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# Lesson 5: Functions and surfaces

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A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.

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### Lesson 5: Functions and surfaces

1. 1. Section 9.6 Functions and Surfaces Math 21a February 13, 2008 Announcements Oﬃce Hours Tuesday, Wednesday, 2–4pm (SC 323) All homework on the website No class Monday 2/18
2. 2. Outline Functions of more than one variable Domain and Range Graphs Traces Quadric Surfaces Review of Conic Sections Examples of Quadric Surfaces
3. 3. What is a function? A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output.
4. 4. What is a function? A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output. The temperature in this room is a function of position and time. The production of an economy is a function of capital (money and goods invested) and labor I derive utility (happiness) from eating bacon and eggs for breakfast.
5. 5. Deﬁnition A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y ) in a set D a unique real number denoted by f (x, y ). The set D is the domain of f and its range is the set of values that f takes on. That is { f (x, y ) | (x, y ) ∈ D }.
6. 6. Example Example √ Find the domain and range of f (x, y ) = xy .
7. 7. Example Example √ Find the domain and range of f (x, y ) = xy . Solution Working from the outside in, we see that xy must be nonnegative, which means x ≥ 0 and y ≥ 0 or x ≤ 0 and y ≤ 0. Thus the domain is the union of the coordinate axes, and the ﬁrst and third quadrants. The range of f is the set of all “outputs” of f . Clearly the range of f is restricted to the set of nonnegative numbers. To make sure that we can get all nonnegative numbers x, notice x = f (x 2 , 1).
8. 8. Worksheet #1
9. 9. Outline Functions of more than one variable Domain and Range Graphs Traces Quadric Surfaces Review of Conic Sections Examples of Quadric Surfaces
10. 10. Deﬁnition If f is a function of two variables with domain D, then the graph of f is the set of all points (x, y , z) in R3 with z = f (x, y ) and (x, y ) ∈ D.
11. 11. Deﬁnition If f is a function of two variables with domain D, then the graph of f is the set of all points (x, y , z) in R3 with z = f (x, y ) and (x, y ) ∈ D. Functions of one variable are easy to graph on the Cartesian plane. Functions of two variables need a three-dimensional space. Our goal is to understand functions of two variables and how to graph them.
12. 12. Example (Worksheet 2(i)) Sketch the graph of the function f (x, y ) = 3.
13. 13. Example (Worksheet 2(i)) Sketch the graph of the function f (x, y ) = 3. Example (Worksheet 2(ii)) Sketch the graph of the function f (x, y ) = 6 − 3x − 2y . Example (Worksheet 2(iii)) Sketch the graph of the function f (x, y ) = x 2 + y 2 .
14. 14. Traces A trace of a surface is the intersection of it with a plane. The result is a curve. Multiple traces give multiple curves which help sketch the function choices for traces: coordinate planes x = 0, y = 0, z = 0 parallel planes, e.g., z = k for many k
15. 15. Worksheet #3–4
16. 16. Outline Functions of more than one variable Domain and Range Graphs Traces Quadric Surfaces Review of Conic Sections Examples of Quadric Surfaces
17. 17. Conic Sections Circle Ellipse Parabola Hyperbola
18. 18. Quadric Surfaces Sphere Ellipsoid Elliptic paraboloid Hyperbolic paraboloid Hyperboloid of one sheet Hyperboloid of two sheets