Problem 36 - Calculus an Applied Approach

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ \text {Function } \quad \text { c-Values } $$ $$ f(x, y)=x^{2}+y^{2} \quad c=0,2,4,6,8 $$

Step-by-Step Solution

Answer

The level curves of function \(f(x, y)=x^{2}+y^{2}\) for c = 0,2,4,6,8 are circles centered at the origin with radii of 0, \(sqrt(2)\), 2, \(sqrt(6)\), and \(sqrt(8)\) respectively.

Step 1: Interpret the given function

Recognize the function \(f(x, y)=x^{2}+y^{2}\) as the equation of a circle in the xy-plane, with the origin (0, 0) as the center. When this function is equal to a constant c, the resulting set of points (x, y) forms a circle.

Step 2: Calculate each level curve for the given c-values

Set up the equation \(f(x, y)=x^{2}+y^{2} = c\) for each given c-value. For c = 0,2,4,6,8, the respective level curves are: \n\n- \(x^{2} + y^{2} = 0\), which is a single point at the origin (0, 0) \n- \(x^{2} + y^{2} = 2\), which forms a circle with radius \(sqrt(2)\) \n- \(x^{2} + y^{2} = 4\), which forms a circle with radius 2 \n- \(x^{2} + y^{2} = 6\), which forms a circle with radius \(sqrt(6)\) \n- \(x^{2} + y^{2} = 8\), which forms a circle with radius \(sqrt(8)\)

Step 3: Sketch the level curves

On a graph in the xy-plane, draw the level curves calculated in step 2. The level curve for c = 0 is a single point at the origin (0, 0). The other curves are circles centered at the origin with radii increasing for increasing c-value.

Key Concepts

Multivariable Functions

In the realm of mathematics, functions like \( f(x, y) = x^2 + y^2 \) are known as multivariable functions. Unlike single-variable functions, which depend on a single input, multivariable functions depend on two or more inputs. For instance, in the function given here, the outputs depend on both \( x \) and \( y \). This is why you might encounter these in subjects that model real-world phenomena where multiple factors affect the outcome.
To understand these functions better, consider them as surfaces hovering over a coordinate plane. The \( z \)-value, or output, changes with varying \( x \) and \( y \) values, which is why these visuals often look like surfaces rather than simple lines. A key aspect of examining these functions is understanding how changes in input values affect the overall shape.

You'll frequently find level curves useful when analyzing these functions. They provide insights into how the function behaves across different inputs.

Graphing Techniques

Visualizing multivariable functions requires a unique set of graphing techniques. One popular method is using the xy-plane to draw level curves or contour lines, which help interpret these complex surfaces.
For the function \( f(x, y) = x^2 + y^2 \), the equation represents a set of circles centered at the origin. Here's why: each level curve corresponds to the points that sum to a particular constant \( c \). For instance, the circle with \( x^2 + y^2 = 4 \) has a radius of 2, as the components add up to the same constant.

• When graphing these, first identify your c-values. These are constants like 0, 2, 4, etc.
• Plot each circle using radii derived from these values: \( \sqrt{c} \).
• Note that each circle expands as \( c \) increases, indicating where that constant value is shared across the surface.

Use graphing tools or simply sketch these with a ruler and compass for easier visualization.

Contour Plots

Another crucial concept in understanding multivariable functions is the use of contour plots. These are essentially a way to project 3D surfaces onto a 2D plane using lines that represent constant values, or level curves.
For our function, contour plots show a series of concentric circles around the origin. Each circle corresponds to a different \( c \)-value, depicting the set of all \( (x, y) \) pairs that give the same output. This results in a map-like depiction where you can see how the function's surface undulates over the xy-plane.

• The closer the lines, the steeper the slope of the function in that area.
• For \( f(x, y) = x^2 + y^2 \), these plots exhibit symmetry, providing a clear representation of how the function spreads out uniformly from the center.
• Contour plots are particularly useful when dealing with more complex functions where direct graphing or visualization in 3D isn’t feasible.

This technique aids in predicting behavior by observing the pattern and distribution of these contour lines.