Cogito
Multivariable Calculus · Chapter 3 · Lesson 2
Level Curves and Contour Maps
The surface, seen from directly above.
12 problems · about 23 minutes · F-IF.B.4, F-IF.C.7
What this lesson teaches
The student reads a contour map and connects contour spacing to steepness.
- A level curve joins every point where the function has one fixed value.
- A contour map is the surface seen from above, and contours never cross.
- Crowded contours mean steep; widely spaced contours mean nearly flat.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe level curve of x² + y² at height 16. What is its radius?
Answer 4
Why 4.
What do closely spaced contour lines mean?
Answer The surface is steep there.
Why Steep.
The level curve of x² + y² at height 36. What is its radius?
Answer 6
Why Square root of 36.
Can two contour lines of the same function cross? 1 yes, 0 no.
Answer 0
Why One point, one height.
Contours are very close together. Is the slope steep? 1 yes, 0 no.
Answer 1
Why A big height change over a short distance.
Build It Up
The same ideas with more to keep track of.
3 problemsWalking along a contour, how much does your height change?
Answer 0
Why That is the definition of a level curve.
f(x, y) = 2x + 3y. Its level curves are which shape? 1 lines, 2 circles, 3 parabolas.
Answer 1
Why 2x + 3y = k is a linear equation.
Contours at 100, 110 and 120 metres over 40 horizontal metres each. What is the rise per metre, as a decimal?
Answer 0.25
Why 10 divided by 40.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each contour pattern by what it shows.
Answer A peak or a pit: Nested closed rings, Rings shrinking toward one point · A saddle: Contours splitting into opposing pairs, Two contours crossing at a pass
Why Closed rings surround a single extreme.
A region of a map has no contours at all. Is it nearly flat? 1 yes, 0 no.
Answer 1
Why Nothing crossed means no height change.
The Map: Contour lines are drawn every 10 metres. Between two adjacent lines, how much does the height change?
Answer 10
Why 10 metres.
The Circle: The level curve of x² + y² at height 49 is a circle. What is its radius?
Answer 7
Why 7.