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Chapter 11: Problem 35

Graph the curves described by the following functions, indicating the positive orientation. $$\mathbf{r}(t)=e^{-t / 20} \sin t \mathbf{i}+e^{-t / 20} \cos t \mathbf{j}+t \mathbf{k}, \text { for } 0 \leq t<\infty$$

Short Answer

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Short Answer: Given the vector function \(\mathbf{r}(t)=e^{-t / 20} \sin t \mathbf{i}+e^{-t / 20} \cos t \mathbf{j}+t \mathbf{k}\), its behavior for every t value can be analyzed by separating its x, y, and z components. The x and y components have a common decaying exponential term and oscillate due to sine and cosine, while the z component increases linearly. The overall curve behavior will show a spiral that moves closer to the z-axis as t increases, displaying a clear positive orientation along the z-axis. To graph this curve, use a 3D graphing tool to visualize how the curve behaves as t increases.

Step by step solution

01

Analyze the given function

The given function is written as: $$ \mathbf{r}(t)=e^{-t / 20} \sin t \mathbf{i}+e^{-t / 20} \cos t \mathbf{j}+t \mathbf{k} $$ Where \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the unit vectors in the x, y, z directions respectively. It is important to understand the t range goes from \(0 \leq t < \infty\). This will help us analyze the function's behavior.
02

Break-down the function into x, y, and z components

Let's separate the x, y, and z components for a clearer understanding: $$ \begin{cases} x(t) = e^{-t / 20} \sin t \\ y(t) = e^{-t / 20} \cos t \\ z(t) = t \end{cases} $$ Now that we have the components, we can see that all of them are functions of t.
03

Identify patterns in the components of the function

Analyzing the components: 1. Both x(t) and y(t) have a common term, \(e^{-t / 20}\), which is a decaying exponential function as t goes to infinity. This means that both x(t) and y(t) will start positive and decay towards zero. 2. In x(t), we have a sine function, and in y(t), we have a cosine function. These two functions will oscillate between -1 and 1. This means that as t goes to infinity, the oscillations of both the x and y components eventually get smaller. 3. The z(t) component is a simple linear function. This means that as t increases, z(t) increases linearly.
04

Use component analysis to determine the overall curve behavior

We can combine our observations from step 3 to determine the overall behavior of the curve: 1. As t goes to infinity, the x and y components will decay towards zero and oscillate because of their sine and cosine terms. 2. At the same time, the z component will keep increasing linearly. Putting this all together, we can visualize the curve as a spiraling motion towards the z-axis, and as t goes to infinity, the curve will approach the z-axis.
05

Graph the curve

Now we are ready to graph the curve. You can use a graphing tool to sketch the curve for a better visualization: 1. The x(t) and y(t) components will spiral inward towards the origin due to their oscillating and decaying nature. 2. The z(t) component will increase linearly as t increases. In conclusion, the graph will be a spiral that moves closer to the z-axis as t increases, showing a clear positive orientation along z-axis. To visualize this better, use a 3D graphing tool to see how the curve behaves as t increases.

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Most popular questions from this chapter

Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in the plane. a. Use the Triangle Rule for adding vectors to explain why \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}| .\) This result is known as the Triangle Inequality. b. Under what conditions is \(|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}| ?\)

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