf{j}\right) + \left(3 \cos t \mathbf{k}\right)$$ At \(t=0\), we have \(\mathbf{r'}(0) = -\frac{1}{10} \mathbf{i} - 3 \mathbf{j} + 3 \mathbf{k}\). We can see that all components of this tangent vector are negative or positive, which means the curve has a positive orientation.

Now that we have the coordinate functions, boundaries and orientation, we can plot the curve. 1. Start by plotting the \(x\)-function, $$x(t)=e^{-t/10}$$, which should give an exponential decrease along the \(x\)-axis as \(t\) increases. 2. Plot the \(y\)-function, $$y(t)=3\cos t$$, which should give a simple harmonic oscillation along the \(y\)-axis. 3. Plot the \(z\)-function, $$z(t)=3\sin t$$, which should similarly give a simple harmonic oscillation along the \(z\)-axis. 4. Combine all three coordinate functions by varying \(t\) from \(0\) to a large value, and observe the resulting curve. This spiral-shaped curve will have a positive orientation. 5. Finally, indicate the positive orientation by drawing small arrows along the curve to indicate the direction in which \(t\) increases. Now, you should have a graph of the given curve with the positive orientation indicated.