Problem 5 Express the parametric equations... [FREE SOLUTION]

Chapter 14: Problem 5

Express the parametric equations as a single vector equation of the form $\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}$ or $\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}$\ $$x=3 \cos t, y=t+\sin t$$

Short Answer

Expert verified

$\mathbf{r}(t) = 3\cos t\, \mathbf{i} + (t + \sin t)\, \mathbf{j}$.

Step by step solution

Understanding the Problem

We are given two parametric equations: $ x = 3 \cos t $ and $ y = t + \sin t $. Our goal is to express these parametric equations as a single vector equation.

Identify Components of the Vector Equation

To express the parametric equations as a vector equation of the form $ \mathbf{r} = x(t) \mathbf{i} + y(t) \mathbf{j} $, we identify $ x(t) = 3 \cos t $ and $ y(t) = t + \sin t $.

Formulate Vector Equation

Using the components identified, we can write the vector equation: $ \mathbf{r}(t) = 3 \cos t \mathbf{i} + (t + \sin t) \mathbf{j} $.

Verify the Vector Equation

Check that the vector equation correctly represents the parametric equations. $ x(t) = 3 \cos t $ corresponds to the $ \mathbf{i} $ component, and $ y(t) = t + \sin t $ corresponds to the $ \mathbf{j} $ component of the vector equation, verifying the solution is correct.

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Key Concepts

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

Parametric Equations

Parametric equations are a powerful mathematical tool used to describe a set of related quantities. In simple terms, they define a relationship between variables using a third variable, often called a parameter. For example, consider two parametric equations: $ x = f(t) $ and $ y = g(t) $. Here, $ t $ is the parameter, and $ f(t) $ and $ g(t) $ express $ x $ and $ y $ as functions of $ t $. This method is particularly useful for modeling curves, where the standard $ y=f(x) $ form does not suffice.

When using parametric equations:

Each parameter represents a different aspect of the system, allowing complex shapes and paths to be defined.
They can help find intersections, tangents, and lengths of curves in calculus.
Often used in physics and engineering to describe motion or paths.

Vector Equation

Vector equations combine parametric equations by using vectors to express them as a single statement. In the context of our example, each function, $ x(t) $ and $ y(t) $, becomes a component of a vector. That vector represents a point or a moving particle's position in a two-dimensional space. So, for our given equations, we express them in the form $ \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} $.

Here's the breakdown:

$ \mathbf{i} $ and $ \mathbf{j} $ are unit vectors pointing in the direction of the x-axis and y-axis, respectively.
The functions $ x(t) = 3 \cos t $ and $ y(t) = t + \sin t $ tell us how the x and y coordinates change as $ t $ varies.
This form captures the trajectory of a point in the plane as $ t $ changes.

Vector equations make understanding and visualizing the movement described by parametric equations easier.

Trigonometric Functions

Trigonometric functions are a category of mathematical functions crucial in studying cycles and periodic phenomena. Handling periodic behavior, like that shown in $ x = 3 \cos t $, is a classic application of these functions. Let's break down their relevance:

They are used to describe oscillations, rotations, and waves, essential in physics and engineering.
In terms of parametric equations, trig functions help depict curved paths, such as circles or spirals.
Functions like $ \cos t $ repeat values over intervals, making them predictable for cyclical movements.

Amidst calculus problem-solving, understanding how these functions operate can simplify solving complex problems involving periodicity.

Calculus Problem Solving

Calculus problem solving revolves around using derivatives and integrals to model and interpret real-world situations. Parametric equations and trigonometric functions are tools often employed in these scenarios. They allow for a more dynamic modeling of motion and change. Whether tracking a particle's motion along a path or analyzing forces in engineering, calculus provides the framework.

To effectively solve calculus problems:

Understand the relationships between the parametric expressions and the real-world phenomena they represent.
Employ derivatives to study the rate of change, such as analyzing the velocity from displacement.
Use vectors to simplify multidimensional problems, allowing a systematic approach to complex movement simulations.

Mastery in calculus involves knowing how and when to apply these mathematical tools effectively.

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91影视

Express the parametric equations as a single vector equation of the form \(\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}\) or \(\mathbf{r}=x(t) \mathbf{i}+y(t) \mathbf{j}+z(t) \mathbf{k}\)\ $$x=3 \cos t, y=t+\sin t$$

Short Answer

Step by step solution

Understanding the Problem

Identify Components of the Vector Equation

Formulate Vector Equation

Verify the Vector Equation

Key Concepts

Parametric Equations

Vector Equation

Trigonometric Functions

Calculus Problem Solving

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