A contour map of f(x, y) shows contours at values 0, 10, 20, and 30. In region A the contour lines are widely spaced; in region B they are tightly packed. What can you conclude?
ARegion B has larger output values than region A
BThe function changes more rapidly (steeper slope) in region B than in region A
CRegion A contains a local maximum of the function
DThe map has been drawn incorrectly — contour spacing should be uniform across the domain
The fundamental reading rule: contour spacing encodes steepness. Because output values are equally spaced (increments of 10), closely packed lines mean the function traverses 10 units of output over a short horizontal distance — steep slope. Widely spaced lines mean the same 10-unit change occurs over a long horizontal distance — gentle slope. Region B is steeper. Output value magnitude is not related to contour spacing: you cannot tell from spacing alone whether values are large or small.
Question 2 Multiple Choice
Two contour lines on a contour map appear to intersect at a point P. What does this imply?
AThe function f(x, y) has a saddle point at P
BThe function achieves a local maximum at P
CThis is impossible — a single point (x, y) cannot have two different output values, so contour lines for distinct values can never cross
DThe function is not differentiable at P
Contour lines are level curves: each line represents all points where f equals a specific constant. If two contour lines with different values crossed at P, then f(P) would simultaneously equal both constants — a contradiction, since f is a function (one output per input). This is a strict geometric constraint: crossing contours are logically impossible for any well-defined function, not just for 'nice' functions. The topology of contour lines therefore directly encodes function structure.
Question 3 True / False
On a contour map where output values increase toward the center, a series of nested closed loops converging inward indicates a local extremum at the center.
TTrue
FFalse
Answer: True
Nested closed loops converging inward are the signature of a peak or bowl in the function surface. Each loop represents a level curve at a higher output value (if a maximum) or lower value (if a minimum); as you move inward the values increase or decrease monotonically until the extreme point at the center. This is one of the most useful pattern-recognition skills on contour maps: before doing any calculus, you can identify the rough locations of local maxima and minima just from the topology of the contour lines.
Question 4 True / False
The gradient vector at any point on f(x, y) points along the nearest contour line in the direction of increasing values.
TTrue
FFalse
Answer: False
The gradient vector points PERPENDICULAR to contour lines, in the direction of steepest ascent. Contour lines are curves of constant output — moving along a contour produces no change in f. The direction of maximum rate of change must therefore be perpendicular to constant-f curves. This perpendicularity relationship is one of the most important facts about gradients and will be essential when you study directional derivatives and optimization in this course.
Question 5 Short Answer
Why do closely spaced contour lines indicate a steep slope? Use the equal-spacing property of contour values in your explanation.
Think about your answer, then reveal below.
Model answer: On a contour map, adjacent lines represent equally spaced output values — for example, increments of 10. The horizontal distance between those lines in the xy-plane represents how far you must travel to achieve that fixed output change. If the lines are closely spaced, you achieve the same 10-unit output change over a short horizontal distance: that is a steep slope. If the lines are widely spaced, the same 10-unit output change happens over a large horizontal distance: that is a gentle slope. The equal-spacing convention is what makes the spacing geometrically meaningful — it is the denominator in the rise-over-run ratio.
The insight is that the contour map compresses 3D information (rise and run) into a 2D diagram by fixing the 'rise' (equal output increments) and letting the 'run' (horizontal distance between lines) vary. Close spacing means small run for fixed rise — steep. Wide spacing means large run for fixed rise — gentle. This is exactly how topographic maps encode terrain steepness, and it transfers directly to the mathematical analysis of scalar fields.