11 1 Functions of Several Variables There are four points to study functions of two or more variables 1 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 Representations Example p Sketch the graph of g x y 9 x2 y 2 Find and sketch the domain of the function f x y ln 9 x2 9y 2 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 equations f 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 a given value k shows where the graph of f has height k Example Draw a contour map of the function f x y y 2x 2 showing several level curves 1 Draw a contour map of the function f x y yex showing several level curves Functions of Three or More Variables Definition 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 Example 2 2 Let f x y z e z x y Evaluate f 2 1 6 Find the domain of f Find the range of f Describe the level surfaces of the function f x y z x 3y 5z 2 11 2 Limits and Continuity Limits Definition We write lim f x y L and we say that the limit of f x y as x y x y a b approaches a b is L if we can make the values of f x y as close to L as we like by taking the 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 is not the case for multivariables Fact If f x y L1 as x y a b along a path C1 and f x y L2 as x y a b along a path C2 where L1 6 L2 then the limit does not exist Example Find the limit if it exists or show that the limit does not exist a lim x5 4x3 y 5xy 2 x y 5 2 b xy cos y x y 0 0 3x2 y 2 lim Note All of the Limit Laws Section 2 3 including the Squeeze Theorem can be extended to functions of two variables Example Find the limit if it exists or show that the limit does not exist xy lim p x y 0 0 x2 y 2 3 Continuity Definition A function f of two variables is called continuous at a b if lim f x y x y a b f a b Definition A polynomial function of two variables is a sum of terms of the form cxm y n where c is a constant and m and n are nonnegative integers Definition A rational function is a ratio of polynomials Example Determine the set of points at which F x y x3 y 3 Use polar coordinates to find lim x y 0 0 x2 y 2 4 sin xy is continuous ex y 2 11 3 Partial Derivatives If 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 with respect to x at a b Denoted fx a b g 0 a where g x f x b The limit definition is f a h b f a b h Similarly the partial derivative of f with respect to y at a b is given by fx a b lim h 0 f a b h f a b h 0 h fy a b lim In general if f is a function of two variables its partial derivatives are defined by f x h y f x y h 0 h fx x y lim f x y h f x y h 0 h fy x y lim Notations for Partial Derivatives If z f x y we write fx x y fx z f f x y f1 D1 f Dx f x x x fy x y fy f z f x y f2 D2 f Dy f y y y Rule For finding partial derivatives of z f x y 1 To find fx regard y as a constant and differentiate f x y with respect to x 2 To find fy regard x as a constant and differentiate f x y with respect to y Example Find the first partial derivatives of the function a f x y 3x 2y 4 5 b f x y x y x y c f x y z xy 2 z 3 3yz d Find fx 3 4 where f x y p x2 y 2 For implicit functions where we cannot solve for one variable we use implicit differentiation Example Use implicit differentiation to find z z and for x2 y 2 z 2 3xyz x y Interpretations of Partial Derivatives The partial derivatives fx a b and fy a b can be interpreted geometrically as the slopes of the tangent lines at a point P a b c to the traces C1 and C2 of S in the planes y b and x a Can also be interpreted as rates of change 6 Higher Derivatives If f is a function of two variables then its partial derivatives fx and fy are also functions of two variables so we can consider their partial derivatives which are called the second partial derivatives Notation 2f f fx x fxx f11 x x x2 f 2f fx y fxy f12 y x y x 2f f fy x fyx f21 x y x y f 2f fy y fyy f22 y y y 2 2z x2 2z y x 2z x y 2z y 2 Example Find all the second partial derivatives for f x y x4 3x2 y 3 Note fxy fyx Clairaut s Theorem Suppose f is defined on a disk D that contains the point a b If the functions fxy and fyx are both continuous on D then fxy a b fyx a b 7 11 4 Tangent Planes and Linear Approximations Recall In single variable calculus we approximate a differentiable function with its tangent line For a function f x y at x0 y0 the tangent line equation is given by y y0 f 0 x x x0 Since y0 f x0 then we can rewrite this equation as y f x0 f 0 x x x0 giving us the linear approximation …