Problem 6 - Calculus Early Transcendentals

Question

Explain how to graph the level curves of a surface \(z=f(x, y)\).

Step-by-Step Solution

Answer

Answer: Level curves, also known as contour lines, are curves on a surface where the function's value remains constant. They represent a set of points (x, y) that satisfy the equation z = f(x, y) = c, where c is a constant value. Level curves provide a two-dimensional representation of a three-dimensional surface. To graph the level curves of a surface, follow these steps: 1. Set the function f(x, y) equal to a constant c to form the equation representing a level curve. 2. Choose several different values for the constant c, evenly spaced and spanning the range of possible values for the function. 3. Solve the equation f(x, y) = c for each chosen constant value, obtaining the equations of the level curves. 4. Graph each level curve in the xy-plane using the equations found in the previous step. 5. Examine the graph and interpret the shape and structure of the surface based on the level curves' spacing and patterns.

Step 1: Understanding level curves

Level curves, also known as contour lines, are curves on a surface where the function's value remains constant. In other words, a level curve is a set of points \((x, y)\) that satisfy the equation \(z = f(x, y) = c\), where \(c\) is a constant value. For example, if \(f(x, y) = x^2 + y^2\), a level curve might represent all points where \(f(x, y) = 1\), or \(x^2 + y^2 = 1\). Graphing these level curves will provide us with a two-dimensional representation of the three-dimensional surface.

Step 2: Setting the function equal to a constant

First, set the function \(f(x, y)\) equal to a constant \(c\): \(f(x, y) = c\). This equation represents a level curve of the surface.

Step 3: Choosing constant values for level curves

Choose several different values for the constant \(c\). Generally, these values should be evenly spaced and span the range of possible values for the function \(f(x, y)\). For instance, if \(f(x, y)\) has a range of \([-3,3]\), you might choose \(c=-3,-2,-1,0,1,2,3\). Each value of \(c\) corresponds to a level curve.

Step 4: Solving the equations for each constant value

For each chosen value of the constant \(c\), plug it into the equation \(f(x, y) = c\) from Step 2, and solve the equation for either \(x\) or \(y\). This will give you the equations of the level curves.

Step 5: Graphing the level curves

Using the equations found in Step 4, graph each level curve in the \(xy\)-plane. It can be helpful to use different colors for each level curve corresponding to different constant values.

Step 6: Interpreting the level curves

Examine the graph of the level curves to understand the overall shape and structure of the surface. Close level curves indicate steep regions on the surface, and sparse level curves represent gentle slopes.