A level curve is every point where f has one fixed value. Drawing several of them gives a contour map.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Where you have seen this
A hiking map, a weather chart, an ocean depth chart. All of them are contour maps of a function of two variables.
Spacing is steepness
Contours crowded together mean the surface climbs fast. Contours far apart mean it is nearly flat.
They never cross
A point cannot have two different heights. Two contours meeting would say exactly that.
Reading the shape
Nested closed rings mean a single peak or pit. Contours splitting into opposing pairs mean a saddle.
Walking along one
Walking along a contour changes nothing about your height. That fact becomes the key to the gradient in Chapter 5.
The surface seen from above
A level curve joins points where f has a fixed value. Drawing several for evenly spaced values produces a contour map, which represents a three-dimensional surface on flat paper.
Spacing shows steepness
Contours close together mean a steep slope; widely spaced ones mean gentle terrain. Reading steepness from spacing is exactly how walkers read an ordnance map.
The gradient is perpendicular to contours
The direction of steepest ascent crosses the level curves at right angles, because moving along a contour changes nothing. That fact links this lesson directly to the gradient chapter.
Contours cannot cross
A crossing would mean one point having two different function values, which no function permits. Contours that appear to meet indicate a cliff or a discontinuity in the underlying data.
Step 2: Try It Yourself
Tap and try it out.
Nested closed rings around one point mean a single peak or valley.
Contours that separate into opposing pairs mean a saddle: uphill one way, downhill the other.
Step 3: Watch an Example
One step at a time.
Watch Devon Identify a Level Curve
Devon finds the level curve of f(x, y) = x² + y² at height 25.
- Step 1
He sets the function equal to the height: x² + y² = 25.
Step 4: Your Turn
Practice makes it stick.
The Map
Problem 1 of 2
Contour lines are drawn every 10 metres. Between two adjacent lines, how much does the height change?
The Circle
Problem 2 of 2
The level curve of x² + y² at height 49 is a circle. What is its radius?
Read the Map
1 of 8
The level curve of x² + y² at height 36. What is its radius?
2 of 8
Can two contour lines of the same function cross? 1 yes, 0 no.
3 of 8
Contours are very close together. Is the slope steep? 1 yes, 0 no.
4 of 8
Walking along a contour, how much does your height change?
5 of 8
f(x, y) = 2x + 3y. Its level curves are which shape? 1 lines, 2 circles, 3 parabolas.
6 of 8
Contours at 100, 110 and 120 metres over 40 horizontal metres each. What is the rise per metre, as a decimal?
7 of 8
Sort each contour pattern by what it shows.
Tap something to move it.
- Empty
- Empty
8 of 8
A region of a map has no contours at all. Is it nearly flat? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
The level curve of x² + y² at height 16. What is its radius?
Question 2 of 2
What do closely spaced contour lines mean?
What You Learned
- 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.