On a topographic map, contour lines (level sets of elevation) are packed closely together on one side of a mountain and spread widely apart on the other. What does this tell you about the terrain?
AThe packed side has lower elevation and the spread side has higher elevation
BThe packed side is steeper and the spread side has a gentler slope
CThe map is inaccurate — contour spacing should be uniform
DThe packed side has more surface area than the spread side
Contour lines are level sets of elevation — each line connects points at the same height. Closely packed contours mean many level sets in a small horizontal distance: elevation changes rapidly, indicating a steep slope. Widely spaced contours mean elevation changes slowly — a gentle grade. This is exactly why topographic maps are useful: contour spacing encodes steepness in a 2D diagram without requiring a 3D drawing.
Question 2 Multiple Choice
Why can't we draw the graph of f(x, y, z) the way we draw the graph of f(x, y)?
AThree-variable functions are not well-defined mathematically without additional constraints
BThe graph of f(x, y, z) would require four dimensions — three inputs plus one output — which cannot be visualized
CThree-variable functions only have level sets, not graphs
DThe computational complexity makes it impractical
The graph of f(x, y) is the set of points (x, y, f(x,y)) in 3D space — one dimension beyond the 2D domain. For f(x, y, z), the graph would be points (x, y, z, f(x,y,z)) in 4D space, which cannot be embedded in our 3D world. This is why level sets become essential: the level set f(x, y, z) = c is a surface in 3D space, which can be visualized — giving us useful 3D snapshots of an inherently 4D object.
Question 3 True / False
The level set of f(x, y) = x² + y² at the value c = 9 is a circle of radius 3 centered at the origin.
TTrue
FFalse
Answer: True
The level set at c = 9 is all (x, y) satisfying x² + y² = 9, which is by definition a circle of radius √9 = 3. This illustrates how level sets reduce a surface to curves: instead of the full paraboloid z = x² + y², each level set gives one ring at a specific height. Circular level sets tell you the function is radially symmetric — its value depends only on the distance from the origin.
Question 4 True / False
To verify that a multivariable function has a limit at a point, it is sufficient to check that the limit is the same along most straight-line path through that point.
TTrue
FFalse
Answer: False
This is a dangerous misconception. A multivariable limit requires the function to approach the same value along ALL paths — not just straight lines. Functions exist where every straight-line limit yields the same value, yet curved paths give a different value, so the true limit fails to exist. The classic example: f(x, y) = x²y/(x⁴ + y²) has limit 0 along every line through the origin, but limit 1/2 along the parabolic path y = x². Checking only straight lines is insufficient.
Question 5 Short Answer
Why are level sets a more useful visualization tool for f(x, y) than attempting to describe the graph of f directly?
Think about your answer, then reveal below.
Model answer: The graph of f(x, y) is a surface in 3D space, which is hard to read precisely from a static 2D drawing. Level sets project information onto the 2D domain: each curve shows all points where f takes a specific value, and the spacing between curves encodes the rate of change. This produces a complete, readable picture — like a topographic map — without the distortion of a 3D projection. Level sets also generalize to three-variable functions, where the graph is 4-dimensional but level sets are 3D surfaces.
The level-set representation packs full information about a function's behavior into a 2D diagram. Engineers and scientists use contour plots routinely for this reason. The technique generalizes: for f(x, y, z), the graph lives in 4D and is impossible to visualize, but level surfaces f(x, y, z) = c are 3D objects that can be rendered and analyzed.