14 Partial Derivatives Copyright Cengage Learning All rights reserved 1 14 1 Functions of Several Variables Copyright Cengage Learning All rights reserved Functions of Several Variables Learning objectives Interpret description of function of two or more variables verbal description graph or level curves explicit formula Find the domain of a function of several variables 3 Functions of Several Variables In this section we study functions of two or more variables 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 or level curves 4 Functions of Two Variables 5 Functions of Several Variables The temperature T at a point on the surface of the earth at any given time depends on the longitude x and latitude y of the point We can think of T as being a function of the two variables x and y or as a function of the pair x y We indicate this functional dependence by writing T T x y Other examples see Section 14 1 in your textbook volume V r2h of a cylinder r radius and h height wind chill index W W T v T actual temperature and v wind speed Cobb Douglas production function 6 Functions of Several Variables We often write z f x y to make explicit the value taken on by f at the general point x y The variables x and y are independent variables and z is the dependent variable Compare this with the notation y f x for functions of a single variable 7 Graphs 8 Graphs One way of visualizing the behavior of a function of two variables is to consider its graph Just as the graph of a function f of one variable is a curve C with equation y f x the graph of a function f of two variables is a surface S with equation z f x y 9 Graphs We can visualize the graph S of f as lying directly above or below its domain D in the xy plane see Figure 5 Figure 5 10 Graphs The function 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 so it is a plane In much the same way that linear functions of one variable are important in single variable calculus we will see that linear functions of two variables play a central role in multivariable calculus 11 Example Sketch the graph of Solution The graph has equation We square both sides of this equation to obtain z2 9 x2 y2 or y2 z2 9 which we recognize as an equation of the sphere with center the origin and radius 3 x2 But since z 0 the graph of g is just the top half of this sphere see Figure 7 Graph of Figure 7 12 Level Curves 13 Level Curves So far we have two methods for visualizing functions arrow diagrams and graphs A third method borrowed from mapmakers is a contour map on which points of constant elevation are joined to form contour lines or 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 In other words it shows where the graph of f has height k 14 Level Curves You can see from Figure 11 the relation between level curves and horizontal traces Figure 11 15 Level Curves The level curves f x y k are just the traces of the graph of f in the horizontal plane z k projected down to the xy plane So if you draw the level curves of a function and visualize them being lifted up to the surface at the indicated height then you can mentally piece together a picture of the graph The surface is steep where the level curves are close together It is somewhat flatter where they are farther apart 16 Level Curves One common example of level curves occurs in topographic maps of mountainous regions such as the map in Figure 12 Figure 12 17 Level Curves The level curves are curves of constant elevation above sea level If you walk along one of these contour lines you neither ascend nor descend Another common example is the temperature function that we discussed earlier Here the level curves are called isothermals and join locations with the same temperature 18 Level Curves Figure 13 shows a weather map of the world indicating the average January temperatures The isothermals are the curves that separate the colored bands World mean sea level temperatures in January in degrees Celsius Figure 13 19 Level Curves For some purposes a contour map is more useful than a graph It is true in estimating function values Figure 19 shows some computer generated level curves together with the corresponding computer generated graphs Figure 19 20 Level Curves Figure 19 c Figure 19 d Notice that the level curves in Figure 19 c crowd together near the origin That corresponds to the fact that the graph in Figure 19 d is very steep near the origin 21 Functions of Three or More Variables 22 Functions of Three or More Variables A function of three variables f is a rule that assigns to each ordered triple x y z in a domain a unique real number denoted by f x y z For instance the temperature T at a point on the surface of the earth depends on the longitude x and latitude y of the point and on the time t so we could write T T x y t 23 Example Find the domain of f if f x y z ln z y xy sin z Solution The expression for f x y z is defined as long as z y 0 so the domain of f is D x y z z y This is a half space consisting of all points that lie above the plane z y 24 Functions of Three or More Variables It is very difficult to visualize a function f of three variables by its graph since that would lie in a four dimensional space However we do gain some insight into f by examining its level surfaces which are the surfaces with equations f x y z k where k is a constant If the point x y z moves along a level surface the value of f x y z remains fixed Functions of any number of variables can be considered A function of n variables is a rule that assigns a number z f x1 x2 xn to an n tuple x1 x2 xn of real numbers We denote by the set of all such n tuples 25 Functions of Three or More Variables For example if a company uses n different ingredients in making a food product ci is the cost per unit of the i th ingredient and xi units of the i th ingredient are used then the total cost C of the ingredients …
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