11.1 Functions of Several VariablesThere are four points to study functions of two or more variables1. verbally (by a description of words)2. numerically (by a table of values)3. algebraically (by an explicit formula)4. visually (by a graph or level curves)Visual RepresentationsExampleSketch the graph of g(x, y) =p9 − x2− y2.Find and sketch the domain of the function f(x, y) = ln(9 − x2− 9y2).Another method for visualizing functions is to draw contour lines, or level curves.Definition: The level curves of a function f of two variables are the curves with equationsf(x, y) = k where k is a constant (in the range of f).A level curve f(x, y) = k is the set of all points in the domain of f at which f takes on agiven value k (shows where the graph of f has height k).ExampleDraw a contour map of the function f(x, y) = (y − 2x)2showing several level curves.1Draw a contour map of the function f(x, y) = yexshowing several level curves.Functions of Three or More VariablesDefinition: A function of three variables, f, is a rule that assigns to each ordered triple(x, y, z) in a domain D a unique real number denoted by f(x, y, z).ExampleLet f(x, y, z) = e√z−x2−y2. Evaluate f(2, −1, 6). Find the domain of f. Find the range off.Describe the level surfaces of the function f(x, y, z) = x + 3y + 5z.211.2 Limits and ContinuityLimitsDefinition: We write lim(x,y)→(a,b)f(x, y) = L and we say that the limit of f (x, y) as (x, y)approaches (a, b) is L if we can make the values of f(x, y) as close to L as we like by takingthe point (x, y) sufficiently close to the point (a, b), but not equal to (a, b).Recall: For functions of a single variable we let x approach a in two directions, but this isnot the case for multivariables.Fact: If f(x, y) → L1as (x, y) → (a, b) along a path C1and f(x, y) → L2as (x, y) → (a, b)along a path C2where L16= L2then the limit does not exist.ExampleFind the limit, if it exists, or show that the limit does not exist.(a) lim(x,y)→(5,−2)x5+ 4x3y −5xy2(b) lim(x,y)→(0,0)xy cos y3x2+ y2Note: All of the Limit Laws (Section 2.3) including the Squeeze Theorem can be extendedto functions of two variables.ExampleFind the limit, if it exists, or show that the limit does not exist.lim(x,y)→(0,0)xypx2+ y23ContinuityDefinition: A function f of two variables is called continuous at (a, b) if lim(x,y)→(a,b)f(x, y) =f(a, b).Definition: A polynomial function of two variables is a sum of terms of the form cxmyn,where c is a constant and m and n are nonnegative integers.Definition: A rational function is a ratio of polynomials.ExampleDetermine the set of points at which F (x, y) =sin(xy)ex− y2is continuous.Use polar coordinates to find lim(x,y)→(0,0)x3+ y3x2+ y2.411.3 Partial DerivativesIf f is a function of two variables x and y, suppose we let only x vary while keeping y fixed(y = b). Then we really have a function of a single variable x, g(x) = f(x, b).Definition: If g has a derivative at a then we call it the partial derivative of f withrespect to x at (a, b). Denoted: fx(a, b) = g0(a) where g(x) = f (x, b).The limit definition isfx(a, b) = limh→0f(a + h, b) − f(a, b)hSimilarly, the partial derivative of f with respect to y at (a, b) is given byfy(a, b) = limh→0f(a, b + h) − f(a, b)hIn general if f is a function of two variables, its partial derivatives are defined byfx(x, y) = limh→0f(x + h, y) − f(x, y)hfy(x, y) = limh→0f(x, y + h) − f (x, y)hNotations for Partial DerivativesIf z = f(x, y), we writefx(x, y) = fx=∂f∂x=∂∂xf(x, y) =∂z∂x= f1= D1f = Dxffy(x, y) = fy=∂f∂y=∂∂yf(x, y) =∂z∂y= f2= D2f = DyfRule: For finding partial derivatives of z = f(x, y)1. To find fxregard y as a constant and differentiate f (x, y) with respect to x.2. To find fyregard x as a constant and differentiate f(x, y) with respect to y.ExampleFind the first partial derivatives of the function.(a) f(x, y) = 3x − 2y45(b) f(x, y) =x − yx + y(c) f(x, y, z) = xy2z3+ 3yz(d) Find fx(3, 4) where f(x, y) =px2+ y2For implicit functions (where we cannot solve for one variable), we use implicit differentiation.ExampleUse implicit differentiation to find∂z∂xand∂z∂yfor x2+ y2+ z2= 3xyz.Interpretations of Partial Derivatives• The partial derivatives fx(a, b) and fy(a, b) can be interpreted geometrically as theslopes of the tangent lines at a point P (a, b, c) to the traces C1and C2of S in theplanes y = b and x = a.• Can also be interpreted as rates of change6Higher DerivativesIf f is a function of two variables, then its partial derivatives fxand fyare also functionsof two variables, so we can consider their partial derivatives, which are called the secondpartial derivatives.Notation:(fx)x= fxx= f11=∂∂x∂f∂x=∂2f∂x2=∂2z∂x2(fx)y= fxy= f12=∂∂y∂f∂x=∂2f∂y∂x=∂2z∂y∂x(fy)x= fyx= f21=∂∂x∂f∂y=∂2f∂x∂y=∂2z∂x∂y(fy)y= fyy= f22=∂∂y∂f∂y=∂2f∂y2=∂2z∂y2ExampleFind all the second partial derivatives for f(x, y) = x4− 3x2y3.Note: fxy= fyxClairaut’s Theorem: Suppose f is defined on a disk D that contains the point (a, b). If thefunctions fxyand fyxare both continuous on D, then fxy(a, b) = fyx(a, b).711.4 Tangent Planes and Linear ApproximationsRecall: In single-variable calculus, we approximate a differentiable function with its tangentline. For a function f(x, y) at (x0, y0) the tangent line equation is given by y −y0= f0(x)(x−x0).Since y0= f(x0) then we can rewrite this equation as y − f(x0) = f0(x)(x − x0) giving usthe linear approximation of f near (x0, y0) as L(x, y) = f (x0) + f0(x)(x − x0).Idea: as we zoom in on a point on a surface that is a graph of a differentiable function, thesurface looks more and more like a plane.Tangent PlanesFor two variable functions we have 3D surfaces which we approximate by tangent planesprovided our function has continuous partial derivatives.Definition: The tangent plane to the surface S at the point P is defined to be the planethat contains both tangent lines T1and T2.To find the equation of the tangent plane, we need a point on the plane and a normal vector.To find the equation of the normal plane we take the cross product of h1, 0, fxi and h0, 1, fyiboth of which are parallel to the plane and get the normal vector h−fx, −f, 1i.Suppose f has continuous partial derivatives. An equation of the tangent plane to the surfacez = f(x, y) at P (x0, y0, z0) is z −z0= fx(x0, y0)(x − x0) + fy(x0, y0)(y −