SOLUTION: NYU Multivariable Functions & Constrained Optimization Example Worksheet
Constrained Optimization Example
A troop effects the similar fruit at three di↵erent factories. Denote
these quantities (in kg ) by x, y , z. The consume of submissive each is
C1 (x) = x 2 + 4x + 5,
C2 (y ) = y 2 ,
C3 (z) = (z + 3)2
1.
What amount should they effect at each factory in ordain to effect 28 kg
of the fruit at minimal consume?
28 x y
z
:
28

xtytz
Constant
consume :
C,
t
Cz
C,
t
=
=
C (x
=
,
y
,
z
[y ]
'd
[444×+5]
=


+
+
[
( zest

I
]
)
X2t4xt5tyZtf@g_xy1t3TZ_17CCx.y
ccx y ,
Gpiohd
28
,
)
:
Cx
Cy
=
=
2x
2g

x

t
)
=
X2t4×ty2+
=
c
y
4
2.
2/31
+
(
31

x

x
y)

C
( 31

y) C D=


t
)
=

x

62

y)
58
+
2+4
+
dy
4x
t
t
2g
2x
=
o
→
*:÷÷÷f"
4×+29=58
"

(
(9,11 )
=
92+419 )
lit 131
Look at
boundary
Check
+
2x
y

9 a)


o
"
2+4
xey=28
ytz


28
is
'
=
y=zs
z
28


x
y

:i%i
363
Fadden
=
12=28
=o
"
constraint

xty
"
get to:
28
Iµ
xh

28
2=28 "
at
I
¥j¥
µ¥→×
→
xez
log! ]
test
y
"
×
,
Y
•
2
x
and
x=o
CCO
,
y)
02+4

fly )=4y
ft 't )
f (
)
o
few)
=
=
=

o
②
28
gey EE
=
fly )=2y2

EYE
0
①
62g

2.
co>
965
797
2
(
tyZ
)
31

o

4=0 01×128
?
g) 2+4 =2y
9615 62g

+965
y
62
(E)
28
y=zsµg¥g×l¥8
xey=
=o
y③y=o2!
=
¥= Iz
6431) +965=484.5
=
=
Cco ,
C
(
o
o
,
,
28)
28,0)
=
c
(o
,
15.5, 12.5 )
② 4=0
and
CCX , o )
gcx)


X
=
28
E
ex
o
74×+02+131
g
14.5)
( o)
)
gL28
x

032+4=744×+961
2×258×+965
X
84×1=4×58
g(

=
=
=
544.5
965
909
=
=
.
=
c.
C
c
(
(
(
548=14.5
_
14.5,
o
o
,
25,
o
zsg
,
o
13.5)
,
,
o
)

62×1×24
③
((
28
y=
x , 28


and
X
) =x44x
X
+
E
o
( 28×12
=
hcx )= 2×2

52
X
t
X
E
28
+
( 31

x

784
2144×1

797
13h43
(
h' 1×1=4×52
)
h ( 07
h
( 28)
=
=
=
459
X=
=
797
=
909
=
.
(
C
C
=
15
13
(
o
)
o
,
,
,
28,0)
( 28 ,
O
,
O
)
[ 28×3774
56×1×243174
Constrained Optimization Example
A fellowsubject has 4000 ft of sheet and wants to stave o↵ a across ground that
borders a undeviating large stream. He needs no stave parallel the large stream. What are the
dimensions of the ground that has the largest area?




4000
{
=
2x
+
T
A
=
x.
y
y
y

=
A' txt
"
A G)
=
4 ooo
4

4×
Lo
x
=
[ 4000
Ly ooo x
critical
concave
4000
#
X=
down


2x

2X
]
2×2
1000
everywhere
.
Abs.maxareaBwhenX=coooardy=2oo+
Announcements
I No further written homework.
I Homework 11 is merely for performance.
I Final Exam: Friday, May 15th
I It consists of everything we perceiveing up tend (including)
Optimization of Multivariable Functions.
I Last Quiz this Friday
jealousy
I Particular Derivatives, Linearization, Di↵erential, Elasticity,
Min/Max, Discriminating Points, Second Derivative Test
I Recitation Handout 13
I WebAssign
WA16 due May 11
Important Topics (everything!)
Important Topics (everything!)
Part I: The Basics
1. Functions
I Finding inclosure
I Exponential offices, straight and quadratic offices, polynomials,
rational offices
I Cost, retaliate, profit; Supply and Demand
Important Topics (everything!)
Part I: The Basics
1. Functions
I Finding inclosure
I Exponential offices, straight and quadratic offices, polynomials,
rational offices
I Cost, retaliate, profit; Supply and Demand
2. Limits
Direct supply property; name laws; divorce & expand by conjugate
3. Continuity
Definition of continuity; included compute theorem
4. Derivative
I Definition of the derivative in stipulations of name
I Interpretations (trounce of change; slope; final ...)
I Basic di↵erentiation rules
I Product, quotient, security rules
5. Higherordain derivatives and convexity
Part II: Further Fun delay Derivatives
1. Derivatives of exponential and logarithmic offices
2. Implicit di↵erentiation
3. Logarithmic di↵erentiation
4. Derivatives of inverse offices:
(f
1
(x))0 =
f 0 (f
1
1 (x))
5. Straight adits; Di↵erentials
6. Elasticity
7. Singlevariable optimization
I Local/Absolute max/min; Discriminating number; firstderivative test;
secondderivative test; Extreme compute theorem; Closed Interval
Method.
I Applications to Retaliate and Cost.
I Constrained optimization; Method of supply
Part III: Functions of two (or further) variables
1. Inclosure of f (x, y ), graph of f (x, y ), crosssections of f (x, y ).
2. Outcontinuity (flatten deflexion) diagrams of f (x, y )
Interpreting outcontinuity diagrams and consideration of computes
3. Particular Derivatives; Higherordain particular derivatives
I Name definition; Interpretation
I Computing particular derivatives using di↵erentiation rules
I Estimating particular derivatives from outcontinuity diagrams/tables of
values
4. Particular elasticity
5. Linearization, Di↵erentials
6. Optimization
I Definition of absolute max/min, topical max/min
I Finding discriminating aims
I Second derivative test
MFE1 Jeopardy! (Final Exam Review Edition)
Functions
Derivatives
Optimization
Applications
Miscellaneous
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
Functions 100
Which of the subjoined is the inclosure of the office
p
4 x2
f (x) =
?
ln(x + 1)
A. ( 1, 2]
B. [ 2, 2]
C. (0, 2)
D. ( 1, 2)
E. None of the above
Back to board
Functions 200
Suppose that f (x) is a natural, onetoone office,
and f (4) = 3, f (3) = 5, f 0 (3) = 4, f 0 (4) = 0.5.
Let g (x) personate the inverse of f (x). Which of the subjoined is the equation for
the continuity that is tangent to the graph of g (x) at x = 3?
A. y =
0.25x + 4.75
B. y =
4x + 17
C. y =
2x + 10
D. y =
4x + 10
E. None of the above
Back to board
Functions 300
The subjoined hidden country is the inclosure of which one of the offices underneath?
2
1
2
1
1
2
1
2
p
p
4
x2
B. f (x, y ) = ln(x 2 + y 2
4)
C. f (x, y ) = ln(x 2 + y 2
p
D. f (x, y ) = x 2 + y 2
4)
A. f (x, y ) =
y+
E. None of the above
Back to board
4
y2
ln(x 2 )
Derivatives 100
Which of the subjoined is the particular derivative of f (x, y ) = xe y delay regard to
x?
A. fx (x, y ) = lim
h!0
B. fx (x, y ) = lim
he y
h
he y
xe y
h
(x + h)e y
C. fx (x, y ) = lim
h!0
h
h!0
(x + h)e y +h
h
E. None of the above
D. fx (x, y ) = lim
h!0
Back to board
xe y
xe y
Derivatives 200
Use the consideration underneath to estimate the particular derivative of fx (0, 1).
y=
x=
1
x =3
9
5
9
E. None of the above
Back to board
4
11
C. fx (0, 1) ⇡
1 y =1
2
6
3
D. fx (0, 1) ⇡ 0
3
x =0
A. fx (0, 1) ⇡
B. fx (0, 1) ⇡ 4
1
2 y=
9
Derivatives 300
Use logarithmic di↵erentiation to meet the particular derivative of
f (x, y ) = (y 2 + 1)xy delay regard to y .
A. fy (x, y ) = 2xy 2 (y 2 + 1)xy 1
⇣ 2
⌘
2
B. fy (x, y ) = (y 2 + 1)xy y2xy
2 +1 + x ln(y + 1)
C. fy (x, y ) =
D. fy (x, y ) =
2xy 2
+ x ln(y 2 + 1)
y 2 +1
(y 2 + 1)xy ln(y 2 + 1)
E. None of the above
Back to board
Optimization 100
Consider the office f (x) whose inclosure is ( 1, 1) and whose first
derivative is graphed underneath.
y
a
b
c
d
e
f
y = f 0 (x)
x
Which one of the subjoined is gentleman?
A. f (x) has a topical insufficiency at x = b.
B. The discriminating aggregate of f (x) are x = a, c, f (and dot else)
C. f (x) has topical minima at x = c and x = f
D. On the spacebetween (c, d), f (x) is excavated downward (concave)
E. None of the above
Back to board
Optimization 200
A across storage container delay an perceiven top is to bear a
volume of 10 m3 . The diffusiveness of its sordid is twice the width.
Material for the sordid consumes $10 per balance meter. Material for the
sides consumes $5 per balance meter. Meet the consume of materials for the
cheapest such container.
Optimization 300
Find all discriminating aims of f (x, y ) = xy (1
↳
§
y
=
y

x

=
( tx y)

(
i

x

g)
+
t
xy
C c)
xyc



y ).
=
=
3y
y

X
zxy

4g
=

I

XZ


y
Zxy
=
G

x

2g)
so
y= 2×+1
X=o
4=2*1
I
y
B. (0, 1), (1, 0), and (0, 0) merely
C. (0, 1), (1, 0), (1/3, 1/3), and (0, 0) merely
O
Back to board
x
(12×4)=0
or
_
(
D. (1/3, 1/3) and (0, 0) merely
y
=
A. (0, 0) merely
E. None of the above
gOgre
'
D=
21 Zytl ) tf
x
I
2ft )tl= }
x=
0,0 )
,
(
'
so)
°
(
( t
'
,
,
t)
)
Applications 100
The insist for potatoes in the United States from 1927 to 1941 was
estimated to be
q(p, m) = Ap 0.28 m0.34 ,
where p is appraisement of potatoes and m is average allowance. Meet the elasticity of
insist delay regard to appraisement and the elasticity of insist delay regard to
income.
Back to board
Applications 200
Consider the CobbDouglas fruition office f (x, y ) = Ax 2/3 y 1/3 . Meet the
straight adit of f (x, y ) at the aim (1, 8).
Back to board
Applications 300
Use straightization to approach the compute of f (1.02, 1.09) where
f (x, y ) = 3x 2 + xy
Back to board
y 2.
Miscellaneous 100
True or bogus?
The name
lim
h!0
p
9+h
h
does not await.
A.
B.
C.
D.
True, and I perceive precisely why
True, but I’m not thoroughly sure
False, and I perceive precisely why
False, but I’m not thoroughly sure
Back to board
3
Miscellaneous 200
True or bogus?
The equation
32x + x = 1
has
A.
B.
C.
D.
a key in the spacebetween [ 2, 2].
True, and I perceive precisely why
True, but I’m not thoroughly sure
False, and I perceive precisely why
False, but I’m not thoroughly sure
Back to board
Miscellaneous 300
True or bogus?
¥; ,fcH=¥
The office
f (x) =
✓
(
'
=4
,
1×32+44=4
x2 4
x 2
x
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