PowerPoint PresentationSlide 2Functions of Several VariablesSlide 4Slide 6Slide 7Slide 8GraphsSlide 10Slide 11ExampleSlide 13Level CurvesSlide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Functions of Three or More VariablesSlide 24Slide 25Slide 26Slide 27Slide 2811Copyright © Cengage Learning. All rights reserved. 14Partial DerivativesCopyright © Cengage Learning. All rights reserved. 14.1Functions of Several Variables33Functions of Several VariablesLearning 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 variables44Functions of Several VariablesIn 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)55Functions of Two Variables66Functions of Several VariablesThe 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 function77Functions of Several VariablesWe often write z = f (x, y) to make explicit the value takenon 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.]88Graphs99GraphsOne 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).1010GraphsWe can visualize the graph S of f as lying directly above or below its domain D in the xy-plane (see Figure 5).Figure 51111GraphsThe 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 = 0so 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.1212ExampleSketch the graph ofSolution:The graph has equation We square both sides of this equation to obtain z2 = 9 – x2 – y2, or x2 + y2 + z2 = 9, which we recognize as an equation of the sphere with center the origin and radius 3.But, since z 0, the graph of g is just the top half of this sphere (see Figure 7).Figure 7Graph of1313Level Curves1414Level CurvesSo 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.1515Level CurvesYou can see from Figure 11 the relation between level curves and horizontal traces.Figure 111616Level CurvesThe 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.1717Level CurvesOne common example of level curves occurs in topographic maps of mountainous regions, such as the map in Figure 12.Figure 121818Level CurvesThe 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.1919Level CurvesFigure 13 shows a weather map of the world indicating the average January temperatures. The isothermals are the curves that separate the colored bands.Figure 13World mean sea-level temperatures in January in degrees Celsius2020Level CurvesFor 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 192121Level CurvesNotice 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.Figure 19(c)Figure 19(d)2222Functions of Three or More Variables2323Functions of Three or More VariablesA 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).2424ExampleFind the domain of f iff (x, y, z) = ln(z – y) + xy sin zSolution:The expression for f (x, y, z) is defined as long as z – y > 0, so the domain of f isD = {(x, y, z) | z > y}This is a half-space consisting of all points that lie above the plane z = y.2525Functions of Three or More VariablesIt 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.2626Functions of Three or
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