Ch11 Partial Derivatives

7/29/2019 Ch11 Partial Derivatives

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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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Ch11 Partial Derivatives

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