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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

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mathematics for business finance

Mathematics for Business(Finance)

Instructor: Prof. Ken Tsang

Room E409-R11

Email: [email protected]

slide3

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

slide4

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.

slide11

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).

slide14

The graph of h(x, y)=4x2+y2

is formed by lifting the level curves.

slide15

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
slide17

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.

slide19

NOTATIONS FOR PARTIAL DERIVATIVES

If z=f (x, y) , we write

slide20

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.
slide21

The partial derivatives of f at (a, b) are

the slopes of the tangents to C1 and C2.

slide24

The second partial derivatives of f.

If z=f (x, y), we use the following notation:

slide25

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

slide26

The tangent plane contains the

tangent lines T1 and T2

slide27

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

slide28

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

slide29

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

slide32

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

slide33

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.

slide34

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.

slide36

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.

slide37

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.

slide38

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.
slide39

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:

slide43

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.

slide44

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
Double integrals over rectangles

Suppose f(x) is defined on a interval [a,b].

Recall the definition of definite integrals of

functions of a single variable

slide46
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

volume and double integral
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 = ?

slide49
Double integral of a function of two variables defined on a closed rectangle like the following

Taking a partition of the rectangle

slide51

z

y

(xi, yj)

Rij

f (xij*, yij*)

y

Sample point (xij*, yij*)

x

x

Δ x

Δ y

ij’s column:

Area of Rijis ΔA = Δ x Δ y

Volume of ij’s column:

Total volume of all columns:

definition
Definition:

The double integral

of f over the rectangle R is

if the limit exists

Double Riemann sum:

example 1
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

slide55

m=n=16

m=n=4

m=n=8

V≈46.46875

V≈41.5

V≈44.875

V=48

Exact volume?

integrals over arbitrary regions
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)

0

R

properties
Properties

Linearity

Comparison

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

slide59

Additivity

A2

A1

If A1 and A2 are non-overlapping regions then

Area

computation

y

d

c

x

a

b

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

y

fixed

fixed

x

slide61
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.

slide62
Note

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

slide64

where

EXAMPLE 2

Evaluate the double integral

(See the blackboard)

slide65
EXAMPLE 4 Find the volume of the solid S that is bounded by the elliptic paraboloid , the plane and , and three coordinate planes.
slide66

Solution

and the above the square

We first observe that S is the solid that lies under the surface

more general case
More general case
  • If f (x,y) is continuous onA={(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
Similarly
  • If f (x,y) is continuous onA={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral

y

d

A

y

g(y)

h(y)

c

x

type i regions
Type I regions

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

then

slide70

Type II regions

(4)

(5)

where D is a type II region given by Equation 4

slide72

Find the volume of the solid under the

Example 2

and above the region D in the

paraboloid

xy-plane bounded by the line

and the parabola

slide74

Type I

Solution 1

Type II

Solution 2

slide75
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

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