Problem 28 - 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. $$z=f(x, y)=\sqrt{x^{2}+y^{2}}, \quad c=3$$

Step-by-Step Solution

Answer

The level curve is a circle centered at (0,0) with radius 3.

Step 1: Understand the level curve concept

A level curve of a function of two variables is the set of all points \( (x, y) \) where the function takes on a constant value \( c \). For the function \( z = f(x, y) = \sqrt{x^2 + y^2} \), this means solving the equation \( \sqrt{x^2 + y^2} = c \).

Step 2: Set the function equal to c

Substitute the given \( c = 3 \) into the equation: \[ \sqrt{x^2 + y^2} = 3. \]

Step 3: Eliminate the square root

Square both sides of the equation to eliminate the square root: \[ x^2 + y^2 = 9. \]

Step 4: Interpret the equation

The equation \( x^2 + y^2 = 9 \) represents a circle centered at \( (0, 0) \) with a radius of \( \sqrt{9} = 3 \).

Step 5: Verify solution

Check that the steps correctly represent the level curve as \( z = \sqrt{x^2 + y^2} \) when \( c = 3 \). The level curve is indeed a circle with radius 3.

Key Concepts

Understanding Multivariable Calculus

Imagine trying to picture a landscape made of gentle hills and valleys. Multivariable calculus lets us work with functions that describe such landscapes. This branch of mathematics extends the ideas of calculus to functions with more than one variable. It helps us understand how these functions behave when they depend on multiple inputs, typically using variables like \( x \) and \( y \). We study how changes in these variables affect the outcome, similar to how a change in your position affects your elevation on a hike.

Key aspects of multivariable calculus include:

• Partial Derivatives: They measure how a function changes as one variable changes while the others are held constant.
• Gradient: This is a vector that points in the direction of the greatest rate of increase of the function, telling us how to "climb" the steepest path.
• Tangent Planes: A generalization of tangent lines that gives us a linear approximation of the function surface at a particular point.

These elements allow us to model complex systems and solve real-world problems that involve multiple inputs.

Exploring Functions of Two Variables

Functions of two variables are like maps that use two inputs to determine a single output. They are often written as \( f(x, y) \), where \( x \) and \( y \) are the variables. These functions form a surface in three-dimensional space, making them a powerful tool to study real-world phenomena.

For instance:

• Temperature Maps: Imagine using latitude and longitude to determine temperature at various points on Earth.
• Economic Models: Functions can represent how two factors, like supply and demand, influence market prices.
• Physics Simulations: They help model how conditions like pressure and volume influence a system’s state.

When you graph these functions, you create a surface in 3D space, and the level curves are slices of this surface that represent constant values of the function output.

Visualizing Functions Through Level Curves

Visualizing a function of two variables can be quite enlightening. One effective method is to use level curves, which are like contour lines on a topographical map. These curves join points where the function has the same value, providing insights into the shape and structure of the function's graph.

In the exercise solution provided:

• The function \( z = \sqrt{x^2 + y^2} \) gives us a series of concentric circles, representing level curves for various values of \( c \).
• These circles illustrate the fact that as you move away from the origin, the function value \( z \) increases.
• Level curves simplify complex 3D surfaces into clear and manageable 2D visualizations.

Understanding these curves helps not only in visualizing functions but also in applications such as geographical mapping, where elevation contours represent constant altitudes on maps.