# ESSENTIAL CALCULUS CH11 Partial derivatives by sEELs8Fe

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```									ESSENTIAL
CALCULUS

CH11 Partial
derivatives
In this Chapter:
 11.1 Functions of Several Variables
 11.2 Limits and Continuity
 11.3 Partial Derivatives
 11.4 Tangent Planes and Linear Approximations
 11.5 The Chain Rule
 11.6 Directional Derivatives and the Gradient Vector
 11.7 Maximum and Minimum Values
 11.8 Lagrange Multipliers
Review
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, f ( x, y) ( x, y)  D .

Chapter 11, 11.1, P593
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.

Chapter 11, 11.1, P593
Chapter 11, 11.1, P593
x  y 1
Domain of f ( x, y ) 
x 1

Chapter 11, 11.1, P594
Domain of   f ( x, y )  x ln( y 2  x)

Chapter 11, 11.1, P594
Domain of g ( x, y)  9  x  y
2    2

Chapter 11, 11.1, P594
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.

Chapter 11, 11.1, P594
Chapter 11, 11.1, P595
Graph of   g ( x, y)  9  x 2  y 2
Chapter 11, 11.1, P595
h ( x, y )  4 x 2  y 2
Graph of
Chapter 11, 11.1, P595
 x2  y 2
(a) f ( x, y)  ( x  3y )e
2       2      x2  y 2
(b) f ( x, y)  ( x  3y )e
2      2

Chapter 11, 11.1, P596
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).

Chapter 11, 11.1, P596
Chapter 11, 11.1, P597
Chapter 11, 11.1, P597
Chapter 11, 11.1, P598
Contour map of   f ( x, y)  6  3x  2 y

Chapter 11, 11.1, P598
Contour map of g ( x, y)  9  x 2  y 2
Chapter 11, 11.1, P598
The graph of h (x, y)=4x2+y2
is formed by lifting the level curves.
Chapter 11, 11.1, P599
Chapter 11, 11.1, P599
Chapter 11, 11.1, P599
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
lim              f ( x, y )  L
( x , y ) ( a ,b )

if for every number ε> 0 there is a corresponding
number δ> 0 such that
If ( x, y)  Dand 0  ( x  a)2  ( y  b)2   then f ( x, y)  L  

Chapter 11, 11.2, P604
Chapter 11, 11.2, P604
Chapter 11, 11.2, P604
Chapter 11, 11.2, P604
If f( x, y)→L1 as (x, y)→ (a ,b) along a path C1
and f (x, y) →L2 as (x, y)→ (a, b) along a path C2,
where L1≠L2, then lim (x, y)→ (a, b) f (x, y) does not
exist.

Chapter 11, 11.2, P605
4. DEFINITION A function f of two variables
is called continuous at (a, b) if

lim              f ( x, y )  f (a, b)
( x , y ) ( a ,b )

We say f is continuous on D if f is continuous
at every point (a, b) in D.

Chapter 11, 11.2, P607
5.If f is defined on a subset D of Rn, then lim x→a
f(x) =L means that for every number ε> 0 there
is a corresponding number δ> 0 such that

If x  D and 0  x  a   then f ( x)  L  

Chapter 11, 11.2, P609
4, If f is a function of two variables, its partial
derivatives are the functions fx and fy defined by

f ( x  h, y )  f ( x, y )
f x ( x, y )  lim
h 0             h
f ( x  y, h)  f ( x, y )
f y ( x, y )  lim
h 0             h

Chapter 11, 11.3, P611
NOTATIONS FOR PARTIAL DERIVATIVES If
Z=f (x, y) , we write

f                  z
f x ( x, y )  f x         f ( x, y )      f1  D1 f  Dx f
x   x              x

f                  z
f y ( x, y )  f y           f ( x, y )      f 2  D2 f  D y f
y   y              y

Chapter 11, 11.3, P612
RULE FOR FINDING PARTIAL DERIVATIVES
OF z=f (x, y)
1.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.

Chapter 11, 11.3, P612
FIGURE 1
The partial derivatives of f at (a, b) are
the slopes of the tangents to C1 and C2.

Chapter 11, 11.3, P612
Chapter 11, 11.3, P613
Chapter 11, 11.3, P613
The second partial derivatives of f. If z=f (x,
y), we use the following notation:

  f   2 f    2z
( f x ) x  f xx  f11                  
x  x   x 2
x 2

  f    2 f   2z
( f x ) y  f xy  f12                   
y  x  yx    yx

  f     2 f    2 z
( f y ) x  f yx  f 21         y   xy  xy
    
x     

  f   2 f     2 z
( f y ) y  f yy  f 22       
 y   y 2  y 2
y     


Chapter 11, 11.3, P614
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

f xy ( a, b)  f yx ( a, b)

Chapter 11, 11.3, P615
FIGURE 1
The tangent plane contains the
tangent lines T1 and T2
Chapter 11, 11.4, P619
2. 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

z  z0  f x ( x0 , y0 )( x  x0 )  f y ( x0 , y0 )( y  y0 )

Chapter 11, 11.4, P620
The linear function whose graph is this tangent
plane, namely

3.    L( x, y )  f (a, b)  f x (a, b)( x  a)  f y (a, b)( y  b)

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

4.     f ( x, y )  f (a, b)  f x (a, b)( x  a)  f y (a, b)( y  b)

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

Chapter 11, 11.4, P621
7. DEFINITION If z= f (x, y), then f is
differentiable at (a, b) if ∆z can be expressed
in the form

z  f x (a, b)x  f y (a, b)y   1x   2 y

where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0).

Chapter 11, 11.4, P622
8. THEOREM If the partial derivatives fx and fy
exist near (a, b) and are continuous at (a, b),
then f is differentiable at (a, b).

Chapter 11, 11.4, P622
Chapter 11, 11.4, P623
For a differentiable function of two variables, z=
f (x ,y), we define the differentials dx and dy
to be independent variables; that is, they can
be given any values. Then the differential dz,
also called the total differential, is defined by

z   z
dx  f x ( x, y )dx  f y ( x, y )dy  dx  dy
x   y

Chapter 11, 11.4, P623
Chapter 11, 11.4, P624
For such functions the linear approximation is

f ( x, y, z)  f (a, b, c)  f x (a, b, c)(x  a)  f y (a, b, c)( y  b)  f z (a, b, c)(z  c)

and the linearization L (x, y, z) is the right side of
this expression.

Chapter 11, 11.4, P625
If w=f (x, y, z), then the increment of w is

w  f ( x  x, y  y, z  z)  f ( x, y, z )

The differential dw is defined in terms of the
differentials dx, dy, and dz of the independent
variables by
w      w      w
dw     dx     dy     dz
x      y      a

Chapter 11, 11.4, P625
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

dz f dx f dy
     
dt x dt y dt

Chapter 11, 11.5, P627
dz z dx z dy
     
dt x dt y dt

Chapter 11, 11.5, P628
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

dz z dx z dy         dz z dx z dy
                          
dx x ds y ds         dt x dt y dt

Chapter 11, 11.5, P629
Chapter 11, 11.5, P630
Chapter 11, 11.5, P630
Chapter 11, 11.5, P630
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

u u dx1 u x2         u xn
               ‧‧‧

ti x1 dti x2 dti      xn ti
for each i=1,2,‧‧‧,m.

Chapter 11, 11.5, P630
Chapter 11, 11.5, P631
F (x, y)=0. Since both x and y are functions of
x, we obtain
F dx F dy
      0
x dx y dx
But dx /dx=1, so if ∂F/∂y≠0 we solve for
dy/dx and obtain
F
dy
 x   Fx
dx    F     Fy
y

Chapter 11, 11.5, P632
F (x, y, z)=0      F dx F dy F z
               0
x dx y dx z x
                  
But x  ( x)  1 and        ( y)  1
x
so this equation becomes               F F z
      0
x z x
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.
F                F
dz
  x
dz    y
F

dx                 dy    F
Chapter 11, 11.5, P632
z                z
Chapter 11, 11.6, P636
2. DEFINITION The directional derivative
of f at (xo,yo) in the direction of a unit vector
u=<a, b> is
f ( x0  ha, y0  hb)  f ( x0 , y0 )
Du f ( x0 , y0 )  lim
h0                  h
if this limit exists.

Chapter 11, 11.6, P636
3. THEOREM If f is a differentiable function of
x and y, then f has a directional derivative in
the direction of any unit vector u=<a, b> and

Du f ( x, y )  f x ( x, y )a  f y ( x, y )b

Chapter 11, 11.6, P637
8. DEFINITION If f is a function of two
variables x and y , then the gradient of f is
the vector function ∆f defined by

f   f
f ( x, y )  f x ( x, y ), f y ( x, y )  i     j
x y

Chapter 11, 11.6, P638
Du f ( x, y )  f ( x, y )  u

Chapter 11, 11.6, P638
10. DEFINITION The directional derivative
of f at (x0, y0, z0) in the direction of a unit
vector u=<a, b, c> is
f ( x0  ha, y0  hb, z0  hc)  f ( x0 , y0 , z0 )
Du f ( x0 , y0 , z0 )  lim
h 0                        h
if this limit exists.

Chapter 11, 11.6, P639
f ( x0  hu)  f ( x0 )
Du f ( x0 )  lim
h 0           h

Chapter 11, 11.6, P639
f   f   f
f  f x , f y , f z  i     j k
x y     z

Chapter 11, 11.6, P639
Du f ( x, y, z )  f ( x, y, z )  u

Chapter 11, 11.6, P640
15. THEOREM Suppose f is a differentiable
function of two or three variables. The maximum
value of the directional derivative
Du f(x) is │▽f (x)│ and it occurs when u has the
same direction as the gradient vector ▽ f(x) .

Chapter 11, 11.6, P640
The equation of this tangent plane as
F0 ( x0 , y0 , z0 )( x  x0 )  Fy ( x0 , y0 , z0 )( y  y0 )  Fz ( x0 , y0 , z0 )( z  z0 )

The symmetric equations of the normal
line to soot P are
x  x0              y  y0              z  z0
                   
Fx ( x0 , y0 , z0 ) Fy ( x0 , y0 , z0 ) Fz ( x0 , y0 , z0 )

Chapter 11, 11.6, P642
Chapter 11, 11.6, P644
Chapter 11, 11.6, P644
Chapter 11, 11.7, P647
1. 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.

Chapter 11, 11.7, P647
2. 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)=1 and
fy(a, b)=0.

Chapter 11, 11.7, P647
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.

Chapter 11, 11.7, P647
Chapter 11, 11.7, P648
3. SECOND DERIVATIVES TEST Suppose the
second partial derivatives of f are continuous on
a disk with center (a, b) , and suppose that
fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical
point of f]. Let
D  D(a, b)  f xx (a, b) f yy (a, b)  [ f xy (a, b)]2
(a)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.

Chapter 11, 11.7, P648
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:

f xy f xy
D                f xx f yy  ( f xy )   2

f yx f yy

Chapter 11, 11.7, P648
z  x  y  4 xy  1
4   4

Chapter 11, 11.7, P649
Chapter 11, 11.7, P649
Chapter 11, 11.7, P651
4. 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.

Chapter 11, 11.7, P651
5. To find the absolute maximum and minimum
values of a continuous function f on a closed,
bounded set D:
1. Find the values of f at the critical points of in D.
2. Find the extreme values of f on the boundary of D.
3. The largest of the values from steps 1 and 2 is the
absolute maximum value; the smallest of these
values is the absolute minimum value.

Chapter 11, 11.7, P651
Chapter 11, 11.7, P652
Chapter 11, 11.8, P654
f ( x0 , y0 , z0 )  g ( x0 , y0 , z0 )

Chapter 11, 11.8, P655
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
f ( x, y, z)  g ( x, y, z )
and            g ( x, y, z)  k
(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.
Chapter 11, 11.8, P655
Chapter 11, 11.8, P657
Chapter 11, 11.8, P657

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