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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]


Chapter 6 calculus of two variables

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.





Graph of z=f(x,y)



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).



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


The differential of the tangent plane to the surface 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 we define the linear approximation is:


THE CHAIN RULE we define the 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) we define the 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 we define the 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 we define the 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 we define the 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 we define the 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 we define the 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 we define the

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 we define the

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 we define the

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

Recall the definition of definite integrals of

functions of a single variable


Taking a partition P of [ we define the 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 we define the

(1)

(2)


Volume and double integral
Volume and Double Integral we define the

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


z defined on a closed rectangle like the following

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:


double Riemann sum defined on a closed rectangle like the following

Definition


Definition
Definition: defined on a closed rectangle like the following

The double integral

of f over the rectangle R is

if the limit exists

Double Riemann sum:


Example 1
Example 1 defined on a closed rectangle like the following

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 defined on a closed rectangle like the following

m=n=4

m=n=8

V≈46.46875

V≈41.5

V≈44.875

V=48

Exact volume?


Example 2

z defined on a closed rectangle like the following

Example 2


Integrals over arbitrary regions
Integrals over arbitrary regions defined on a closed rectangle like the following

  • 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 defined on a closed rectangle like the following

Linearity

Comparison

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


Additivity defined on a closed rectangle like the following

A2

A1

If A1 and A2 are non-overlapping regions then

Area


Computation

y defined on a closed rectangle like the following

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


Fubini’s Theorem defined on a closed rectangle like the following 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 defined on a closed rectangle like the following

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


EXAMPLE 1 defined on a closed rectangle like the following Evaluate the iterated integrals

(See the blackboard)


where defined on a closed rectangle like the following

EXAMPLE 2

Evaluate the double integral

(See the blackboard)


EXAMPLE 4 defined on a closed rectangle like the following Find the volume of the solid S that is bounded by the elliptic paraboloid , the plane and , and three coordinate planes.


Solution defined on a closed rectangle like the following

and the above the square

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


More general case
More general case defined on a closed rectangle like the following

  • 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 defined on a closed rectangle like the following

  • 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 defined on a closed rectangle like the following regions

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

then


Type II defined on a closed rectangle like the following regions

(4)

(5)

where D is a type II region given by Equation 4


it is Type I region! defined on a closed rectangle like the following


Find the volume of the solid under the defined on a closed rectangle like the following

Example 2

and above the region D in the

paraboloid

xy-plane bounded by the line

and the parabola


Type I defined on a closed rectangle like the following

Solution 1

Type II

Solution 2


Example 3 defined on a closed rectangle like the following 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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