Problem 43 - Calculus Volume 3

Question

Sketch the following by finding the level curves. Verify the graph using technology. $$f(x, y)=2-\sqrt{x^{2}+y^{2}}$$

Step-by-Step Solution

Answer

The level curves of \( f(x, y) = 2 - \sqrt{x^2 + y^2} \) are circles centered at the origin with decreasing radii as \( c \) increases towards 2.

Step 1: Understand the Function

The function given is \( f(x, y) = 2 - \sqrt{x^2 + y^2} \). It represents a surface in three dimensions where each point \((x, y)\) on the surface has the height \(z\) given by the formula.

Step 2: Set Up Level Curves

Level curves are created by setting the function equal to a constant \( c \). So we have \( 2 - \sqrt{x^2 + y^2} = c \). This simplifies to \( \sqrt{x^2 + y^2} = 2 - c \).

Step 3: Solve for Level Curves

Solving for \(x\) and \(y\), we find that \( x^2 + y^2 = (2-c)^2 \). These are equations of circles centered at the origin \((0,0)\) with radius \(|2-c|\).

Step 4: Analyze Validity of \(c\)

Since \( \sqrt{x^2 + y^2} \) is non-negative, \(2 - c\) must also be non-negative. Thus, \( 2 \geq c \).

Step 5: Sketch the Circles

For each valid constant \( c \) from the range \( -\infty < c \leq 2 \), draw a circle of radius \( |2 - c| \) centered at \((0,0)\). As \( c \) increases towards 2, the circle's radius decreases towards 0.

Step 6: Verify Using Technology

Use graphing software such as Desmos or GeoGebra to enter the equation \( f(x, y) = 2 - \sqrt{x^2 + y^2} \) and observe the level curves by setting \( f(x, y) = c \) for different values of \( c \). Confirm the graph shows concentric circles with diminishing radius as predicted.

Key Concepts

Multivariable Calculus

In multivariable calculus, we study functions with two or more variables, like our function \( f(x, y) = 2 - \sqrt{x^2 + y^2} \). Unlike single-variable calculus, multivariable calculus deals with surfaces and curves in higher dimensions. This adds a beautiful complexity to our mathematical toolkit, because rather than just focusing on points along a line, we explore

• Points on surfaces in three-dimensional space
• Curves that form from intersecting these surfaces
• Ways of visualizing gradients and directions along these surfaces

Level curves, in particular, give us a way to "slice" these surfaces along constant heights. This is akin to geographical maps that show elevation levels. Level curves can help identify shapes and structures on these three-dimensional surfaces, making multivariable calculus a powerful tool in fields like physics, engineering, and economics.

Equations of Circles

The original exercise shows that the function \( f(x, y) = 2 - \sqrt{x^2 + y^2} \) leads to level curves that represent circles. These are equations of the form \( x^2 + y^2 = r^2 \), where \( r \) is the radius. To better understand how this connects to our function, notice that by setting \( f(x, y) = c \) and simplifying, we ended up with \( \sqrt{x^2 + y^2} = 2 - c \), which is rearranged to \( x^2 + y^2 = (2-c)^2 \).
Here are the key points:

• The center of the circle is always at \((0,0)\), known as the origin.
• The radius of the circle is determined by the expression \(|2-c|\).
• As \( c \) changes, the size of the circles change, reflecting different heights of the surface.

This transformation of the function into circles helps visualize and understand the surface affected by the variables \( x \) and \( y \). It conveys how shifts in these variables sketch out entire circinal paths at specific heights, demonstrating the intertwined nature of geometry and algebra.

Three-Dimensional Surfaces

Three-dimensional surfaces can initially seem overwhelming, but they're simply an extension of two-dimensional graphs. When dealing with an equation like \( f(x, y) = 2 - \sqrt{x^2 + y^2} \), you are picturing a surface floating in three-dimensional space where every point on the surface corresponds to a pair of \( x \) and \( y \) values and a resulting \( z \) value.
Key aspects to keep in mind:

• Each pair \((x, y)\) corresponds to a specific height \(z\) on the surface.
• Level curves, like the circles we plotted, are the intersections of the surface with horizontal planes of constant \(z\).
• Visualizing these surfaces can reveal insights on slopes and variations across different regions.

Using technology or graphing software aids in visualizing these concepts, providing tangible images to help solidify your understanding. In essence, three-dimensional surfaces enable us to explore complex relationships between variables in spatial dimensions.