Constrained Optimization Example
A company produces the same product at three di↵erent factories. Denote
these quantities (in kg ) by x, y , z. The cost of producing each is
C1 (x) = x 2 + 4x + 5,
C2 (y ) = y 2 ,
C3 (z) = (z + 3)2
1.
What quantity should they produce at each factory in order to produce 28 kg
of the product at minimal cost?
28 x y
z
:
28
-
xtytz
Constant
cost :
C,
t
Cz
C,
t
=
=
C (x
=
,
y
,
z
[y ]
'd
[444×+5]
=
-
-
+
+
[
( zest
-
I
]
)
X2t4xt5tyZtf@g_x-y1t3TZ_1-7CCx.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=74-4×+961
2×2-58×+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×-1×2-43174
Constrained Optimization Example
A farmer has 4000 ft of fencing and wants to fence o↵ a rectangular field that
borders a straight river. He needs no fence along the river. What are the
dimensions of the field 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 more written homework.
I Homework 11 is only for practice.
I Final Exam: Friday, May 15th
I It consists of everything we learned up till (including)
Optimization of Multivariable Functions.
I Last Quiz this Friday
jealousy
I Partial Derivatives, Linearization, Di↵erential, Elasticity,
Min/Max, Critical 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 domain
I Exponential functions, linear and quadratic functions, polynomials,
rational functions
I Cost, revenue, profit; Supply and Demand
Important Topics (everything!)
Part I: The Basics
1. Functions
I Finding domain
I Exponential functions, linear and quadratic functions, polynomials,
rational functions
I Cost, revenue, profit; Supply and Demand
2. Limits
Direct substitution property; limit laws; divide & multiply by conjugate
3. Continuity
Definition of continuity; intermediate value theorem
4. Derivative
I Definition of the derivative in terms of limit
I Interpretations (rate of change; slope; marginal ...)
I Basic di↵erentiation rules
I Product, quotient, chain rules
5. Higher-order derivatives and convexity
Part II: More Fun with Derivatives
1. Derivatives of exponential and logarithmic functions
2. Implicit di↵erentiation
3. Logarithmic di↵erentiation
4. Derivatives of inverse functions:
(f
1
(x))0 =
f 0 (f
1
1 (x))
5. Linear approximations; Di↵erentials
6. Elasticity
7. Single-variable optimization
I Local/Absolute max/min; Critical number; first-derivative test;
second-derivative test; Extreme value theorem; Closed Interval
Method.
I Applications to Revenue and Cost.
I Constrained optimization; Method of substitution
Part III: Functions of two (or more) variables
1. Domain of f (x, y ), graph of f (x, y ), cross-sections of f (x, y ).
2. Contour (level curve) diagrams of f (x, y )
Interpreting contour diagrams and table of values
3. Partial Derivatives; Higher-order partial derivatives
I Limit definition; Interpretation
I Computing partial derivatives using di↵erentiation rules
I Estimating partial derivatives from contour diagrams/tables of
values
4. Partial elasticity
5. Linearization, Di↵erentials
6. Optimization
I Definition of absolute max/min, local max/min
I Finding critical points
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 following is the domain of the function
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 continuous, one-to-one function,
and f (4) = 3, f (3) = 5, f 0 (3) = 4, f 0 (4) = 0.5.
Let g (x) denote the inverse of f (x). Which of the following is the equation for
the line 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 following shaded region is the domain of which one of the functions below?
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 following is the partial derivative of f (x, y ) = xe y with respect 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 table below to estimate the partial 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 find the partial derivative of
f (x, y ) = (y 2 + 1)xy with respect 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 function f (x) whose domain is ( 1, 1) and whose first
derivative is graphed below.
y
a
b
c
d
e
f
y = f 0 (x)
x
Which one of the following is true?
A. f (x) has a local minimum at x = b.
B. The critical numbers of f (x) are x = a, c, f (and nothing else)
C. f (x) has local minima at x = c and x = f
D. On the interval (c, d), f (x) is concave downward (concave)
E. None of the above
Back to board
Optimization 200
A rectangular storage container with an open top is to have a
volume of 10 m3 . The length of its base is twice the width.
Material for the base costs $10 per square meter. Material for the
sides costs $5 per square meter. Find the cost of materials for the
cheapest such container.
Optimization 300
Find all critical points 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) only
C. (0, 1), (1, 0), (1/3, 1/3), and (0, 0) only
O
Back to board
x
(1-2×-4)=0
or
-_
(
D. (1/3, 1/3) and (0, 0) only
y
=
A. (0, 0) only
E. None of the above
g-Ogre
'
D=
-21 Zytl ) tf
x
I
-2ft )tl= }
x=
0,0 )
,
(
'
so)
°
(
( t
'
,
,
t)
)
Applications 100
The demand 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 price of potatoes and m is mean income. Find the elasticity of
demand with respect to price and the elasticity of demand with respect to
income.
Back to board
Applications 200
Consider the Cobb-Douglas production function f (x, y ) = Ax 2/3 y 1/3 . Find the
linear approximation of f (x, y ) at the point (1, 8).
Back to board
Applications 300
Use linearization to approximate the value of f (1.02, 1.09) where
f (x, y ) = 3x 2 + xy
Back to board
y 2.
Miscellaneous 100
True or false?
The limit
lim
h!0
p
9+h
h
does not exist.
A.
B.
C.
D.
True, and I know exactly why
True, but I’m not completely sure
False, and I know exactly why
False, but I’m not completely sure
Back to board
3
Miscellaneous 200
True or false?
The equation
32x + x = 1
has
A.
B.
C.
D.
a solution in the interval [ 2, 2].
True, and I know exactly why
True, but I’m not completely sure
False, and I know exactly why
False, but I’m not completely sure
Back to board
Miscellaneous 300
True or false?
¥; ,fcH=¥
The function
f (x) =
✓
(
'
=4
,
1×32+44=4
x2 4
x 2
x
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