basic curves and basics of partial differentiation
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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 ),,(
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