Lectures 11 and 12 Critical Points Local (or Relative) Extrema Global (or Absolute) Extrema Calculus II Topic 1: Introduction to Optimization Calculus II
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2D Example
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2D Example
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Optimization Problems Notation: đ?’™ = đ?‘Ľ1 , ‌ , đ?‘Ľđ?‘› , đ?’š = đ?‘Ś1 , ‌ , đ?‘Śđ?‘› ∈ đ??ˇ ⊂ â„?đ?‘› , đ??š đ?’™ = đ??š đ?‘Ľ1 , ‌ , đ?‘Ľđ?‘› Problem (P) Find đ?’™ = đ?‘Ľ1 , ‌ , đ?‘Ľđ?‘› ∈ đ??ˇ to be solution of : đ??š đ?’™ ≤đ??š đ?’š
∀đ?’š = đ?‘Ś1 , ‌ , đ?‘Śđ?‘› ∈ đ??ˇ (đ?‘€đ?‘–đ?‘›đ?‘–đ?‘šđ?‘–đ?‘§đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› đ?‘ƒđ?‘&#x;đ?‘œđ?‘?đ?‘™đ?‘’đ?‘š) or
đ??š đ?’™ ≼đ??š đ?’š
∀đ?’š = đ?‘Ś1 , ‌ , đ?‘Śđ?‘› ∈ đ??ˇ (đ?‘€đ?‘Žđ?‘Ľđ?‘–đ?‘šđ?‘–đ?‘§đ?‘Žđ?‘Ąđ?‘–đ?‘œđ?‘› đ?‘ƒđ?‘&#x;đ?‘œđ?‘?đ?‘™đ?‘’đ?‘š)
Remarks: 1. đ??š Objective Function 2. đ??ˇ Set of Constraints a) If đ??ˇ ≥ â„?đ?‘› , (P) is said to be a Unconstrained Optimization Problem b) If đ??ˇ ≢ â„?đ?‘› , (P) is said to be a Constrained Optimization Problem Calculus II
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Local or Relative Extrema
Calculus II
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Local or Relative Extrema
Calculus II
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2D Example
Calculus II
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Critical Points
Calculus II
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Example of Critical Points
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Saddle Points
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Saddle Point Example
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Second Derivative Test
Calculus II
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Second Derivative Test in the Special Case of a Two Variables Function
Calculus II
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Local Minimum Example
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Saddle Point Example
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Saddle Points Examples
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Different Critical Points for the same Function
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Finding Local Extreme Values
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Finding Local Extrema
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Summary of Test for Critical Points
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Second Derivative Test does not always work
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Global or Absolute Extrema
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Global or Absolute Extrema
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Minimum Distance Problems Find the points on the surface đ??š(đ?‘Ľ, đ?‘Ś) = 1/đ?‘Ľđ?‘Ś that are closest to the origin
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Constrained Optimization
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Constrained Optimization Finding Maximum Volume of a Case
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Constrained Optimization Example
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Lecture Review (I)
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Lecture Review (II)
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Lesson Review (III)
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Lecture Review (and IV)
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Homework
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