Mathematics for Business (Finance) Instructor: Prof. Ken Tsang Room E409-R11 Email: kentsang@uic.edu.hk

Chapter 6: Calculus of two variables

In this Chapter: Functions of 2 Variables Limits and Continuity Partial Derivatives Tangent Planes and Linear Approximations The Chain Rule Maximum and Minimum Values Double integrals and volume evaluation

DEFINITION: A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, . We 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.

Domain of

Domain of

Domain of

Graph of z=f(x,y)

Graph of

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).

Contour map of

The graph of h(x, y)=4x2+y2 is formed by lifting the level curves.

DEFINITION Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say that the limit of f (x, y) as (x, y) approaches (a ,b) is L and we write if for every number ε> 0 there is a corresponding number δ> 0 such that If and then

DEFINITION A function f of two variables is called continuous at (a, b) if We say f is continuous on D if f is continuous at every point (a, b) in D.

If f is a function of two variables, its partial derivatives are the functions fx and fy defined by

NOTATIONS FOR PARTIAL DERIVATIVES If z=f (x, y) , we write

RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) 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.

The partial derivatives of f at (a, b) are the slopes of the tangents to C1 and C2.

The second partial derivatives of f. If z=f (x, y), we use the following notation:

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

The tangent plane contains the tangent lines T1 and T2

Suppose f has continuous partial derivatives Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P(xo ,yo ,zo) is

The differential of x is dx=△x, if y=f(x), then dy=f’(x)dx is the differential of y.

For a differentiable function of two variables, z= f (x ,y), we define the differentials dx and dy (i.e. small increments in x & y directions). Then the differential dz (total differential), is defined by

For such functions the linear approximation is:

THE CHAIN RULE Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (t) and y=h (t) and are both differentiable functions of t. Then z is a differentiable function of t and

THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2,‧‧‧,xn and each xj is a differentiable function of the m variables t1, t2,‧‧‧,tm Then u is a function of t1, t2,‧‧‧, tm and for each i=1,2,‧‧‧,m.

DEFINITION A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). [This means that f (x, y) ≤ f (a, b) for all points (x, y) in some disk with center (a, b).] The number f (a, b) is called a local maximum value. If f (x, y) ≥ f (a, b) when (x, y) is near (a, b), then f (a, b) is a local minimum value.

THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then fx(a, b)=0 and fy(a, b)=0.

A point (a, b) is called a critical point (or stationary point) of f if fx (a, b)=0 and fy (a, b)=0, or if one of these partial derivatives does not exist.

SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b) , and suppose that both fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical point of f]. Let If D>0 and fxx (a, b)>0 , then f (a, b) is a local minimum. (b)If D>0 and fxx (a, b)<0, then f (a, b) is a local maximum. (c) If D<0, then f (a, b) is not a local maximum or minimum.

NOTE 1 In case (c) the point (a, b) is called a saddle point of f and the graph of f crosses its tangent plane at (a, b). NOTE 2 If D=0, the test gives no information: f could have a local maximum or local minimum at (a, b), or (a, b) could be a saddle point of f. NOTE 3 To remember the formula for D it’s helpful to write it as a determinant:

EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

Double integrals over rectangles Recall the definition of definite integrals of functions of a single variable Suppose f(x) is defined on a interval [a,b].

Taking a partition P of [a, b] into subintervals: Using the areas of the small rectangles to approximate the areas of the curve sided echelons

and summing them, we have (1) (2)

Volume and Double Integral z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d] S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R} Volume of S = ?

Double integral of a function of two variables defined on a closed rectangle like the following Taking a partition of the rectangle

ij’s column: (xi, yj) f (xij*, yij*) Rij Sample point (xij*, yij*) z Rij (xi, yj) Sample point (xij*, yij*) x y Area of Rij is Δ A = Δ x Δ y Volume of ij’s column: Total volume of all columns:

double Riemann sum Definition

Definition: The double integral of f over the rectangle R is if the limit exists Double Riemann sum:

Example 1 z=16-x2-2y2 0≤x≤2 0≤y≤2 Estimate the volume of the solid above the square and below the graph

m=n=16 m=n=4 m=n=8 V≈46.46875 V≈41.5 V≈44.875 Exact volume? V=48

Example 2 z

Integrals over arbitrary regions A is a bounded plane region f (x,y) is defined on A Find a rectangle R containing A Define new function on R: A f (x,y) R

Properties Linearity Comparison If f(x,y)≥g(x,y) for all (x,y) in R, then

Additivity A1 A2 If A1 and A2 are non-overlapping regions then Area

Computation If f (x,y) is continuous on rectangle R=[a,b]×[c,d] then double integral is equal to iterated integral a b x y c d y fixed fixed x

Fubini’s Theorem If is continuous on the rectangle then More generally, this is true if we assume that is bounded on , is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.

Note If f (x, y) = g (x) h(y) then

EXAMPLE 1 Evaluate the iterated integrals (See the blackboard)

EXAMPLE 2 Evaluate the double integral where (See the blackboard)

EXAMPLE 4 Find the volume of the solid S that is bounded by the elliptic paraboloid , the plane and , and three coordinate planes.

Solution We first observe that S is the solid that lies under the surface and the above the square

More general case If f (x,y) is continuous on A={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral y g(x) A h(x) x a x b

Similarly If f (x,y) is continuous on A={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral y d A y h(y) g(y) c x

Type I regions If f is continuous on a type I region D such that then

Type II regions (4) (5) where D is a type II region given by Equation 4

it is Type I region!

Example 2 Find the volume of the solid under the paraboloid and above the region D in the xy-plane bounded by the line and the parabola

Solution 1 Type I Type II Solution 2

Example 3 Evaluate , where D is the region bounded by the line and the parabola D as a type II D as a type I