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MTH374: Algebra

For

 Master of Mathematics

By

Dr. M. Fazeel AnwarAssistant Professor

Department of Mathematics, CIIT Islamabad

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Lecture 03

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Recap

• Introduction• Mathematical model

Example

• A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000?

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Example..

• A chemical manufacturer sells sulphuric acid in bulk at a price of $100 per unit. If the daily total production cost in dollars for units is The total daily production capacity is at most 7000 units. How many units of sulphuric acid must be manufactured and sold daily to maximize profit.

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Today’s Topics

• Optimization problem• Variables and objective functions• Some optimization from calculus

A general optimization problem

An optimization problem can be stated as follows:

Find which minimizes a function

subject to the constraints

for

for

for .

Some notations

• The variable is called a design vector or decision variable.

• The function is called the objective function.• The functions are called the constraints of the problem.• The problem is called a constrained optimization

problem.• If there are no constraints then the problem is called an

unconstrained optimization problem.

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Optimization in one variable

Example (Revisited)

• A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000?

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Some optimization tools from Calculus

Relative Maxima and Minima

1. A function is said to have a relative maximum at if there is an open interval containing on which is the largest value, that is, for all in the interval.

2. A function is said to have a relative minimum at if there is an open interval containing on which is the smallest value, that is, for all in the interval

Note: If has a relative maximum or a relative minimumat , then is said to have a relative extremum at

Relative Maxima and Minima

Example

Critical PointsA critical point for a function is a point in the domain of at which either the graph of has a horizontal tangent line or is not differentiable.

A critical point is called a stationary point of if

Relative extrema and critical pointsSuppose that is a function defined on an open interval containing the point If has a relative extremum at then is a critical point of i.e. either or is not differentiable at

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Example

Find all critical points of

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Example

Find all critical points of

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Example

xxxf 123)( 2

0123 2 xx

Find all the relative extrema of

0)4(3 xx04or 03 xx

4,0x

Relative max. Relative min.

Critical points

(0) 1f 3 2(4) (4) 6(4) 1 31f

Summary

Thank You