/*! 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 19 Draw the \(x y\) region \(R\) th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 19

Draw the \(x y\) region \(R\) that corresponds to the \(u v\) square \(S\) with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of \(R\) and then its sides (bike a jigsaw puzzle). $$ x=e^{2 u+v}, y=e^{u+2 v} $$

Short Answer

Expert verified
The region \( R \) is defined by the corners: \((1,1), (e^2,e), (e,e^2), (e^3,e^3)\).

Step by step solution

01

Determine the Expression for Corner Points

First, we'll evaluate the given expressions for \( x \) and \( y \) at each corner of the \( uv \) square \( S \). The corners of \( S \) are given as \((0,0), (1,0), (0,1), (1,1)\). Let's substitute each corner into the equations \( x = e^{2u+v} \) and \( y = e^{u+2v} \).
02

Calculate Coordinates for Corner (0,0)

For the corner \((0,0)\), substitute \( u = 0 \) and \( v = 0 \) into the equations:\[ x = e^{2(0) + 0} = e^0 = 1 \]\[ y = e^{0 + 2(0)} = e^0 = 1 \]Thus, the \( xy \) coordinates for the corner \((0,0)\) are \((1,1)\).
03

Calculate Coordinates for Corner (1,0)

For the corner \((1,0)\), substitute \( u = 1 \) and \( v = 0 \) into the equations:\[ x = e^{2(1) + 0} = e^2 \]\[ y = e^{1 + 2(0)} = e \]Thus, the \( xy \) coordinates for the corner \((1,0)\) are \((e^2, e)\).
04

Calculate Coordinates for Corner (0,1)

For the corner \((0,1)\), substitute \( u = 0 \) and \( v = 1 \) into the equations:\[ x = e^{2(0) + 1} = e \]\[ y = e^{0 + 2(1)} = e^2 \]Thus, the \( xy \) coordinates for the corner \((0,1)\) are \((e, e^2)\).
05

Calculate Coordinates for Corner (1,1)

For the corner \((1,1)\), substitute \( u = 1 \) and \( v = 1 \) into the equations:\[ x = e^{2(1) + 1} = e^3 \]\[ y = e^{1 + 2(1)} = e^3 \]Thus, the \( xy \) coordinates for the corner \((1,1)\) are \((e^3, e^3)\).
06

Summary of Transformed Corners

Now we have calculated all transformed \( xy \) corners from the \( uv \) square:- \((0,0)\) becomes \((1,1)\)- \((1,0)\) becomes \((e^2,e)\)- \((0,1)\) becomes \((e,e^2)\)- \((1,1)\) becomes \((e^3,e^3)\)These are the vertices of the region \( R \).
07

Draw the Region \( R \)

Using the transformed vertices, plot the points \((1,1), (e^2,e), (e,e^2), (e^3,e^3)\). Connect these points like a jigsaw to form a region. The outcome is a contrast to the original square shape, but these points represent the boundary of region \( R \) in the \( xy \) 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.

Coordinate Transformation
In multivariable calculus, coordinate transformation is a process where we switch from one coordinate system to another. This transition simplifies complex equations or helps in visualizing problems in a new light. In the given exercise, we transformed a square in the \(uv\) plane into a region in the \(xy\) plane.
  • The original \(uv\) square had corners at \((0,0), (1,0), (0,1), (1,1)\).
  • We used the transformations \(x = e^{2u + v}\) and \(y = e^{u + 2v}\) to find the new points in the \(xy\) plane.
  • This transformation maps each \((u, v)\) point to a new \((x, y)\) coordinate.
After transforming each corner of the \(uv\) square, we observed that the spatial relationship and distances have changed. By analyzing how these points shift, we gain insights into how the transformation reshapes the region. It鈥檚 like placing a puzzle piece into a matching spot in a jigsaw puzzle, ensuring the correct orientation and fit.Coordinate transformation not only allows for visual representation changes but also often simplifies problem-solving in theoretical and applied scenarios.
Exponential Functions
Exponential functions are mathematical functions of the form \(e^x\), where \(e\) is Euler's number, approximately equal to 2.718. They play a critical role in various scientific domains due to their unique growth properties.
  • These functions grow rapidly and are used to model phenomena such as population growth, radioactive decay, and even financial calculations.
  • In our exercise, the equations \(x = e^{2u + v}\) and \(y = e^{u + 2v}\) involve exponential functions with variables \(u\) and \(v\).
  • Here, the exponential terms transform each \(uv\) coordinate pair into an \(xy\) coordinate pair, stretching or compressing based on the exponents.
The exponential nature of these transformations can create wide variances in growth along different axes. This property was observed when distinct changes in the \(uv\) square led to an altered region in the \(xy\) plane. Visualizing these functions graphically shows how rapidly values can scale, demonstrating their impact in mathematics and real-world applications alike.
XY Plane
The \(xy\) plane is a two-dimensional coordinate system crucial in graphing relationships and transformations between different mathematical entities. It consists of two axes:
  • The \(x\)-axis, which runs horizontally.
  • The \(y\)-axis, which runs vertically.
  • Points in this plane are defined by pairs \((x,y)\).
This plane provides a simple, yet powerful, platform for mapping the transformed region resulting from the given exercise. When we plot the transformed points: \((1,1), (e^2,e), (e,e^2), (e^3,e^3)\), we shift from abstract expressions to a visual representation of our calculations.The characteristics of the \(xy\) plane allow for varied visual interpretation of mathematical functions, geometry, and transformations like we did with our coordinate transformation from the \(uv\) system. Understanding the structure and utility of the \(xy\) plane enriches problem-solving skills in calculus, making it invaluable in both education and professional contexts.

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

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Find the limits of integration in \(\iiint d x d y d z\) and the volume of solids. Draw a very rough picture. The volume above \(z=0\) below the cone \(\sqrt{x^{2}+y^{2}}=1-z\).

Compute the double integrals by two integrations. $$ \int_{0}^{1} \int_{1}^{2} y e^{x y} d x d y \text { and } \int_{0}^{1} \int_{0}^{3} d y d x \sqrt{3+2 x+y} $$

From the limits of integration deserihe each region and find its volume. The inner integral has the inner limits. $$ \int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\sin \phi} \rho^{2} \sin \phi d \rho d \phi d \theta $$

Draw the \(x y\) region \(R\) that corresponds to the \(u v\) square \(S\) with corners (0,0),(1,0),(0,1),(1,1) . Locate the corners of \(R\) and then its sides (bike a jigsaw puzzle). $$ x=u, y=v\left(l+u^{2}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure 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.