Chapter 14 – Partial Derivatives 14.1 Functions of Several Variables 1 Objectives: Use differential calculus to evaluate functions of several variables How to produce graphs of functions of two or more variables
Remember In general, we study functions from four points of view: ◦ Verbally(by a description in words) ◦ Numerically(by a table of values) ◦ Algebraically(by an explicit formula) ◦ Visually(by a graph) The same holds true for functions of two variables 14.1 Functions of Several Variables2
Definition – Function of Two Variables We often write z=f(x,y) to make explicit the value taken on by f at the general point (x,y). That means, the variables x and y are independent variables and z is the dependent variable Functions of Several Variables3
Arrow Diagrams One way of visualizing such a function is by means of an arrow diagram, where the domain D is represented as a subset of the xy-plane Functions of Several Variables4
Example 1 Find and sketch the domain of the function given below. What is the range of f ? 14.1 Functions of Several Variables5
Definition - Graphs Another way to visualize the behavior of a function of two variables is to consider its graph Functions of Several Variables6
Definition – Linear Function A function in the form f(x, y) = ax + by + c is called a linear function. The graph of such a function has the equation z = ax + by + c or ax + by – z + c = 0 thus making it a plane Functions of Several Variables7
Example 2 Sketch the graph of the function below Functions of Several Variables8
Graphs of functions in two variables Here is a computer generated graph. Notice that we get an especially good picture of a function when rotation is used to give views from different vantage points Functions of Several Variables9
Example 3 What is the range of Describe the graph of f Functions of Several Variables10
Definition – Level Curves So far we have two ways for visualizing functions: arrow diagrams and graphs. Our third method is borrowed from mapmakers and is called a contour map. This is made of contour curves or level curves Functions of Several Variables11
Domain of Level Curves A level curve f(x, y) = k is the set of all points in the domain of f at which f takes on a given value k. This means it shows where the graph of f has height k Functions of Several Variables12
Level Curves You can see from the figure the relation between level curves and horizontal traces Functions of Several Variables13
Level Curves The surface is: ◦ Steep where the level curves are close together. ◦ Somewhat flatter where the level curves are farther apart Functions of Several Variables14
Visualization Level Curves 14.1 Functions of Several Variables15
Computer generated level curves Level curve Computer graphs 14.1 Functions of Several Variables16
Visualization Level Curves of a Surface 14.1 Functions of Several Variables17
Function of Three Variables A function of three variables, f, is a rule that assigns to each ordered triple (x, y, z) in a domain D R 3 a unique real number denoted by f(x, y, z) Functions of Several Variables18
Functions of Three Variables It’s very difficult to visualize a function f of three variables by its graph. ◦ That would lie in a four-dimensional space Functions of Several Variables19
Example 4 Describe the level surfaces of the function below Functions of Several Variables20
More Examples The video examples below are from section 14.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 6 Example 6 ◦ Example 8 Example 8 ◦ Example 11 Example Functions of Several Variables21
Demonstrations Feel free to explore these demonstrations below. Cross Sections of Graphs of Functions of Two Variables Cross Sections of Graphs of Functions of Two Variables Cobb-Douglas Production Functions Graph and Contour Plots of Functions of Two Variables Graph and Contour Plots of Functions of Two Variables Quadratic Surfaces 14.1 Functions of Several Variables22