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Calculus II – Fall 2014Lecture 2 – Contour Diagrams
Eitan Angel
University of Colorado
Friday, November 21, 2014
E. Angel (CU) Calculus II 21 Nov 1 / 9
Introduction
Today we will continue to discuss functions of two variables and theirgraphs. In particular, we will use level curves to understand the graph oftwo variable functions by drawing contour diagrams or contour maps.
E. Angel (CU) Calculus II 21 Nov 2 / 9
Algebraic Definition
Definition
The level curves of a function f of two variables are the curves withequations f(x, y) = k, where k is a constant (in the range of f). A set oflevel curves for z = f(x, y) is called a contour plot or contour map of f .
We will show some examples.
E. Angel (CU) Calculus II 21 Nov 3 / 9
Graphical Contour Plots
: A contour map of the temperaturefunction T with the domain equal tothe United States. Each contour iscalled an isotherm.
: A contour map of altitudez = h(x, y). Also known as atopographic map.
E. Angel (CU) Calculus II 21 Nov 4 / 9
Sketching Level Curves
Example
Sketch the level curves for k = −6, 0, 6, 12 of the function
f(x, y) = 3− x/2− 3y/4
The level curves are
k = 3− x/2− 3y/4
or
2x+ 3y + (4k − 3) = 0
This is a family of lineswith slope −2
3 .
E. Angel (CU) Calculus II 21 Nov 5 / 9
Sketching Level Curves
Example
Sketch the level curves for k = −6, 0, 6, 12 of the function
f(x, y) = 3− x/2− 3y/4
The level curves are
k = 3− x/2− 3y/4
or
2x+ 3y + (4k − 3) = 0
This is a family of lineswith slope −2
3 .
E. Angel (CU) Calculus II 21 Nov 5 / 9
Sketching Level Curves
Example
Sketch the level curves of
g(x, y) =√
9− x2 − y2 for k = 0, 1, 2, 3
The level curves are
x2 + y2 = 9− k2
which is a family ofconcentric circles withcenter (0, 0) and radius√9− k2.
E. Angel (CU) Calculus II 21 Nov 6 / 9
Sketching Level Curves
Example
Sketch the level curves of
g(x, y) =√
9− x2 − y2 for k = 0, 1, 2, 3
The level curves are
x2 + y2 = 9− k2
which is a family ofconcentric circles withcenter (0, 0) and radius√9− k2.
E. Angel (CU) Calculus II 21 Nov 6 / 9
Sketching Level Curves
Example
Sketch some level curves of h(x, y) = 4x2 + y2.
The level curves are
4x2 + y2 = k
which, for k > 0, describes afamily of ellipses with semiaxes√k/2 and
√k.
E. Angel (CU) Calculus II 21 Nov 7 / 9
Sketching Level Curves
Example
Sketch some level curves of h(x, y) = 4x2 + y2.
The level curves are
4x2 + y2 = k
which, for k > 0, describes afamily of ellipses with semiaxes√k/2 and
√k.
E. Angel (CU) Calculus II 21 Nov 7 / 9
Surface Reconstruction Using Level Curves
Let’s use the level curves/contour plot given in the last example toreconstruct the surface. The first animation shows the planes z = 1, 2, 3, 4.The second animation shows contours on the surface and ends with a viewlooking down the z-axis, i.e., a projection onto the xy-plane.
E. Angel (CU) Calculus II 21 Nov 8 / 9
Surface Reconstruction Using Level Curves
Example
Given the contour diagram below has k = 0, k = 1, k = 2, k = 4, k = −1,k = −2, k = −4. Give a rough sketch of the corresponding surface.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-3
-2
-1
1
2
3
E. Angel (CU) Calculus II 21 Nov 9 / 9
Surface Reconstruction Using Level Curves
Example
Given the contour diagram below has k = 0, k = 1, k = 2, k = 4, k = −1,k = −2, k = −4. Give a rough sketch of the corresponding surface.We can draw these curves in 3-space on the planes z = 0, z = 1, z = 2,z = 4, z = −1, z = −2, z = −4.
E. Angel (CU) Calculus II 21 Nov 9 / 9
Surface Reconstruction Using Level Curves
Example
Given the contour diagram below has k = 0, k = 1, k = 2, k = 4, k = −1,k = −2, k = −4. Give a rough sketch of the corresponding surface.We obtain the following curves in 3-space for k = 0, k = 1, k = 2, k = 4,k = −1, k = −2, k = −4.
E. Angel (CU) Calculus II 21 Nov 9 / 9
Surface Reconstruction Using Level Curves
Example
Given the contour diagram below has k = 0, k = 1, k = 2, k = 4, k = −1,k = −2, k = −4. Give a rough sketch of the corresponding surface.The graph is in fact the function f(x, y) = xy, a hyperbolic paraboloid.
E. Angel (CU) Calculus II 21 Nov 9 / 9
