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Magazine Chapter 9: Problem 18
Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=2 x-y\)
Short Answer
Expert verified
Contour lines are straight, parallel, and evenly spaced with slope 2.
Step by step solution
01
Identify the General Form of Contour Lines
For a function \( f(x, y) = c \), the contour lines satisfy the equation \( 2x - y = c \), where \( c \) is a constant value for each contour line.
02
Choose Several Values for c
Select at least four values for \( c \). For example, choose \( c = -2, 0, 2, 4 \). These values will determine the specific contour lines of the function.
03
Write Equations for Each Contour Line
For each chosen value of \( c \), write the equation of the contour line:- For \( c = -2 \), the equation is \( 2x - y = -2 \).- For \( c = 0 \), the equation is \( 2x - y = 0 \).- For \( c = 2 \), the equation is \( 2x - y = 2 \).- For \( c = 4 \), the equation is \( 2x - y = 4 \).
04
Rearrange Equations to Standard Line Form
Convert the contour line equations to the slope-intercept form \( y = mx + b \) to easily sketch them:- For \( 2x - y = -2 \), rearrange to \( y = 2x + 2 \).- For \( 2x - y = 0 \), rearrange to \( y = 2x \).- For \( 2x - y = 2 \), rearrange to \( y = 2x - 2 \).- For \( 2x - y = 4 \), rearrange to \( y = 2x - 4 \).
05
Sketch the Contour Lines
Plot these lines on a graph. Each of these lines is a straight line with the same slope, which is \( m=2 \), but with differing y-intercepts. Use a different color or notation to label each contour line with its corresponding \( c \) value.
06
Describe the Contour Lines and Their Spacing
The contour lines are straight, parallel lines because their slope \( m = 2 \) is constant. They are spaced evenly vertically, with each line being 2 units away from its neighboring line due to the change in \( c \) values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Contour Lines
In the world of mathematics, contour lines offer a fascinating way to represent the geometry of functions of two variables. Imagine them as topographic lines on a map that show areas of equal elevation. Similarly, contour lines of a function like \( f(x, y) = 2x - y \) represent points where the function has the same value, denoted by \( c \). To sketch contour lines, you start by setting the function equal to various constants. For example, if \( c = 0 \), then you have the equation \( 2x - y = 0 \). Each contour line corresponds to a different value of \( c \). These lines make complex surfaces easier to understand by breaking them down into simpler parts.
- Contour lines help us visualize functions in two dimensions.
- They show where a function holds the same value across a plane.
- These lines are handy for understanding surface elevations, gradients, and more.
Slope-Intercept Form
The slope-intercept form is a super useful format for equations of straight lines, written as \( y = mx + b \). In this equation:- \( m \) represents the slope, which tells you how steep the line is.- \( b \) represents the y-intercept, the point where the line crosses the y-axis.In our contour diagram task, once we set \( f(x, y) = c \), we rearrange the equations to find their slope-intercept forms:
- For \( 2x - y = -2 \), it rearranges to \( y = 2x + 2 \).
- For \( 2x - y = 0 \), it becomes \( y = 2x \).
- For \( 2x - y = 2 \), it turns into \( y = 2x - 2 \).
- For \( 2x - y = 4 \), it becomes \( y = 2x - 4 \).
Function of Two Variables
A function of two variables, such as \( f(x, y) = 2x - y \), is an equation involving two different inputs. This type of function outputs a value based on these inputs and can be visualized as a surface in 3D space.These functions are foundational in multivariable calculus, allowing us to explore complex interactions between variables. As we find contour lines for such a function, we're essentially slicing through the 3D surface to see cross-section lines where the function value is constant.
- Functions of two variables can model many real-world situations like temperature, pressure, or altitude.
- We use contour diagrams to represent variations within these systems on a flat plane.
- This makes it easier to interpret and analyze complex data across various fields.
Graph Sketching
Graph sketching involves drawing a visual representation of functions or equations. By plotting contour lines from a function like \( f(x, y) = 2x - y \), we develop a clearer understanding of its behavior.How to Sketch Contour Lines:1. Rearrange your function into slope-intercept form for easy sketching.2. Use the slope \( m \) to determine line direction and steepness.3. Mark the y-intercept \( b \) to know where to start the line.4. Plot each line using consistent spaces between them, based on \( c \) values.When done, you'll notice:
- Lines are parallel, indicating a uniform slope.
- The space between them represents the gradient of change in function values.
- Each line uniquely identifies a constant level of the function: level curves.
