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ESSENTIALCALCULUS
CH11 Partialderivatives
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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
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Chapter 11, 11.1, P593
DEFINITION A function f of two variablesis a rule that assigns to each ordered pair ofreal numbers (x, y) in a set D a unique realnumber denoted by f (x, y). The set D is thedomain of f and its range is the set of valuesthat f takes on, that is, . Dyxyxf ),(),(
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Chapter 11, 11.1, P593
We often write z=f (x, y) to make explicit thevalue taken on by f at the general point (x, y) .The variables x and y are independentvariables and z is the dependent variable.
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Chapter 11, 11.1, P593
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Chapter 11, 11.1, P594
Domain of1
1),(
x
yxyxf
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Chapter 11, 11.1, P594
Domain of )ln(),(2 xyxyxf
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Chapter 11, 11.1, P594
Domain of229),( yxyxg
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Chapter 11, 11.1, P594
DEFINITION If f is a function of two variableswith 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.
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Chapter 11, 11.1, P595
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Chapter 11, 11.1, P595
Graph of229),( yxyxg
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Chapter 11, 11.1, P595
Graph of22
4),( yxyxh
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Chapter 11, 11.1, P596
22
)3(),()(22 yx
eyxyxfa
22
)3(),()(22 yx
eyxyxfb
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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 off).
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Chapter 11, 11.1, P597
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Chapter 11, 11.1, P597
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Chapter 11, 11.1, P598
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Chapter 11, 11.1, P598
Contour map of yxyxf 236),(
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Chapter 11, 11.1, P598
Contour map of229),( yxyxg
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Chapter 11, 11.1, P599
The graph of h (x, y)=4x2+y2
is formed by lifting the level curves.
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Chapter 11, 11.1, P599
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Chapter 11, 11.1, P599
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Chapter 11, 11.2, P604
1.DEFINITION Let f be a function of twovariables whose domain D includes points
arbitrarily close to (a, b). Then we say that thelimit of f (x, y) as (x, y) approaches (a ,b)is L and we write
if for every number > 0 there is a correspondingnumber > 0 such that
If and then
Lyxfbayx ),(lim ),(),(
Dyx ),( 22 )()(0 byax Lyxf ),(
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P604
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Chapter 11, 11.2, P605
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 L1L2, then lim (x, y) (a, b) f (x, y) does notexist.
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Chapter 11, 11.2, P607
4. DEFINITION A function f of two variablesis called continuous at (a, b) if
We say f is continuous on D if f is continuousat every point (a, b) in D.
),(),(lim),(),(
bafyxfbayx
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Chapter 11, 11.2, P609
5.If f is defined on a subset D of Rn, then lim xaf(x) =L means that for every number > 0 there
is a corresponding number > 0 such that
If and thenDx ax0 Lxf )(
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Chapter 11, 11.3, P611
4, If f is a function of two variables, its partialderivatives are the functions fx and fy defined by
h
yxfyhxfyxf
hx
),(),(lim),(
0
hyxfhyxfyxf
hy ),(),(lim),(
0
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Chapter 11, 11.3, P612
NOTATIONS FOR PARTIAL DERIVATIVES IfZ=f (x, y) , we write
fDfDfx
zyxf
xx
ffyxf xxx
11),(),(
fDfDfy
zyxf
yy
ffyxf yyy
22),(),(
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Chapter 11, 11.3, P612
RULE FOR FINDING PARTIAL DERIVATIVESOF z=f(x, y)
1.To find fx, regard y as a constant and differentiatef (x, y) with respect to x.
2. To find fy, regard x as a constant and differentiate f(x, y) with respect to y.
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Chapter 11, 11.3, P612
FIGURE 1The partial derivatives of f at (a, b) arethe slopes of the tangents to C1 and C2.
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Chapter 11, 11.3, P613
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Chapter 11, 11.3, P613
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Chapter 11, 11.3, P614
2
2
2
2
11)(
xz
xf
xf
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:
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Chapter 11, 11.3, P615
CLAIRAUTS THEOREM Suppose f is definedon a disk D that contains the point (a, b) . If
the functions fxy and fyx are both continuous onD, then
),(),( bafbaf yxxy
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Chapter 11, 11.4, P619
FIGURE 1The tangent plane contains thetangent lines T
1and T
2
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Chapter 11, 11.4, P620
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
))(,())(,( 0000000 yyyxfxxyxfzz yx
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Chapter 11, 11.4, P621
The linear function whose graph is this tangentplane, namely
3.
is called the linearization offat (a, b) and theapproximation
4.
is called the linear approximation or the
tangent plane approximation offat (a, b)
))(,())(,(),(),( bybafaxbafbafyxL yx
))(,())(,(),(),( bybafaxbafbafyxf yx
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Chapter 11, 11.4, P622
7. DEFINITION If z= f (x, y), then f isdifferentiable at (a, b) if z can be expressed
in the form
where 1 and 2 0as (x, y)
(0,0).
yxybafxbafz yx 21),(),(
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Chapter 11, 11.4, P622
8. THEOREM If the partial derivatives fx and fyexist near (a, b) and are continuous at (a, b),
then fis differentiable at (a, b).
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Chapter 11, 11.4, P623
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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 canbe given any values. Then the differential dz,also called the total differential, is defined by
dyy
zdx
x
zdyyxfdxyxfdx yx
),(),(
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Chapter 11, 11.4, P624
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Chapter 11, 11.4, P625
For such functions the linear approximation is
and the linearization L (x, y, z) is the right side ofthis expression.
))(,,())(,,())(,,(),,(),,( czcbafbycbafaxcbafcbafzyxf zyx
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Chapter 11, 11.4, P625
If w=f (x, y, z), then the increment of w is
The differential dw is defined in terms of thedifferentials dx, dy, and dz of the independentvariables by
),,(),,( zyxfzzyyxxfw
dza
wdy
y
wdx
x
wdw
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Chapter 11, 11.5, P627
2. THE CHAIN RULE (CASE 1) Suppose thatz=f (x, y) is a differentiable function of x and y,
where x=g (t) and y=h (t) and are bothdifferentiable functions of t. Then z is adifferentiable function of t and
dt
dy
y
f
dt
dx
x
f
dt
dz
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Chapter 11, 11.5, P628
dtdy
yz
dtdx
xz
dtdz
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Chapter 11, 11.5, P629
3. THE CHAIN RULE (CASE 2) Suppose that z=f(x, y) is a differentiable function of x and y, wherex=g (s, t) and y=h (s, t) are differentiablefunctions ofs and t. Then
ds
dy
y
z
ds
dx
x
z
dx
dz
dt
dy
y
z
dt
dx
x
z
dt
dz
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Chapter 11, 11.5, P630
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Chapter 11, 11.5, P630
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Chapter 11, 11.5, P630
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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 differentiablefunction of the m variables t1, t2,,tm Then u is afunction of t1, t2,, tm and
for each i=1,2,,m.
i
n
niii tx
xu
dtx
xu
dtdx
xu
tu
2
2
1
1
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Chapter 11, 11.5, P631
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Chapter 11, 11.5, P632
F (x, y)=0. Since both x and y are functions ofx, we obtain
But dx /dx=1, so if F/y0 we solve for
dy/dx and obtain
0
dxdy
yF
dxdx
xF
y
x
F
F
y
Fx
F
dx
dy
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Chapter 11, 11.5, P632
F (x, y, z)=0
But and
so this equation becomes
If F/z0 ,we solve for z/x and obtain thefirst formula in Equations 7. The formula forz/y is obtained in a similar manner.
0
x
z
z
F
dx
dy
y
F
dx
dx
x
F
1)(
x
x1)(
y
x
0
x
z
z
F
x
F
z
FxF
dx
dz
z
F
yF
dy
dz
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Chapter 11, 11.6, P636
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Chapter 11, 11.6, P636
2. DEFINITION The directional derivativeof f at (xo,yo) in the direction of a unit vectoru= is
if this limit exists.
h
yxfhbyhaxfyxfD
hu
),(),(lim),( 0000
000
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Chapter 11, 11.6, P637
3. THEOREM If f is a differentiable function ofx and y, then f has a directional derivative in
the direction of any unit vector u= and
byxfayxfyxfD yxu ),(),(),(
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Chapter 11, 11.6, P638
8. DEFINITION If f is a function of twovariables x and y , then the gradient of f isthe vector function f defined by
j
y
fi
x
fyxfyxfyxf yx
),(),,(),(
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Chapter 11, 11.6, P638
uyxfyxfDu ),(),(
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Chapter 11, 11.6, P639
10. DEFINITION The directional derivativeof f at (x0, y0, z0) in the direction of a unit
vector u= is
if this limit exists.
h
zyxfhczhbyhaxfzyxfD
hu
),,(),,(lim),,( 000000
0000
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Chapter 11, 11.6, P639
h
xfhuxf
xfD hu
)()(
lim)(
00
00
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Chapter 11, 11.6, P639
kz
fj
y
fi
x
fffffzyx
,,
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Chapter 11, 11.6, P640
uzyxfzyxfDu ),,(),,(
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Chapter 11, 11.6, P640
15. THEOREM Suppose f is a differentiablefunction of two or three variables. The maximum
value of the directional derivativeDu f(x) is f (x) and it occurs when u has thesame direction as the gradient vector f(x) .
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Chapter 11, 11.6, P642
))(,,())(,,())(,,( 0000000000000 zzzyxFyyzyxFxxzyxF zy
The symmetric equations of the normalline to soot P are
),,(),,(),,( 000
0
000
0
000
0
zyxF
zz
zyxF
yy
zyxF
xx
zyx
The equation of this tangent plane as
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Chapter 11, 11.6, P644
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Chapter 11, 11.6, P644
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Chapter 11, 11.7, P647
f f bl
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Chapter 11, 11.7, P647
1. DEFINITION A function of two variableshas a local maximum at (a, b) if f (x, y) f(a, b) when (x, y) is near (a, b). [This meansthatf (x, y) f (a, b) for all points (x, y) in somedisk with center (a, b).] The number f (a, b) iscalled a local maximum value. If f (x, y) f(a, b) when (x, y) is near (a, b), then f (a, b) isa local minimum value.
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Chapter 11, 11.7, P647
2. THEOREM If f has a local maximum orminimum at (a, b) and the first order partialderivatives of f exist there, then fx(a, b)=1 andfy(a, b)=0.
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Chapter 11, 11.7, P647
A point (a, b) is called a critical point (orstationary point) of f if fx (a, b)=0 and fy (a,
b)=0, or if one of these partial derivatives doesnot exist.
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Chapter 11, 11.7, P648
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Chapter 11, 11.7, P648
3. SECOND DERIVATIVES TEST Suppose thesecond partial derivatives of f are continuous on
a disk with center (a, b) , and suppose thatfx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical
point of f]. Let
(a)If D>0 and fxx (a, b)>0 , then f (a, b) is a localminimum.
(b)If D>0 and fxx (a, b)
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Chapter 11, 11.7, P648
NOTE 1 In case (c) the point (a, b) is called asaddle point of f and the graph of f crosses itstangent plane at (a, b).NOTE 2 If D=0, the test gives no information:f could have a local maximum or localminimum at (a, b), or (a, b) could be a saddlepoint of f.NOTE 3 To remember the formula for D itshelpful to write it as a determinant:
2)( xyyyxxyyyx
xyxy fffff
ffD
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Chapter 11, 11.7, P649
1444 xyyxz
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Chapter 11, 11.7, P649
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Chapter 11, 11.7, P651
O O
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Chapter 11, 11.7, P651
4. EXTREME VALUE THEOREM FORFUNCTIONS OF TWO VARIABLES If f is
continuous on a closed, bounded set D in R2
,then f attains an absolute maximum valuef(x1,y1) and an absolute minimum value f(x2,y2)at some points (x1,y1) and (x2,y2) in D.
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Chapter 11, 11.7, P651
5. To find the absolute maximum and minimumvalues 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 thesevalues is the absolute minimum value.
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Chapter 11, 11.7, P652
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Chapter 11, 11.8, P654
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Chapter 11, 11.8, P655
),,(),,( 000000 zyxgzyxf
METHOD OF LAGRANGE MULTIPLIERS To find
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Chapter 11, 11.8, P655
METHOD OF LAGRANGE MULTIPLIERS To findthe 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 g0 on thesurface 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 themaximum value of f; the smallest is theminimum value of f.
),,(),,( zyxgzyxf
kzyxg ),,(
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Chapter 11, 11.8, P657
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