Problem 15 - 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. \(f(x, y)=x+y\)

Step-by-Step Solution

Answer

The contour diagram consists of parallel lines (with slope -1), equally spaced vertically by 1 unit.

Step 1: Understanding the Contour

The function given is a simple linear function: \( f(x, y) = x + y \). A contour line for \( z = c \) (where \( c \) is a constant) represents all points \((x, y)\) such that \( x + y = c \). Thus, each contour line can be represented as \( y = c - x \).

Step 2: Identifying Contour Lines

Since the contours represent the equation \( y = c - x \), each contour is a line. To draw at least four contours, select four different constant values for \( c \). For example, let's take \( c = 1, c = 2, c = 3, c = 4 \). This gives us four lines: \( y = 1 - x \), \( y = 2 - x \), \( y = 3 - x \), and \( y = 4 - x \).

Step 3: Drawing the Contour Lines

On a Cartesian plane, draw the four contour lines. For \( c = 1 \), the line \( y = 1 - x \) can be drawn by plotting two points such as \((0, 1)\) and \((1, 0)\), and drawing a line through them. Repeat similarly for \( c = 2 \), \( c = 3 \), and \( c = 4 \) to get respective lines \( y = 2 - x \), \( y = 3 - x \), and \( y = 4 - x \).

Step 4: Describing the Contours

Each contour is a straight line with a negative slope of -1. As the constant \( c \), which represents the function's value, increases, the line shifts upwards parallel to the previous one. The spacing between each of the contour lines is equal because the increase in \( c \) by 1 results in a vertical shift by 1 unit parallelly upwards.

Key Concepts

Linear Function

A linear function is one of the simplest types of functions in mathematics. It's called "linear" because its graph is a straight line. In our context, the function is defined as \( f(x, y) = x + y \). This particular linear function represents a plane in 3-dimensional space. The relationship between the inputs \( x \) and \( y \) is straightforward, as each unit increase in either \( x \) or \( y \) results in a corresponding increase in the function's value.Linear functions can be recognized by their characteristic form:

• They have constants and variables only raised to the power of one.
• They graph as straight lines on a 2D plane.

Understanding linear functions is crucial because they form the foundation for more complex mathematical concepts.

Constant Value

When we talk about a 'constant value' in the context of contour diagrams, we are referring to a fixed number that a line represents on the graph. Here, for the function \( f(x, y) = x + y \), the constant value represented by each contour line is \( c \), where \( c \) is any real number.In this scenario:

• The contour lines are the result of setting the linear function equal to these constant values.
• Each line represents the equation \( x + y = c \), which can be rewritten as \( y = c - x \).
• The constant value \( c \) dictates how the lines are spaced along the graph.

Constant values help in visualizing and understanding how the function behaves at different levels and are key in plotting contour diagrams.

Cartesian Plane

A Cartesian plane is a two-dimensional plane defined by two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). It is used extensively in math to plot equations and graph relationships between variables.For drawing contour lines:

• The x-axis and y-axis divide the plane into four quadrants.
• Points are specified by coordinates \((x, y)\).

In our contour diagram, the Cartesian plane is where we plot the lines arising from the function \( f(x, y) = x + y \).By choosing different constant values for \( c \), say \( c = 1, 2, 3, 4 \), and plotting the corresponding lines, we're able to illustrate the distribution and spacing of these contours. This plane serves as the groundwork for visual representation in mathematics.

Contour Lines

Contour lines are lines on a graph that illustrate locations where the function holds a constant value. In the case of our problem, these contour lines result from the function \( x + y = c \).Few key points about contour lines:

• They are always straight for linear functions like \( f(x, y) = x + y \).
• Each line’s placement is determined by a different constant value \( c \).
• For this function, the lines slope downward at a constant rate because this is a simple linear relationship.

In a contour diagram:- As \( c \) increases, each new line shifts parallelly upwards by one unit.- The spacing between these lines is uniform, showing a clear visual progression of the function over the Cartesian plane.This visualization helps to deeply grasp how a function behaves under different conditions and is crucial away to interpret multivariable functions visually.