Problem 19 - APEX Calculus

Question

Describe in words and sketch the level curves for the function and given \(c\) values. $$ f(x, y)=\frac{2 x-2 y}{x^{2}+y^{2}+1} ; c=-1,0,1 $$

Step-by-Step Solution

Answer

Level curves for \(c=-1,0,1\) are circles for \(c=-1\) and \(c=1\), and the line \(y=x\) for \(c=0\).

Step 1: Understand Level Curves

Level curves of a function are the curves along which the function takes a constant value. For the given function \( f(x, y) = \frac{2x - 2y}{x^2 + y^2 + 1} \), we need to find the sets of points \((x, y)\) such that \( f(x, y) = c \) for different \(c\) values: -1, 0, and 1.

Step 2: Set up the Equations

For each \(c\) value, set the function equal to \(c\):- \( f(x, y) = -1 \) implies \( \frac{2x - 2y}{x^2 + y^2 + 1} = -1 \).- \( f(x, y) = 0 \) implies \( \frac{2x - 2y}{x^2 + y^2 + 1} = 0 \).- \( f(x, y) = 1 \) implies \( \frac{2x - 2y}{x^2 + y^2 + 1} = 1 \).

Step 3: Solve for Each Level Curve: \(c = -1\)

For \( c = -1 \):\[2x - 2y = -1(x^2 + y^2 + 1)\]Simplifying, we get:\[2x - 2y = -x^2 - y^2 - 1\]Rearrange terms:\[x^2 + y^2 + 2x - 2y + 1 = 0\]This equation represents a circle.

Step 4: Solve for Each Level Curve: \(c = 0\)

For \( c = 0 \):\[2x - 2y = 0(x^2 + y^2 + 1)\] which simplifies directly to:\[2x = 2y\]Thus, \(x = y\), which represents the line \(y = x\).

Step 5: Solve for Each Level Curve: \(c = 1\)

For \( c = 1 \):\[2x - 2y = 1(x^2 + y^2 + 1)\]Simplifying, we have:\[2x - 2y = x^2 + y^2 + 1\]Rearrange terms:\[x^2 + y^2 - 2x + 2y + 1 = 0\]This equation represents another circle.

Step 6: Sketch the Level Curves

With the equations derived:- For \(c = -1\), the level curve is a circle centered at some point shifted 1 unit.- For \(c = 0\), the level curve is the line \(y = x\).- For \(c = 1\), the level curve is another circle centered differently from the first.Draw these on an \((x, y)\) coordinate plane for visualization.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface that holds the coordinates of any point represented by \((x, y)\). Coordinates serve as a useful tool for graphing and visualizing mathematical functions. This plane is essentially a flat surface sliced vertically by the "y-axis" and horizontally by the "x-axis".

• The x-axis runs left to right, acquiring negative-to-positive values from left to right.
• The y-axis runs up and down and takes on negative-to-positive values from bottom to top.

Positions on the plane are marked by coordinates. For example, point \((3, 2)\) represents a position 3 units to the right on the x-axis and 2 units up on the y-axis.
The origin point, \((0, 0)\), represents the center of this plane where the x-axis and y-axis intersect. This coordinate plane is essential for plotting functions like the level curves in our exercise.

Level Curve Equations

Level curves help visualize functions on the coordinate plane. They represent points where the function has a constant value, noted by \(f(x, y) = c\), where \(c\) is the constant. This simplifies the visualization of functions with two variables into two-dimensional graphs.
When crafting level curve equations like in the given exercise, you start by equating the function to different constant values. For example, substituting \(c = -1, 0, 1\) into the function \(f(x, y) = \frac{2x - 2y}{x^2 + y^2 + 1}\), sets up different scenarios to map out each curve:

• If \(c = -1\), solving yields an equation expressing a circular path.
• If \(c = 0\), the resulting equation simplifies to \(x = y\), a simple straight line.
• If \(c = 1\), another circle emerges with a different center than the first.

Graphing these equations on the coordinate plane allows you to better understand how the function behaves across different constant values.

Constant Values in Functions

Constant values in functions allow us to isolate specific characteristics of the function by maintaining specific values for output. When dealing with level curves, constants \(c\) are pivotal in configuring equations for the curves.
When a function of two variables \(f(x, y)\) is equated to a constant \(c\), it becomes easier to explore how the function operates under fixed conditions:

• Setting \(f(x, y) = c\) lets you find the path in the plane where the output of the function remains steady at \(c\).
• Each constant value separates the function's output into clearer picture forms, transforming complex multi-dimensional data into two-dimensional, easily-read forms like circles or lines.

This approach helps to understand the distribution and impact of a function across its domain, making complex calculations and analyses simpler and more intuitive.