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Super klein structures, level curves
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Level Curves and Graphs forFunctions of Two Variables
Notes written by: Silvius Klein
The function f (x1, x2) =√
x21 + x2
2Graph: a cone
-4-2
02
4x1
-4
-2
0
2
4x2
0
1
2
3
y
The function f (x1, x2) =√
x21 + x2
2Level curves: concentric circles
1
2
3
455
55
-4 -2 0 2 4
-4
-2
0
2
4
The function g(x1, x2) = x1 · x2
Graph
0 10 20 30
x1
05
1015
20x20
200
400
600
y
The function g(x1, x2) = x1 · x2
Level curves: hyperbolas
100
200300 400
500
0 5 10 15 20 25 30
0
5
10
15
20
The production function P(L, K ) = 1.01K 0.75 · L0.25
Graph
020
40
60
80
L
0
20
40K
0
20
40
60
P
The production function P(L, K ) = 1.01K 0.75 · L0.25
Level curves
10
20
30
40
50
60
70
0 20 40 60 80
0
10
20
30
40
50
The cost function C (L, K ) = 1000 · L + 5000 · K
Graph: a plane
0
20
40
60
80
L
0
20
40
K
0
100 000
200 000
300 000
P
The cost function C (L, K ) = 1000 · L + 5000 · KLevel curves: parallel lines of negative slope
50 000 100 000
150 000
200 000
250 000300 000
0 20 40 60 800
10
20
30
40
50
The function f (x1, x2) = sin x21 + cos x3
2
Graph
0
1
2
3
x
-3
-2
-1
0
1
y
-2
-1
0
1
2
z
The function f (x1, x2) = sin x21 + cos x3
2
Level curves
-1.5
-1.5
-1.5
-1.5-1.5
-1.5
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-0.5
-0.5
-0.5
-0.5-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
0
0
0 0
0 0
00
0
0
0
0
00 0 00 00 0 00.50.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.511
1
1 1
1 1
1 1
1 1
1 1 11.5 1.5
1.5 1.5
1.5
1.5
1.5 1.5
1.5
1.5 1.5
1.5 1.5
1.5
1.51.5 1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-3
-2
-1
0
1
Topographic map of a mountain region
Level surfaces for F (x1, x2, x3) = x21 + x2
2 + x23
-2
0
2
-2
0
2
-2
0
2
