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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)

Instructor: Prof. Ken Tsang

Room E409-R11

Email: [email protected]

### 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.

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.

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 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

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

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 [a, b] into subintervals:

Using the areas of the small rectangles to approximate the areas of the curve sided echelons

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

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:

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

V=48

Exact volume?

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

Linearity

Comparison

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

Additivity

A2

A1

If A1 and A2 are non-overlapping regions then

Area

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

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

where

EXAMPLE 2

Evaluate the double integral

(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

and the above the square

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

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
• 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

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

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

Type I

Solution 1

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