Problem 17 - Applied Calculus

Question

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the con- tours and how they are spaced. \(f(x, y)=x+y+1\)

Step-by-Step Solution

Answer

The contours for \( f(x, y) = x + y + 1 \) are parallel lines with slope -1, spaced equally apart.

Step 1: Understanding the Function

The given function is a plane given by \( f(x, y) = x + y + 1 \). This is a linear function where each contour line represents a specific constant value of \( f(x, y) \).

Step 2: Identify Contours

To sketch the contour diagram, set \( f(x, y) = k \), where \( k \) is a constant. For example, choose \( f(x, y) = 0, 1, 2, 3 \) to have at least four contours. Substituting these, we have contour equations: \( x + y + 1 = 0 \), \( x + y + 1 = 1 \), \( x + y + 1 = 2 \), \( x + y + 1 = 3 \).

Step 3: Solve for y in terms of x

Rearrange each contour equation to express \( y \) in terms of \( x \):\( y = -x - 1 \) for \( k = 0 \), \( y = -x \) for \( k = 1 \), \( y = -x + 1 \) for \( k = 2 \), and \( y = -x + 2 \) for \( k = 3 \).

Step 4: Sketch the Contours

Plot each of these lines on the xy-plane. All lines are parallel and have a slope of \(-1\). The line \( y = -x - 1 \) intercepts the y-axis at \(-1\), \( y = -x \) at \(0\), and so forth. The spacing between lines is constant because the difference between contour levels \( k \) is constant (1 unit).

Step 5: Interpret the Diagram

The contour diagram consists of parallel lines equally spaced. Each line corresponds to a constant value of \( f(x, y) \) which increases by 1 unit as you move from one line to the next upward. This indicates the consistent change in \( f(x, y) \) across the plane.

Key Concepts

Linear Functions

A linear function is a mathematical expression that creates a straight line when graphed. These functions have constant rates of change and are typically expressed in the form \( f(x, y) = ax + by + c \). In this specific scenario, our function is expressed as \( f(x, y) = x + y + 1 \). This means the function simply adds the values of \( x \) and \( y \), before further adding 1.

• The expression \( x + y + 1 \) indicates that the output or value of \( f(x, y) \) is influenced equally by \( x \) and \( y \).
• The addition of a constant (1 in this case) shifts the entire line without changing its slope.
• The function is characterized as linear because each variable increases or decreases by a constant rate.

Understanding linear functions is crucial as they are foundational to more advanced mathematical concepts such as calculus and differential equations. This simplicity also makes them quite visible in real-world applications.

Contour Lines

Contour lines are used to represent constant values of a function on a graph. When dealing with a function of two variables, like \( f(x, y) \), contour lines are drawn to show elevation, levels, or values over a two-dimensional plane.

• In our example, each contour line represents situations where \( f(x, y) \) equals some constant \( k \).
• These lines illustrate where the function shares the same output value, essentially 'slicing' through the graph at different heights.
• The contour lines \( y = -x - 1 \), \( y = -x \), \( y = -x + 1 \), \( y = -x + 2 \) are parallel, indicating uniform slope.
• In this linear function, it's easier to interpret because lines do not curve, illustrating how the function consistently behaves across its field.

Drawing such lines on diagrams enhances understanding of the function's behavior across different inputs, particularly how changes in inputs relate to changes in outputs.

Constant Value

Constant values in contour diagrams signify levels where the function \( f(x, y) \) remains unchanged, despite differing \( x \) and \( y \). In contour maps, these constants \( k \) mark specific horizontal slices through a three-dimensional graph.

• Given our function \( f(x, y) = x + y + 1 \), selecting different constants like 0, 1, 2, 3 helps plot the contours.
• The constant value is crucial as it defines each contour line, dictating where on the plane the function maintains a specific outcome.
• The chosen constants lead to the equations \( x + y + 1 = k \), forming distinct, parallel lines for each value.
• These constant values simplify complex graphs, transforming them into understandable and manageable sections.

Within mathematical and real-world contexts, constant values assist in maintaining clarity when interpreting comprehensive data sets, enabling easier analysis of continuous functions over diverse platforms.