Problem 16 - Calculus Volume 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$g(x, y)=x^{2}+y^{2} ; c=4, c=9$$

Step-by-Step Solution

Answer

The level curves for \(c = 4\) and \(c = 9\) are circles with radii 2 and 3, respectively.

Step 1: Understand the Level Curves

The level curve of a function at a certain value of \(c\) is obtained by setting the function equal to \(c\). For the function \(g(x, y) = x^2 + y^2\), level curves are circles centered at the origin \((0,0)\).

Step 2: Substitute the First Value of c

To find the level curve where \(c = 4\), set the function equal to 4: \(x^2 + y^2 = 4\).

Step 3: Identify the Geometry for c=4

The equation \(x^2 + y^2 = 4\) represents a circle centered at the origin with a radius of 2.

Step 4: Substitute the Second Value of c

Similarly, for \(c = 9\), set the function equal to 9: \(x^2 + y^2 = 9\).

Step 5: Identify the Geometry for c=9

The equation \(x^2 + y^2 = 9\) represents a circle centered at the origin with a radius of 3.

Step 6: Visualize Level Curves

The level curves of \(g(x, y) = x^2 + y^2\) are concentric circles. Specifically, for \(c = 4\), it is a circle with radius 2, and for \(c = 9\), it is a circle with radius 3.

Key Concepts

Level Curve Visualization

Level curves, also known as contour lines, are key to visualizing a multi-variable function. They represent the set of all points \(x, y\) where the function has a constant value. For our specific function \(g(x, y) = x^2 + y^2\), this involves solving for curves along which this sum remains constant.
Imagine a topographical map where each contour line represents regions of equal height. Similarly, each level curve of a function visualizes where the output value is unchanged. These curves help us understand the behavior of a function in 3D space by projecting it onto a 2D plane, making it easier to interpret complex functions.
To visualize level curves for given values of \(c\), we set \(x^2 + y^2 = c\) and find the equations representing these curves. The original exercise does this by setting \(c = 4\) and \(c = 9\) to get specific curves to visualize.

Concentric Circles

In the context of level curves for the function \(g(x, y) = x^2 + y^2\), we find that these curves are concentric circles. Concentric circles are circles that share the same center but have different radii. Here, the common center is the origin point (0,0).
For \(c = 4\), the equation \(x^2 + y^2 = 4\) describes a circle with radius 2. Similarly, for \(c = 9\), the circle described is larger, with a radius of 3.

• Both circles are centered at the origin, highlighting their concentric nature.
• By using different values of \(c\), we uncover an infinite series of these concentric circles, mapping out the surface of the function in layers.

Visualizing this helps in understanding how the function behaves as it moves away from the origin, showcasing growth in radius as the \(c\) value increases.

Function Geometry

Functions like \(g(x, y) = x^2 + y^2\) can be visualized with level curves to reveal their geometric nature. In this example, the function’s geometry is that of a paraboloid, which is like an infinitely stretched out bowl. The further one travels from the center of the function, the higher the value of \(g(x,y)\) becomes.

• Each level curve represents a particular height in this 3D space. For \(c=4\), the circle at height 4, has radius 2, while for \(c=9\), the circle at height 9, has radius 3.
• The geometry of this function is symmetric around the center, making it easy to predict level curves at other values of \(c\).

Understanding the geometry of functions allows you to predict and sketch level curves for any value. Here, the paraboloid nature illustrates how level curves can map a 3D shape’s surface onto a 2D plane effectively, giving insight into its overall shape and behavior.