Contour COrsiter the following f(xi4): x² +442 Draw a showing the leuce f(x,4)=K,where K=4,K:9 ¢ k=16 mep (urves correspording to
Contour COrsiter the following f(xi4): x² +442 Draw a showing the leuce f(x,4)=K,where K=4,K:9 ¢ k=16 mep (urves correspording to
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Topic: Level Curves and Contour Maps**
**Problem Statement:**
Consider the following function \( f(x, y) = x^2 + 4y^2 \). Draw a contour map showing the level curves corresponding to \( f(x, y) = k \), where \( k = 4 \), \( k = 9 \), and \( k = 16 \).
**Explanation:**
To create a contour map for the function \( f(x, y) = x^2 + 4y^2 \), follow these steps:
1. **Identify the Function:**
- The function given is a quadratic form, which suggests that its level curves will be ellipses centered at the origin.
2. **Level Curves:**
- For \( k = 4 \), the level curve is described by the equation \( x^2 + 4y^2 = 4 \).
- For \( k = 9 \), the equation becomes \( x^2 + 4y^2 = 9 \).
- For \( k = 16 \), the level curve is \( x^2 + 4y^2 = 16 \).
3. **Draw the Contour Map:**
- Each equation represents an ellipse on the coordinate plane.
- The contour map will illustrate these ellipses corresponding to each value of \( k \).
- The axes of the ellipses come from solving for \( x \) and \( y \) in terms of \( k \). Specifically, solve for \( y = \pm \sqrt{\frac{k - x^2}{4}} \) and \( x = \pm \sqrt{k} \) to find the bounds for plotting.
By plotting these ellipses, the contour map will visually represent how \( k \) influences the layout and size of level curves for the function \( f(x, y) \).
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