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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