Transcript lecture 1 - FMT-HANU

LECTURE 10 &11
Functions of Several Variables
Objectives
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



Functions of Several Variables
Partial Derivatives
Second Partial Derivatives
Optimization
Functions of Several Variables
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
A function f(x,y) of two variables x and y is a rule that
assigns a number to each pair of values for the
variables
 f(x,y)
= ex(x2+2y)
 f(3,5)=
 f(5,3)=
 f(x,y,z)
= 5xy2z
 f(1,3,5)=
 f(1,5,3)=
Functions of Several Variables
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

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Example: A store sells butter at $2.50 per kg and
margarine at $1.40 per kg.
The revenue from the sale of x kg of butter and y kg
of margarine is given by the function
f(x,y) = 2.5x + 1.4y;
Determine and interpret f(200,300).
Domain
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
Z= x2+y2
𝑥
 Z=
𝑦

Z= 1 − (x2+y2)
Graph – 3D co-ordinate system.
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Production Functions in Economics
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
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Cost of manufacturing process can be classified into
two types: Labor (L) and Capital (K).
A manufacturer normally has control over the relative
portion s of labor and capital.
 Maximize
labor usage, reduce capital
 Minimize labor, need more capital.


f(L,K) : the number of units produced using L labor and
K capital  f(L,K) = CLAK(1-A)
Such function is called Cobb-Douglas production
function
Example
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
Suppose that during a certain time period the number
of units of goods produced when utilizing x units of
labor and y units of capital is f(x,y) = 60x3/4y1/4.
 How
many units of goods will be produced by using 81
units of labor and 16 units of capital
 Show that whenever the amounts of labor and capital
being used are doubled, so is the production. (Economists
say that the production function has “constant returns to
scale”).
Solution
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Level Curve
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Example
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Partial Derivatives
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
𝑧 = 𝑓 𝑥, 𝑦
𝜕𝑧
𝑓 𝑥+∆𝑥,𝑦 −𝑓 𝑥,𝑦

= lim
𝜕𝑥 ∆𝑥→0
∆𝑥
𝜕𝑧 𝜕𝑓

, , 𝑧 x, 𝑓 x
𝜕𝑥 𝜕𝑥
𝜕𝑧
𝑓 𝑥,𝑦+∆𝑦 −𝑓 𝑥,𝑦

=𝑓 = lim
𝜕𝑦 𝑦 ∆𝑦→0
∆𝑦
𝜕𝑧 𝜕𝑓

, ,
𝜕𝑦 𝜕𝑦
𝑧y, 𝑓y
Partial Derivatives
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


The derivative of f(x,y) with respect to x is the
derivative of f(x,y) where y is treated as a constant
and f(x,y) is considered as a function of x alone.
Similar definition for partial derivative of f(x,y) with
respect to y.
Partial Derivative as Rate of Change.
Example
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
𝑓 𝑥, 𝑦 = 2𝑥 + 3𝑦
 𝑓𝑥 =
 𝑓𝑦=

𝑓 𝑥, 𝑦 = 2𝑥2 + 3𝑥𝑦 + 𝑦
 𝑓𝑥 =
 𝑓𝑦=

𝑓 𝑥, 𝑦 = 3𝑥𝑒𝑦
 𝑓𝑥 =
 𝑓𝑦=
Example
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Production function f(x,y) = 60x3/4y1/4.
Find partial derivatives with respect to x and y.
Evaluate them at x = 81, y = 16
Interpret the numbers computed above.
 Marginal
productivity of labor
 Marginal productivity of capital
Solution
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Partial Derivatives
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
Let f(x,y) be a function of two variables. Then if h and
k are small we have
f
( a, b).h
x
f
f ( a, b  k )  f ( a, b) 
( a, b).k
y
f ( a  h, b)  f ( a, b) 
Example
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
A farmer can produce f(x,y) = 200 (6x2+y2)1/2
units of products by utilizing x units of labor and y
units of capital. (The capital is sued to rent or
purchase land, materials and equipment).
 Calculate
the marginal productivities of labor and capital
when x = 10 and y = 5.
 Let h be a small number. Determine the approximate
effect on production of changing labor from 10 to 10 + h
units while keeping capital fixed at 5 units.
 Estimate the change in production when labor decreases
from 10 to 9.5 units and capital stays fixed at 5 units.
Solution
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Second Derivative
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

The partial derivative of fx with respect to x is
the second derivative of f(x,y).
We have similar definitions (and their notations) for
2 f
2 f
2 f
2 f
,
,
,
.
2
2
x
y
xy yx

Remark: Almost all functions f(x,y) encountered in
applications have the
property
2
2
 f
 f

xy yx
Example
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𝑓 𝑥, 𝑦 = 3𝑥𝑒𝑦
𝑓𝑥=
𝑓𝑦=
𝑓𝑥𝑥=
𝑓𝑥𝑦=
𝑓𝑦𝑥=
𝑓𝑦y=
Extreme Points
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
If f(x,y) has either a local maximum or minimum at
(x,y) = (a,b) then
f
(a, b)  0 and
x
f
( a, b)  0
y
Example 1
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f(x,y) =x2+y2
Example 2
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
f(x,y) =1-(x2+y2)
Example 3
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
f(x,y) =x2-y2
Second Derivative Test
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Suppose f(x,y) is a function and (a,b) is a point at which
and let
f
f
( a, b )  0 &
(a, b)  0,
x
y
 f  f  2 f 

D( x, y )  2 * 2  
x
y
 xy 
2
2
2
Second Derivative Test
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If D(a,b) > 0 and second derivative of x > 0 then
f(x,y) has local minimum at (a,b)
If D(a,b) > 0 and second derivative of x < 0 then
f(x,y) has local maximum at (a,b)
If D(a,b) < 0 then f(x,y) is not local at (a,b)
If D(a,b) = 0 then no conclusion can be drawn.
Second Derivative Test
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
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fx(a,b)=0
D>0
A<0
fy(a,b)=0
A=fxx(a,b)
D>0 A>0
B=fxy(a,b)
C=fyy(a,b) D<0
D=0
Example 1
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f(x,y) =x2+y2
Example 2
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f(x,y) =1-(x2+y2)
Example 3
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f(x,y) =x2-y2
Example 4
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f(x,y) = x3 – 3xy + 0.5y2 + 8
Example 5
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A monopolist manufactures and sells two competing
products, call then I and II, that cost $30 and $20 per
unit, respectively, to produce. The revenue from
marketing x units of product I and y units of product II
is 98x + 112y – 0.04xy – 0.1x2 – 0.2y2. Find the
values of x and y that maximize the monopolist’s
profits.
Solution
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Example 6
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(Price Discrimination): A monopolist markets his product
into two countries and can charge different amounts in
each country. Let x be the number of units to be sold
in the first country and y be the number of units to be
sold in the second country. Due to the laws of demand,
the monopolist must set the price at 97 – (x/10)
dollars in the first country and 83 – (y/20) dollars in
the second country in order to sell al the units. The cost
of producing these units is 20,000 + 3(x+y). Find the
values of x and y that maximize the profit.
Solution
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Constrained Optimization
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

Problem: Let f(x,y) and g(x,y) be functions of two
variables. Find values of x and y that maximize (or
minimize) the objective function f(x,y) and that also
satisfy the constraint equation g(x,y) = 0.
E.g.1: Minimize 42x + 28y, subject to the constraint
600 – xy = 0, x,y>0
Constrained Optimization
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

We can solve the equation g(x,y) = 0 for one variable
in terms of the other and substitute the resulting
expression into f(x,y)
If we cannot solve the equation g(x,y) = 0 for one
variable in terms of the other.
Lagrange: Invent “Lagrange multipliers”
Lagrange multipliers
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

Basic idea: Replace f(x,y) by an auxiliary function of
three variables F(x,y,λ) defined as: F(x,y,λ) = f(x,y) +
λg(x,y).
Theorem: Suppose that, subject to the constraint g(x,y)
= 0, the function f(x,y) has a relative maximum or
minimum at (x,y) = (a,b). Then there is a value of λ,
say λ = c, such that the partial derivatives of F(x,y, λ)
all equal to zero at (x,y, λ) = (a,b,c).
Example 1
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
Using Lagrange multipliers, minimize 42x + 28y,
subject to the constraint 600 – xy = 0, where x and y
are restricted to positive values.
Example 2
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Suppose that x units of labor and y units of capital
can produce f(x,y) = 60x3/4y1/4 units of a certain
product. Also suppose that each unit of labor costs
$100, whereas each unit of capital costs $200.
Assume that $30,000 is available to spend on
production. How many units of labor and how many
units of capital should be utilized in order to maximize
production.
Solution
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Example
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Lagrange multiplier λ can be interpreted as the
marginal productivity of money. That is, if one
additional dollar is available, then approximately λ
units of the product can be produced.
Example
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Economics law: If labor and capital are at their
optimal levels, then the ratio of their marginal
productivities equals the ratio of their unit costs.
Example 3
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Use Lagrange multipliers to find the three positive
numbers whose sum is 15 and whose product is as
large as possible.
Utility Maximization Problem
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

A consumer makes consumption decision based on his
preferences (utility function) given his limited income.
Let
 Good
1 consumption: x
 Good 2 consumption: y
 Price for good 1: px
 Price for good 2: py
 Income level: M

Total expenditure: xpx + ypy
Utility maximization problem
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

Total expenditure: xpx + ypy has to be within income
 xpx + ypy ≤ M. If the consumer spends all his
income in good 1 and good 2 then xpx + ypy = M.
This is the budget constraint.
Regions:
 Inside
budget line: feasible.
 Outside budget line: infeasible.

Problem: The consumer maximizes u = U(x,y) subject to
the budget constraint xpx + ypy = M.
Utility Maximization Problem
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
Example: Consider an agent who consumes Good 1
and Good 2. We denote the amount of those goods
by x and y, respectively. The utility function of this
agent is given by
U
= U(x,y) = x1/2y1/2
This agent has income of 12 and does not save. If the
prices of Good 1 and Good 2 are px = 2 and py = 1,
respectively, what is the amount of goods this agent
chooses to consume?
Conclusion
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Functions of Several Variables
Partial Derivatives
Second Partial Derivatives
Optimization