PRELIMINARY MATHEMATICS

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PRELIMINARY MATHEMATICS

• LECTURE 7
• Revision

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On the pre-session exam

• 10-13.00 Friday 2nd October 2009Room V211
• You will be given two answer books one for math
  and one for statistics.
• The math part of the exam will be 1 1/2 hours,
  followed by another 90mins of statistics
  exam.For the math part, you will be given 5
  questions, out of which you will be required to
  answer 3 questions.
• You will be allowed to use your electronic
  calculator in this examination provided that it
  cannot store text. The make and type of
  calculator must be stated clearly on the front of
  your answer book.

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Equations confirmable for matrix representation
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1. Simultaneous linear equations
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1. Simultaneous linear equations
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Quadratic forms

• For example, given
• For ease of matrix representation, we can rewrite
  this polynomial as

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Quadratic forms

• The symmetric matrix representation can be
  obtained as

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Other functional forms

• Consider

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Other type of functions
Consider We can rewrite this as
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Other functional forms

• And express as

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Other functional forms

• On the other hand,
• would not be suitable for matrix representation
  because we would end up with 4 variables and 3
  equations.

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Other functional forms

• In a similar spirit
• can be transformed to a system of linear equations

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Other functional forms

• and hence be expressed as

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Derivatives, differentials, total differentials,
and second order total differentials
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Derivatives
In a function y f (x) the derivative denotes
the limit of as ? x approaches zero.
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Differentials
The differential of y can be expressed as
which measures the approximate change in y
resulting from a given change in x.
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Partial derivative

• In a function, such as z f (x, y), z is the
  dependent variable and x and y are the
  independent variables, partial derivative is used
  to measure the effect of changes in a single
  independent variable (x or y) on the dependent
  variable (z) in a multivariate function.
• The partial derivatives of z with respect to x
  measures the instantaneous rate of change of z
  with respect to x while y is held constant

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Partial derivative

• In a function, such as z f (x, y), z is the
  dependent variable and x and y are the
  independent variables, partial derivative is used
  to measure the effect of changes in a single
  independent variable (x or y) on the dependent
  variable (z) in a multivariate function.
• The partial derivatives of z with respect to y
  measures the instantaneous rate of change of z
  with respect to y while x is held constant

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Total differentials
For a function of two or more independent
variables, the total differential measures the
change in the dependent variable brought about by
a small change in each of the independent
variables. If z f (x, y), the total
differential dz is expressed as
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Determinantal test
We can also derive the second-order total
differential as
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Positive/ negative definiteness of q d 2 z and
the second order sufficient conditions for
relative extrema
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Positive and negative definiteness

• A quadratic form is said to be
• positive definite if
• positive semi-definite if
• negative semi-definite if
• negative definite if
• regardless of the values of the variables in the
  quadratic form, not all zero.

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Positive and negative definiteness

• A quadratic form is said to be
• positive definite if
• positive semi-definite if
• negative semi-definite if
• negative definite if
• If q changes signs when the variables assume
  different values, q is said to be indefinite

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Second-order sufficient conditions for minimum
and maximum

• If d 2 z gt 0 (positive definite), local minima
• If d 2 z lt 0 (negative definite), local maxima

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Second-order necessary conditions for minimum and
maximum

• If d 2 z 0 (positive semi-definite), local
  minima
• If d 2 z 0 (negative semi-definite), local
  maxima

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Indefinite q d 2 z and saddle point

• When q d 2 z is indefinite, we have a saddle
  point

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Conversion and inversion of log base
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Conversion of log base

• This rule can be generalised as
• where b ? c

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Inversion of log base

• Also,

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Example

• Find the derivative of
• We know that given,

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Example

• Using the rule
• and

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On the sign of lambda in the Lagrangian
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The Lagrange multiplier method
Max/Min z f (x, y) (1) s.t. g (x, y)
c (2) Step 1. Rearrange the constraint in
such a way that the right hand side of the
equation equals a zero. Setting the constraint
equal to zero c g (x, y) 0
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The Lagrange multiplier method
Max/Min z f (x, y) (1) s.t. g (x, y)
c (2) Step 2. Multiply the left hand side of
the new constraint equation by ? (Greek letter
lambda) and add it to the objective function to
form the Lagrangian function Z. Z f (x,
y) ? c g (x, y)
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The Lagrange multiplier method
Z f (x, y) ? c g (x, y) Step 3. The
necessary condition for a stationary value of Z
is obtained by taking the first order partial
derivatives, set them equal to zero, and solve
simultaneously Zx fx ? gx 0 Zy fy ?
gy 0 Z? c g (x, y) 0
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The Lagrange multiplier method
Max/Min z f (x, y) (1) s.t. g (x, y)
c (2) Alternatively if we express the
Lagrangian function Z as. Z f (x, y) ?
c g (x, y)
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The Lagrange multiplier method
Z f (x, y) ? c g (x, y) In Step 3.
The first order necessary conditions are now Zx
fx ? gx 0 Zy fy ? gy 0 Z? c g
(x, y) 0 Is this a problem?
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Example of cost minimizing firm
Min c 8 x2 xy 12y2 s.t. x y 42 Form
the Lagrangian function C. C 8 x2 xy 12y2
? (42 x y)
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Example of cost minimizing firm
C 8 x2 xy 12y2 ? (42 x y) Obtain
the first order partial derivatives, C x 16 x
y ? C y x 24y ? C ? 42 x y
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Solution using matrix

• 16 x y ? 0
• x 24y ? 0
• x y 42
• Since the three first order conditions are linear
  equations, we can use matrix to obtain solutions

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Solution using matrix
Using the Cramers rule
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Solution using matrix
Using the Cramers rule
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Solution using matrix
Using the Cramers rule
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Solution using matrix
Using the Cramers rule
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Solution using matrix
Note that the sign of the determinants all
changed except for Recall from lecture
2 Property of the determinant (3) The
multiplication of any one row (or one column) by
a scalar k will change the value of the
determinant k-fold.
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Solution using matrix
Using the Cramers rule Solution for x and
y are unchanged. ? is the same numerical value
with a negative sign.