/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Graph the curves described by th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 40

Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility. $$\mathbf{r}(t)=\cos t \sin 3 t \mathbf{i}+\sin t \sin 3 t \mathbf{j}+\sqrt{t} \mathbf{k}, \text { for } 0 \leq t \leq 9$$

Short Answer

Expert verified
Answer: The anticipated shape of the curve is a spiral or helix moving upward in the positive z direction as it progresses along the x and y plane in a periodic motion.

Step by step solution

01

Analyze the x component

The x component of the function is given by $$x(t) = \cos t \sin 3t$$. We can see that the cosine function has a periodicity of $$2\pi$$ and the sine function has a periodicity of $$\frac{2\pi}{3}$$. Combining these two functions, the x component will exhibit a periodic behavior but with a different periodicity. The amplitude of the x component will not exceed 1, since both cosine and sine have maximum absolute values of 1.
02

Analyze the y component

The y component of the function is given by $$y(t) = \sin t \sin 3t$$. Like in the x component, the y component also shows periodic behavior due to the presence of sine functions. The maximum amplitude for the y component will also not exceed 1.
03

Analyze the z component

The z component of the function is given by $$z(t) = \sqrt{t}$$. Unlike the x and y components, the z component is increasing monotonically as t increases. This means that as the curve progresses along the x and y plane, it will also move upwards in the positive z direction.
04

Anticipate the shape of the curve

Based on the analysis, we can anticipate that the curve will exhibit a periodic behavior along the x and y plane, with its amplitude not exceeding 1. At the same time, the z component will increase with t. Thus, the overall shape of the curve will be like a spiral or helix moving upward in the positive z direction as it progresses along the x and y plane in a periodic motion.
05

Graph the curve using a graphing utility

Utilize a graphing utility or software (such as Desmos, GeoGebra, or WolframAlpha) to plot the given vector-valued function in three-dimensional space. Enter the function components x(t), y(t), and z(t) separately, as well as the parameter range $$0 \leq t \leq 9$$. The resulting graph should confirm the anticipated shape of the curve as a spiraling upward curve with periodic motion along the x and y plane.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D Graphing
Three-dimensional graphing brings mathematical functions to life by giving them depth and volume. Unlike two-dimensional graphs that are limited to width and height, 3D graphs allow us to explore the interplay of three components: the x, y, and z axes. This can add complexity but also a new level of understanding by visualizing how functions and equations behave in space rather than just on a flat plane.

Visualizing in 3D:
  • **x-axis** is typically the horizontal component.
  • **y-axis** adds vertical variation.
  • **z-axis** brings depth, making the object appear from different angles.
In 3D graphing, we often use vector-valued functions, which take a parameter that passes through multiple values to generate points along a curve or surface. A good grasp of 3D graphing helps us to predict function behavior even before graphing, as with the exercise above. By picking apart each component, you can form a clear visualization of the curve's movement through space. Technology tools like Desmos, GeoGebra, and WolframAlpha help bring these ideas into view, making complex shapes more accessible and easier to analyze.
Parametric Curves
In vector calculus, parametric curves are powerful tools for representing complex shapes and movements. Instead of expressing functions using x and y relationships, parametric equations use a parameter like 't', which can represent time, producing a series of x, y, and sometimes z coordinates.

Understanding how they work:
  • Each coordinate \( (x, y, z) \) is given by a function of 't', such as \( x(t) \) or \( y(t) \).
  • As 't' varies, it defines a path along which the point \( (x, y, z) \) moves.
  • Offers significant flexibility and insight into how objects behave over time.
Parametric curves allow us to model real-world phenomena more effectively than standard function plots. In the given exercise, the parametric definitions help anticipate the curve's shape as it winds through space, showing periodic behaviors in x and y while spiraling upwards through z. This helps form a clear mental image of how the curve behaves even before graphing.
Helix
A helix is a type of three-dimensional curve resembling a spiral staircase or the winding of a spring. In mathematics, helixes are categorized based on their reactions to component changes across x, y, and z dimensions. The defining characteristic of a helix is its spiral formation and constant ascent or descent in three-dimensional space.

What makes a helix:
  • **Periodicity in x and y:** Usually involves sinusoidal functions leading to circular or elliptical paths around the z-axis.
  • **Consistent change in z:** As t increases, z often changes at a regular rate, creating the helical rise or fall.
  • **Visual Model:** Think of the inner workings of a spring, coiling upwards with each turn.
The curve in the exercise exhibits similar traits. It spirals due to periodic influences from x and y components, formed from sine and cosine functions. The increasing z component, due to \( \sqrt{t}\), ensures a continuous upward climb over time, perfectly embodying a helix. Helixes are vital in many fields, from engineering to biology, offering a clear view of intricate processes and phenomena.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(\mathbb{R}^{3}\) Consider the vectors \(\mathbf{I}=\langle 1 / 2,1 / 2,1 / \sqrt{2}), \mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}, 0\rangle,\) and \(\mathbf{K}=\langle 1 / 2,1 / 2,-1 / \sqrt{2}\rangle\) a. Sketch I, J, and K and show that they are unit vectors. b. Show that \(\mathbf{I}, \mathbf{J},\) and \(\mathbf{K}\) are pairwise orthogonal. c. Express the vector \langle 1,0,0\rangle in terms of \(\mathbf{I}, \mathbf{J},\) and \(\mathbf{K}\).

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Show that for real numbers \(u_{1}, u_{2},\) and \(u_{3},\) it is true that \(\left(u_{1}+u_{2}+u_{3}\right)^{2} \leq 3\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)\). (Hint: Use the Cauchy-Schwarz Inequality in three dimensions with \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) and choose \(\mathbf{v}\) in the right way.)

Parabolic trajectory Consider the parabolic trajectory $$ x=\left(V_{0} \cos \alpha\right) t, y=\left(V_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} $$ where \(V_{0}\) is the initial speed, \(\alpha\) is the angle of launch, and \(g\) is the acceleration due to gravity. Consider all times \([0, T]\) for which \(y \geq 0\) a. Find and graph the speed, for \(0 \leq t \leq T.\) b. Find and graph the curvature, for \(0 \leq t \leq T.\) c. At what times (if any) do the speed and curvature have maximum and minimum values?

Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle\), and \(\mathbf{w}=\) \(\left\langle w_{1}, w_{2}, w_{3}\right\rangle\). Let \(c\) be a scalar. Prove the following vector properties. \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\)

Let \(D\) be a solid heat-conducting cube formed by the planes \(x=0, x=1, y=0, y=1, z=0,\) and \(z=1 .\) The heat flow at every point of \(D\) is given by the constant vector \(\mathbf{Q}=\langle 0,2,1\rangle\) a. Through which faces of \(D\) does \(Q\) point into \(D ?\) b. Through which faces of \(D\) does \(\mathbf{Q}\) point out of \(D ?\) c. On which faces of \(D\) is \(Q\) tangential to \(D\) (pointing neither in nor out of \(D\) )? d. Find the scalar component of \(\mathbf{Q}\) normal to the face \(x=0\). e. Find the scalar component of \(\mathbf{Q}\) normal to the face \(z=1\). f. Find the scalar component of \(\mathbf{Q}\) normal to the face \(y=0\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.