Reading surfaces through level curves

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

A contour map of f(x, y) = x² + y² shows a circle of radius 3. What is the level value of this contour?
Answer: ______________
On a contour map with equal height intervals and the same horizontal scale, Ben says where contour lines bunch together, the surface has a larger average height change per horizontal distance. Is Ben right?
Circle one: True False
Two contour maps have identical unlabelled contour geometry. Must they have the same high and low orientation?
Circle one: True False
A contour map uses equal height intervals. Across a 2 metre horizontal distance the height changes by 6 metres, and across a 3 metre horizontal distance it changes by 6 metres. Which has the larger average height change per metre?
- the 2 metre stretch
- the 3 metre stretch
- they are equal
What shape are the level curves of f(x, y) = x + y?
- circles
- parallel straight lines
- parabolas
On a contour map with equal height intervals and the same horizontal scale, what does wide spacing between lines indicate?
- a smaller average height change per horizontal distance
- a larger average height change per horizontal distance
- an exactly flat region
What is the domain of g(x, y) = √(9 - x² - y²) / x?
- the open disk x² + y² < 9
- the whole plane except x = 0
- the closed disk x² + y² ≤ 9 with x ≠ 0
For f(x, y) = √(9 - x² - y²), what is the radius of the domain disk?
Answer: ______________
Consider g(x, y) = √(9 - x² - y²) / x. Alex says the pair (3, 0) is in the domain and the pair (0, 0) is also in the domain. Is Alex right?
Circle one: True False
A contour map shows only these labelled contours: an outer loop at 100 m and an inner loop at 140 m. What does this establish?
- inner contour is 40 m higher; a unique peak is not established
- A unique central peak sits at the centre of the inner loop
- The height rises by 140 m from the outer loop to the inner loop

Answer key

For grown-ups. Fold this page away before handing over the rest.

Reading surfaces through level curves W1-mt_u04fZ_XiJ8-s1

9 · On the circle of radius 3, x² + y² = 9.
True · Under equal intervals and scale, bunched lines pack more height change into less ground, a larger relative change.
False · The functions f(x, y) = x² + y² and g(x, y) = -(x² + y²) share the same circular contours. One has a valley at the centre, the other a hill. Identical unlabelled geometry does not determine orientation.
the 2 metre stretch · Average height change per metre is 6/2 = 3 for the 2 metre stretch and 6/3 = 2 for the 3 metre stretch.
parallel straight lines · x + y = c rearranges to y = -x + c, lines of slope -1. Different c values shift the line, giving parallel straight lines.
a smaller average height change per horizontal distance · Wide spacing means less height change per step of ground under these conditions. It does not prove an exactly flat region.
the closed disk x² + y² ≤ 9 with x ≠ 0 · The square root needs x² + y² ≤ 9 and the denominator needs x ≠ 0.
3 · The root needs 9 - x² - y² ≥ 0, a disk of radius 3.
False · The pair (3, 0) has radicand 0 and denominator 3, so it is in the domain. The pair (0, 0) has denominator 0, so the fraction is undefined and it is outside the domain.
inner contour is 40 m higher; a unique peak is not established · The labels give a net rise of 140 - 100 = 40 m from the outer to the inner contour. Only these two contours are shown, so a unique central peak is not established.