math 38 unit 2

7/26/2019 MATH 38 UNIT 2

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MATH 38

UNIT 2. DIRECTIONAL DERIVATIVES AND GRADIENTS

Objectives: Upon the completion of the course, the student must be able to1. find directional derivatives (if they eist!"#. find e$uations of tan%ent plane and normal line to a surface"3. find etreme values of functions (constrained or unconstrained case!" and

&evie' )ectors, *$uations of +ines and lanes in -pace

Outline: efinitionsTan%ent lanes and /ormals to -urfaces*trema of 0unctions of T'o )ariables+a%ran%e Multipliers

Reference: 1#. to 1#.2, T4

Revie N!tes

onsider a vector y , x A = (in position representation!.

Ma%nitude ## y x A +=

irection An%le, Aθ (in standard position! x

y tan A =θ

Unit )ector in the direction of A A

y ,

A

x U A =

j sini cosU A A A θ+θ=

'here 51 ,i = and 15 ,

( )b ,.af

x

f x =

∂

∂ slope of the line tan%ent to the curve of intersection of the surface ( )y , x f z =

and the plane by = at ( )( )b ,af ,b ,aP .

( )b ,.af y

f y =

∂

∂ slope of the line tan%ent to the curve of intersection of the surface ( )y , x f z=

and the plane a x = at ( )( )b ,af ,b ,aP

2." Directi!n#l Deriv#tives #n$ Gr#$ients

irectional derivatives are used to determine rate of chan%e to'ards a certain direction. 6n our case,vectors 'ill be used to represent directions.

Definiti!n %!f $irecti!n#l $eriv#tive&.

+et f be a function of t'o variables x and y . 6f u is a unit vector j sini cos θ+θ ,

then the $irecti!n#l $eriv#tive of f in the direction of u, denoted by f Du is %iven by

( ) ( ) ( )

h

y , x f sinhy ,cosh x f limy , x f Dh

u−θ+θ+

=→5

if this limit eists.

Re'#r(: ( )oou y , x f D is the rate of chan%e of f at the point ( )oo y , x in the direction of

u .

1

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MATH 38

T)e!re'. 6f f is a differentiable function of x and y , and jsinicosu θ+θ= , then

( ) ( ) ( ) θ+θ= siny , x f cosy , x f y , x f D y x u .

Definiti!n %!f *r#$ient&.

+et f be a function of t'o variables x and y such that x f and y f eist. The *r#$ient

of f denoted by f ∇ (read as 7del f 7! is defined by

( ) ( ) ( ) j y , x f i y , x f y , x f y x +=∇ .

Re'#r(s:

1. ( ) ( )y , x f Uy , x f Du ∇⋅= ( /T for%et that U is a unit vector.!

#. ( )oo y , x f ∇ is perpendicular to the level curve of f throu%h ( )oo y , x .

3. ( )oo y , x f ∇ is the direction of the steepest ascent .

#

E+#',le.

onsider the function defined by ( ) ### y x y , x f += .

etermine f Du if U is the unit vector in the direction of

a.3

π=θ b. π=θ

E+ercise.

onsider the function defined by ( ) ##

93 y x y , x f −= .

etermine f Du if U is the unit vector in the direction of

a.9

3π=θ b.

#

π=θ

E+#',le.

onsider the function defined by ( ) ### y x y , x f += .

etermine ( )y , x f ∇ .

E+#',le.

onsider the function defined by ( ) ## y x y , x f += .

etermine( )39 − ,f D

u 'here U is the unit vector in the direction:1# − , .

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MATH 38

%-!r functi!ns !f t)ree v#ri#bles.&

Definiti!n.

+et f be a function of t'o variables x , y and z . 6f u is a unit vector

k cos j cosi cos γ +β+α , then the $irecti!n#l $eriv#tive of f in the direction of u,

denoted byf D

u is %iven by

( ) ( ) ( )

h

z ,y , x f cosh z ,coshy ,cosh x f lim z ,y , x f Dh

u−γ +β+α+

=→5

if this limit

eists.

T)e!re'. 6f f is a differentiable function of x , y and z , and k cos j cosi cosu γ +β+α= ,

then

( ) ( ) ( ) ( ) γ +β+α= cos z ,y , x f cosy , x f cos z ,y , x f z ,y , x f D zy x u .

Definiti!n.

+et

f

be a function of t'o variables x , y and z such that x f ,

y f and zf eist. The *r#$ient of f denoted by f ∇ is defined by

( ) ( ) ( ) ( )k y , x f j y , x f i y , x f z ,y , x f zy x ++=∇ .

3

E+ercises.

etermine the directional derivative of f at point P in the direction of

A

1. ( ) y x y , x f #= , ( )#1 ,P , 93 , A −=

#. ( ) y siney , x f x = ,

π

95 ,P , 31 , A =

1. ( ) ###3 y xy x y , x f +−= , ( )#1 ,P − , 1# −= , A

;;;;;;;;;;;;;;;

#. -<etch the level curve of ( )#

x

y y , x f = that %oes throu%h the point

( )#1 ,P . -ho' that the vector ( )#1 ,f ∇ (dra'n 'ith initial

point at ( )#1 , , /T in its position representation! is perpendicular

to the level curve.

E+#',le.

onsider the function defined by ( ) yzsin xy cos z ,y , x f += .

etermine ( )35#

− ,f Du 'here U is the unit vector in the direction##1 , ,− .

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MATH 38

T)e!re' %!n '#+i'u' #n$ 'ini'u' r#te !f c)#n*e&.

+et f be a function of t'o variables x and y that is differentiable at ( )oo y , x , 'here

( ) 5≠∇ oo y , x f .

1. The maimum value of ( )oou y , x f D is ( )oo y , x f ∇ . This value is attained in the direction of

( )oo y , x f ∇ .

#. The minimum value of ( )oou y , x f D is ( )oo y , x f ∇− . This value is attained in the

opposite the direction of ( )oo y , x f ∇ .

9

E+ercises.

etermine the directional derivative of f at point P in the direction of

A

3. ( ) ##3 z y y x z ,y , x f −= , ( )31# , ,P − , ##1 , , A −=

9. ( ) ### zy x ln z ,y , x f ++= , ( )#31 , ,P , 111 −−= , , A

E+#',les.

5. etermine the maimum and minimum values of f DU for

( ) y tan Arcey , x f x = at ( )15 ,P .

1. The density of a rectan%ular plate at a point in the − xy plane is %iven

by( )

3

1

## ++

=ρ

y x

y , x . etermine the ma%nitude and the unit

vector in the direction of the %reatest rate of chan%e of ρ at ( )#3 ,

.

E+ercise.

The temperature T (in de%rees! at a point ( ) z ,y , x on a three=

dimensional space is %iven by ( ) xy x z ,y , x T #3 # += . etermine the

follo'in%

1. rate of chan%e of T at ( )##3 , ,− in the direction of the vector

k j i 3# −+−

#. the ma%nitude and the unit vector on the direction of the direction of

the %reatest rate of chan%e of T at ( )##3 , ,−

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MATH 38

2.2 T#n*ent l#nes #n$ N!r'#l t! Surf#ces

Rec#ll:

0or the e$uation of a plane

+et ( )5555 z ,y , x P be a point on a plane and c ,b ,aN is a vector normal to the

plane.

6f ( ) z ,y , x P is another point on the plane, then→PP5

is perpendicular to N .

Hence, 55 =⋅→NPP .

This %ives the standard equation of the plane: ( ) ( ) ( ) 5555 =−+−+− z zcy y b x x a

0or the e$uation of a line

+et ( )5555 z ,y , x P be a point on a line that is parallel to the vector c ,b ,aM .

6f ( ) z ,y , x P is another point on the line, then Mt PP ⋅=→5

.

This %ives the parametric equation of the line: ct z z ;bt y y ;at x x =−=−=− 555 .

Alternatively, if b ,a and c are all non=>ero,

thenc z z

by y

a x x 555 −=−=− is the symmetric equation of the line.

;;;;;;;;;;;;;;;;;;;;;;;

Definiti!n %!f # n!r'#l vect!r&

A vector ortho%onal to a tan%ent vector of every curve C throu%h a point 5P on a surface

is called as a n!r'#l vect!r to at 5P .

T)e!re'.

6f a surface is %iven by the e$uation ( ) 5= z ,y , x ! and ! is differentiable and

y x ! ,! and z ! are not all >ero at 5P on , then ( )5P!∇ is a normal vector to

at 5P .

Re'#r(:

The tan%ent plane to at 5P is a plane containin% the point 5P havin% ( )5P!∇ as a normal

vector.Hence, the e$uation of tan%ent plane is %iven by

( ) ( ) ( ) ( ) ( ) ( ) 5555555555555 =−⋅+−+−⋅ z z z ,y , x !y y z ,y , x ! x x z ,y , x ! zy x .

:

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MATH 38

The vector equation is %iven by ( ) ( ) ( ) ( )[ ] 55555 =−+−+−⋅∇ k z z j y y i x x P! .

Re'#r(:

The normal line to at 5P is a line throu%h the point 5P in the direction of the normal vector

( )5P!∇ . Hence, the parametric equation of the line is %iven by

( ) ( ) ( )555555555555 z , y , x F t z z ; z , y , x F t y y ; z , y , x F t x x z y x ⋅=−⋅=−⋅=− .

Also, if y x ! ,! and z ! are all non=>ero, the symmetric equation of the line is %iven by

( ) ( ) ( )555

5

555

5

555

5

z ,y , x !

z z

z ,y , x !

y y

z ,y , x !

x x

zy x

−

=

−

=

−

E+#',les. etermine the e$uation of the tan%ent plane to the %ivensurface at the indicated point.

1. #3## =−+ z y x at the point ( )9# , ,−−

#. z cosey x = at the point ( )51 ,e ,

E+#',le.

etermine a parametric and symmetric e$uation of the normal line to the

surface %iven by 5#### =+++ z z y x at the point ( )151 − , .

E+ercises.

1. etermine the e$uation of the tan%ent plane to the surface

x cose z y ## 3= at the point π 153 , , .

#. 0ind a point on the surface ##3# y x z += 'here the tan%ent plane

is parallel to the plane 538 =−− z y x .

3. -ho' that the e$uation of the tan%ent plane to the ellipsoid

1#

#

#

#

#

#

=++

c

z

b

y

a

x at ( )555 z ,y , x can be 'ritten in the form

1#

5

#

5

#

5=++

c

z z

b

y y

a

x x

9. ?ive a parametric and a symmetric e$uation of the normal line to the

surface 1### =++ zy x at the point ( 33# , ,

:. -ho' that any normal line to a point ( )555 z ,y , x on the sphere

#### a zy x =++ passes throu%h the center of the sphere.

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MATH 38

Re'#r(:

All tan%ent lines to a surface at a point 5P lie on the tan%ent plane to the surface at 5P .

S!lvin* f!r t)e t#n*ent line t! # curve !f intersecti!n

+et 5P be a point on the curve of intersection,C , of surfaces ( ) 5= z ,y , x ! and

( ) 5= z ,y , x " .

• The tan%ent line to C at 5P lies in each tan%ent plane to the surfaces.

• ( )51 P!N ∇= and ( )5# P"N ∇= are both ortho%onal to the unit vector to C .

• 6f 1N is /T parallel to #N , then #1 NN × is the direction of the tan%ent line to the

curve of intersection, C .

Definiti!n %!f t#n*enc/ #t # ,!int&

6f t'o surfaces have a common tan%ent plane, the t'o surfaces are said to be t#n*ent at thatpoint.

Re'#r(: 6f ( ) 5= z ,y , x ! and ( ) 5= z ,y , x " are tan%ent at a point 5P ,

then ( ) ( )55 P"k P! ∇⋅=∇ , for some constant k .

-!r t#n*ent line t! # curve !n # ,l#ne %!,ti!n#l&

T)e!re'.

4

E+#',le.

etermine the e$uation of the tan%ent line to the curve of intersection of

the surfaces defined by8

##

=−+ z y x and#

##

−=+− z y x at thepoint ( )5## , ,− .

E+#',le.

-ho' that the surfaces defined by 59## =++ y z x and

54### =+−++ z z y x are tan%ent at the point ( )#15 , ,− .

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MATH 38

6f C is a curve in ## %iven by ( ) 5=y , x ! such that ! is differentiable and y x ! ,!

are not both >ero at 5P on C , then ( )5P!∇ is a normal vector to C at 5P .

Re'#r(:

6n ## , the e$uation of the tan%ent line to the curve ( ) 5=y , x ! at ( )555 y , x P is %iven by

( ) ( ) ( ) ( ) 5555555 =−⋅+−⋅ y y y , x ! x x y , x ! y x .

000000000000000000

2.1 E+tre'# !f -uncti!ns !f !re T)#n One V#ri#ble

Definiti!n %!f rel#tive e+tre'#&

+et f be a function of t'o variables x and y .

f has a rel#tive 'ini'u' v#lue at ( )b ,a if there eists an open ball $ containin%

( )b ,a such that ( ) ( )y , x f b ,af ≤ for all ( ) $y , x ∈ .

f has a rel#tive '#+i'u' v#lue at ( )b ,a if there eists an open ball $ containin%

( )b ,a such that ( ) ( )y , x f b ,af ≥ for all ( ) $y , x ∈ .

Re'#r(s:

1. A point ( )b ,a is a critic#l ,!int of f if a.! ( ) 5=b ,af x and ( ) 5=b ,af y " or

b.! ( )b ,af x or ( )b ,af y

#. 6f a critical point has no relative etrema, then it is a s#$$le ,!int.

T)e!re' %Sec!n$3Deriv#tive Test&

8

E+ercises.

1. etermine the e$uation of the tan%ent line to the curve of intersection

of the surfaces defined by #+= z siney x and

( ) 31# −+−= x lny z at the point ( )5#5 , , .

#. -ho' that the surfaces defined by 15829 ### =++ z y x and

3= xyz are tan%ent at the point ( )#3 , , .

3. etermine the e$uation of the tan%ent line to the curve

52## 33 =−+ xy y x at the point ( )#1 , .

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MATH 38

+et f be a function of t'o variables x and y such that xy yy xx y x f ,f ,f ,f ,f are continuous

on some open dis< $ containin% ( )b ,a . -uppose further that ( ) 5=b ,af x and

( ) 5=b ,af y .

+et ( ) ( ) ( ) ( )[ ]#

b ,af b ,af b ,af b ,aD xy yy xx −⋅= .

i. 6f ( ) 5>b ,aD and ( ) 5>b ,af xx (or ( ) 5>b ,af yy !, f has a relative minimum value at

( )b ,a .

ii. 6f ( ) 5>b ,aD and ( ) 5<b ,af xx (or ( ) 5<b ,af yy !, f has a relative maximum value at

( )b ,a .

iii. 6f ( ) 5<b ,aD , f has a saddle point at ( )( )b ,af ,b ,a .

iv. /o conclusion re%ardin% relative etrema can be made if ( ) 5=b ,aD .

Definiti!ns.

A re%ion# is b!un$e$ if it is a subre%ion of a closed dis<. The b!un$#r/ of a re%ion # is

the set of all points P such that ( )r ;P$ contains points of # and points not in # , for

any r . A cl!se$ re%ion contains its boundary.

T)e!re' %E+tre'e3V#lue T)e!re'&

2

E+#',le.

etermine the relative etremum values and the point@s 'here they occurof the function defined by ( ) 1 ##3 −+−+= y x y x y , x f . Also,

determine the saddle points, if there are any.

E+ercises. etermine the relative etremum values and the point@s'here they occur of the %iven functions. Also, determine the saddlepoints, if there are any.

1. ( ) x y xy y y x

y , x f 22323#

#3#

−+−++=

#. ( ) #233 #33 +−−++= y x y y x y , x f

;;;;;;;;;;;;;;;;;

3. etermine the minimum distance bet'een the ori%in and the surface

9## += y x z .

Hint *press the distance of a point on the %iven surface as a function

of x and y , only. Also, the point 'here the minimum distance

occurs coincides 'ith the point 'here the square of the minimum

distance occurs.

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MATH 38

+et # be a closed re%ion in the − xy plane and let f be continuous on # . Then, f has

an absolute maimum=value and an absolute minimum value on # .

S!lvin* f!r #bs!lute e+tre'#:

onsider a function f that is continuous on a closed and bounded re%ion # . An absolute

etremum of f occurs at a relative etremum of f or at a point on the boundary of # .

1. -olve for the critical points of f .

#. etermine function values at critical points of f interior to # .

3. etermine etreme values on f on the boundary of # .

9. The lar%est (smallest! function value from (#! and (3! is the absolute maimum (minimum!

value of f on # .

2.4 L#*r#n*e et)!$ %f!r C!nstr#ine$3O,ti'i5#ti!n&

ptimi>ation problems can be distin%uished into t'o the constrained case and theunconstrained (or free etremum! case. Many real 'orld problems (mostly economics in nature! areeamples of the constrained case such as maimi>in% a manufacturers profit but is constrained byfactors such as the amount ra' materials available. 6n this scenario, the profit is our obBective and theamount of ra' material is the constraint. The unconstrained case 'as already considered in theprevious section 'hen the relative etrema of a %iven function are determined.

T)e!re'.

onsider functions f and % 'ith continuous first partial=derivatives. 6f f has a relative

etremum value at ( )55 y , x , subBect to the constraint ( ) 5=y , x % and

( ) 555 ≠∇ y , x % , then there eists a constant λ (read as 7lambda7, called as the

Lagrange multiplier ! such that

( ) ( )5555 y , x %y , x f ∇⋅λ=∇

15

E+#',le.

etermine the absolute maimum and absolute minimum values of the

function defined by ( ) y x y y , x f 3#3 −+= over the re%ion bounded by

( ) 11 ## =−+ y x .

E+ercise.

etermine the absolute maimum and absolute minimum values of

( ) xy x y , x f += #3 on the re%ion bounded by the parabola # x y =

and the line9

=y .

There are t'o distinct boundaries to be considered for this item.

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MATH 38

Re'#r(: ( ) ( )5555 y , x %y , x f ∇⋅λ=∇ indicates the parallelism of the normal lines to surfaces

defined by f and % at ( )55 y , x .

L#*r#n*e 'et)!$ f!r c!nstr#ine$3!,ti'i5#ti!n:

(T'o variables, one constraint!

r!ble': Maimi>e or minimi>e ( )y , x f z = subBect to the constraint ( ) 5=y , x % .

1. 0orm the auiliary function ( ) ( ) ( )y , x %y , x f ,y , x ! ⋅λ+=λ .

#. -et the first=partial derivatives of ! to >ero.

( ) 5=λ ,y , x ! x ( ) 5=λ ,y , x !y ( ) 5=λλ ,y , x !

3. etermine the critical points of ! by solvin% the system in (#!.

9. Amon% the critical points of ! is@are the ordered pairs that %ive the desired etrema.

11

E+#',le.

1. etermine the absolute etrema of the function

( ) ##9 y xy x y , x f ++= subBect to the constraint that =− y x .

#. Use the +a%ran%e Method to determine the point on the plane:#3

=−+ z y x that is closest to the point ( )3#1 , ,− , and find

the minimum distance.

E+ercises.

1. etermine the relative etrema of ( ) xy y , x f = subBect to the

constraint 1#8

##

=+ y x

.

#. etermine the minimum distance of the surface 9## += y x z from

the ori%in.

3. etermine the dimensions of a ri%ht circular cylinder that 'ill %ive the

maimum volume if re$uired surface area is π#9 .

The surface area of a ri%ht circular cylinder is %iven by rhr π+π ## # .

math 38 unit 2

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