Problem 26 - The Calculus with Analytic Geometry

Question

In Exercises 21 through 26 , draw a sketch of a contour map of the function \(f\) showing the level curves of \(f\) at the given numbers. The function \(f\) for which \(f(x, y)=(x-3) /(y+2)\) at \(4,2,1, \frac{1}{2}, \frac{1}{4}, 0,-\frac{1}{4},-\frac{1}{2},-1,-2\), and \(-4\).

Step-by-Step Solution

Answer

Plot lines: \(x = ky + 2k + 3\) for each \(k\) value.

Step 1: Understand the Function and Levels

Identify the function given as \(f(x, y) = \frac{x-3}{y+2}\) and the level curves to be drawn at \([4, 2, 1, \frac{1}{2}, \frac{1}{4}, 0, -\frac{1}{4}, -\frac{1}{2}, -1, -2, -4]\).

Step 2: Set Up Equations for Each Level

For each level \(k\), set up the equation \(\frac{x-3}{y+2} = k\). This leads to the equation \(x - 3 = k(y + 2)\).

Step 3: Rearrange for Each Level

Rearrange the equation for each specific level \(k\): \(x = ky + 2k + 3\).

Step 4: Plot Level Curves

Plot the curves for each specific level value. For example, for \(k = 4\), the curve is \(x = 4y + 8 + 3\), which becomes \(x = 4y + 11\). Continue this process for each level. The specific equations are: \(x = 4y + 11\), \(x = 2y + 7\), \(x = y + 5\), \(x = \frac{1}{2}y + 4\), \(x = \frac{1}{4}y + 3.5\), \(x = 3\) (for \(k = 0\)), \(x = -\frac{1}{4}y + 2.5\), \(x = -\frac{1}{2}y + 2\), \(x = -y + 1\), \(x = -2y - 1\), \(x = -4y - 5\).

Step 5: Draw the Contour Map

On a coordinate plane, draw the lines represented by each equation. These lines are the level curves for the function \(f(x, y)\). They will be straight lines with different slopes intersecting the plane based on their specific values.

Key Concepts

Level Curves

Level curves are a way to represent a three-dimensional surface on a two-dimensional plane. It involves plotting curves where the function holds a constant value, known as a 'level'. For our problem, we analyze the function \(f(x, y) = \frac{x - 3}{y + 2}\). Here, the level curves are specific equations obtained by setting the function to certain constant values. For example, if we set the level \(f = k\), we get \( x = ky + 2k + 3 \).
These equations represent straight lines in the xy-plane. Each line shows where the function \(f(x, y)\) will have a specific value across all coordinates. Drawing level curves helps to understand how the function behaves and changes over its domain.

Two-Variable Functions

A two-variable function, for instance \(f(x, y)\), involves two independent variables, x and y, and one dependent variable, f. In our exercise example \(f(x, y) = \frac{x-3}{y+2}\), the output depends on the values of both x and y. To analyze such functions, we often look at slices (level curves) or use visual tools like contour maps.
Understanding two-variable functions is crucial because they form the basis of more complex concepts in fields like physics, economics, and engineering. These functions can be visualized in three-dimensions with the z-axis representing the function's value, f. Therefore, drawing contour maps (two-dimensional representations) greatly simplifies the study and interpretation of these functions.

Graphing Equations

Graphing equations in the form \(x = ky + 2k + 3 \) for different values of k, as derived from our function \(f(x, y) = \frac{x-3}{y+2}\), helps to plot the level curves. Each level curve in our contour map will be a straight line with a specific slope and y-intercept. For example: \[ x = 4y + 11, \ (k=4), \ x = -2y - 1, \ (k=-2) \].
These lines illustrate how the function's value changes across the xy-plane. To graph these equations, we plot multiple lines on a coordinate plane. Each line represents a different level value and provides a clear visualization of changes in function values. This method is efficient and provides an intuitive grasp of the function's behavior over its domain.

• The slope and y-intercept vary with k.
• Lines may be positive or negative sloped.