Problem 6 - Applied Calculus

Question

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $$f(x, y)=3 x+3 y$$

Step-by-Step Solution

Answer

The contours are evenly spaced parallel lines with a slope of -1, shifting up for increasing constant values.

Step 1: Understanding the Function

Given the function \( f(x, y) = 3x + 3y \), we recognize that this is a linear function. Linear functions in two variables represent straight lines or planes.

Step 2: Setting Up Contour Lines

Contour lines are obtained by setting \( f(x, y) \) equal to a constant, say \( c \). This gives the equation \( 3x + 3y = c \). For a range of constants, such as \( c = 0, 3, 6, \) and \( 9 \), these represent different contour lines.

Step 3: Solving for y in Terms of x

Rearrange the contour equation \( 3x + 3y = c \) to get \( y = -x + \frac{c}{3} \). This shows that each contour line is a straight line with a slope of -1, shifted vertically by \( \frac{c}{3} \).

Step 4: Sketching the Contour Diagram

Plot the contour lines on a graph. For \( c = 0 \), the line is \( y = -x \). For \( c = 3 \), the line is \( y = -x + 1 \). For \( c = 6 \), the line is \( y = -x + 2 \). For \( c = 9 \), the line is \( y = -x + 3 \). Note that all lines are parallel, equally spaced, and increase incrementally on the y-axis as \( c \) increases.

Step 5: Describing the Contours

The contour lines are straight lines with a slope of -1, indicating they descend at a 45-degree angle. Each line shifts up equally, indicating equal spacing between contours, suggesting a uniform rate of change in the function's value.

Key Concepts

Linear Function

A linear function is one of the simplest types of functions you will encounter in mathematics. It is characterized by an equation of the form \( f(x, y) = ax + by + c \), where \( a \), \( b \), and \( c \) are constants. In this exercise, our linear function is \( f(x, y) = 3x + 3y \). This function maps inputs \( x \) and \( y \) to a linear combination of these variables.
This means that, when plotted, the function will create a straight line or a plane in a coordinate system, depending on the number of variables involved. In two dimensions, like \( x \) and \( y \), the graph would represent a flat line. This straightforward nature of linear functions makes them easy to work with, as they follow consistent behaviors across their domain.

Contour Lines

Contour lines are a key concept in understanding how to visualize three-dimensional surfaces on two-dimensional planes. They represent the set of locations at which a function has a constant value.
By setting \( f(x, y) = 3x + 3y = c \), we create contour lines for constant values \( c \). These lines help us "see" how the function behaves without needing a three-dimensional graph.

• For a linear function like \( 3x + 3y \), contour lines are straight and parallel lines.
• Each line corresponds to a specific constant \( c \), giving us a cross-section view of the function.

Contour lines are useful in many fields, such as geography for elevation maps or meteorology for pressure maps. In mathematics, they allow us to analyze how the output of a function changes with different inputs.

Graphing Contours

Graphing contour lines involves plotting the function's response to different constant values on a graph. It visually summarizes the behavior of the function and allows us to analyze changes and patterns.
To sketch the contour lines for \( f(x, y) = 3x + 3y \), we set \( 3x + 3y = c \) and rearrange it to find \( y = -x + \frac{c}{3} \). Here, \( y = -x + \frac{c}{3} \) represents a family of parallel lines with:

• A negative slope of \(-1\), indicating a consistent angle of descent.
• Vertical shifts depending on the value of \( \frac{c}{3} \).

As \( c \) increases by 3, each contour line shifts up by \( 1 \). When plotted, they show equally spaced, parallel lines, making it easy to interpret changes in the function's output. The spacing reflects how the function changes at a constant rate across the plane.

Rate of Change

Understanding the rate at which a function's output changes is crucial in mathematics. For linear functions, this rate is constant, which simplifies our analysis.
Here, the contour lines of the function \( f(x, y) = 3x + 3y \) each have a slope of \(-1\). This slope value indicates the direction and steepness of change in the function as you move across the graph. It tells us that for every unit increase in \( x \), \( y \) decreases by the same amount, and vice versa.

• This equal rate of change keeps the contour lines parallel and equally spaced.
• As \( c \) shifts by 3, the function's output increases or decreases uniformly as displayed by the vertical spacing between contour lines.

This concept clarifies how results are not just dependent on individual values, but also on their rate of change, providing a richer understanding of the function's behavior.