CALCULUS SUB CODE:2110014

DIPAK SINGH 130150111021 ELECTRONICS AND COMMUNICATION

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, .

Dyxyxf ),(),(

We often 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 1

1),(

x

yxyxf

Domain of )ln(),( 2 xyxyxf

Domain of 229),( yxyxg

DEFINITION If f is a function of two variables with domain D, then the graph of is the set of all points (x, y, z) in R3 such that z=f (x, y) and (x, y) is in D.

Graph of 229),( yxyxg

Graph of224),( yxyxh

22

)3(),()( 22 yxeyxyxfa 22

)3(),()( 22 yxeyxyxfb

Contour map of229),( yxyxg

1.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

Lyxfbayx

),(lim),(),(

Dyx ),( 22 )()(0 byax Lyxf ),(

4. 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.

),(),(lim),(),(

bafyxfbayx

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

h

yxfyhxfyxf

hx

),(),(lim),(

0

h

yxfhyxfyxf

hy

),(),(lim),(

0

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

fDfDfx

zyxf

xx

ffyxf xxx

11),(),(

fDfDfy

zyxf

yy

ffyxf yyy

22),(),(

Chapter 11, 11.3, P614

2

2

2

2

11)(x

z

x

f

x

f

xfff xxxx

xy

z

xy

f

x

f

yfff xyyx

22

12)(

yx

z

yx

f

y

f

xfff yxxy

22

21)(

2

2

2

2

22)(y

z

y

f

y

f

yfff yyyy

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

The linear function whose graph is this tangent plane, namely

3.

is called the linearization of f at (a, b) and the approximation

4.

is called the linear approximation or the tangent plane approximation of f at (a, b)

))(,())(,(),(),( bybafaxbafbafyxL yx

))(,())(,(),(),( bybafaxbafbafyxf yx

2. THE CHAIN RULE (CASE 1) 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

dt

dy

y

f

dt

dx

x

f

dt

dz

3. THE CHAIN RULE (CASE 2) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (s, t) and y=h (s, t) are differentiable functions of s and t. Then

ds

dy

y

z

ds

dx

x

z

dx

dz

dt

dy

y

z

dt

dx

x

z

dt

dz

4. 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.

i

n

niii t

x

x

u‧‧‧

dt

x

x

u

dt

dx

x

u

t

u

2

2

1

1

F (x, y)=0. Since both x and y are functions of x, we obtain

But dx /dx=1, so if ∂F/∂y≠0 we solve for dy/dx and obtain

0

dx

dy

y

F

dx

dx

x

F

y

x

F

F

yFxF

dx

dy

F (x, y, z)=0

But and

so this equation becomes

If ∂F/∂z≠0 ,we solve for ∂z/∂x and obtain the first formula in Equations 7. The formula for ∂z/∂y is obtained in a similar manner.

0

x

z

z

F

dx

dy

y

F

dx

dx

x

F

1)( x

x 1)( y

x

0

x

z

z

F

x

F

zFxF

dx

dz

zFyF

dy

dz

METHOD OF LAGRANGE MULTIPLIERS To find the maximum and minimum values of f (x, y, z) subject to the constraint g (x, y, z)=k [assuming that these extreme values exist and ▽g≠0 on the surface g (x, y, z)=k]:(a) Find all values of x, y, z, and such that

and

(b) Evaluate f at all the points (x, y, z) that result from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f.

),,(),,( zyxgzyxf

kzyxg ),,(