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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Sketch the region in the plane consisting of all points (x, y) such that

Chapter 4, Problem 14P, Sketch the region in the plane consisting of all points (x, y) such that

To determine

To sketch: The set of all points (x,y) such that 2xy|xy|x2+y2.

Explanation

Definition used:

An absolute function is defined as, |x|={xif x0xif x<0.

Calculation:

First sketch the region in the plane consisting of all points (x,y) such that 2xy|xy|.

There exist two cases for Part 1.

Case 1:

Let xy0. Then,

2xy|xy|2xyxy

This curve will consists of the region below the line x=y and the region under the curve 2xy=xy that lies in the quadrant IV.

The graph of this case is shown below in Figure 1.

In Figure 1, the shaded region indicates the area required.

Case 2:

Let xy<0 Then,

2xy|xy|2xyyx

This curve will consists of the region above the line x=y and the region above the curve 2xy=yx that lies in the quadrant II.

The graph of this case is shown below in Figure 2.

In Figure 2, the shaded region indicates the area required.

Combine cases 1 and 2 and obtain the graph of 2xy|xy| as shown in below Figure 3.

In Figure 3, the shaded region indicates the set of all points (x,y) such that 2xy|xy|.

Similarly sketch the region in the plane consisting of all points (x,y) such that |xy|x2+y2

There exist two cases for Part 2.

Case 1:

Let xy0

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