Section 12.5

Functions of Several Variables

Visualizing as surfaces, contours, and domains.

1

Introduction

A function takes two inputs and produces a height, creating a surface in space. Just as we read topographical maps, we can understand surfaces using Level Curves (contours) where height is constant.

Interactive: Level Curves Slicing

Change to slice the function at different heights. The red curve is the Level Curve.

2

Key Concepts

Domain

The set of valid input pairs .

  • Even roots :
  • Logs :
  • Denominators :

Level Curves

Curves of constant height .

(Topographical map lines)

Traces

Vertical slices obtained by holding or constant.

Trace x=a:
Trace y=b:
3

Worked Examples

Example 1: Finding Domain (Level 1)

Find and sketch the domain of .

Step 1: Apply Constraints

Inside the square root must be non-negative:

Step 2: Solve inequality

or .

This is the interior of a disk of radius 3 centered at the origin (including boundary).

Example 2: Sketching Level Curves (Level 2)

Sketch the contour map for for .

  • Set f(x,y) = k:
    (Circle radius 1)
    (Circle radius 2)
    (Circle radius 3)
  • Interpretation:
    Concentric circles moving outward as increases. This represents a bowl shape (Paraboloid) opening up.

Example 3: Analyzing Traces (Level 3)

Analyze the traces of (The Saddle).

Vertical Trace x=0

.
Parabola opening up.

Vertical Trace y=0

.
Parabola opening down.

Conclusion: Being convex in one direction and concave in the other creates the saddle shape.

4

Practice Quiz

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