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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

Calculus II

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2D Example

Calculus II

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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

Calculus II

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Saddle Points

Calculus II

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Saddle Point Example

Calculus II

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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

Calculus II

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Saddle Point Example

Calculus II

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Saddle Points Examples

Calculus II

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Different Critical Points for the same Function

Calculus II

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Finding Local Extreme Values

Calculus II

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Finding Local Extrema

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Summary of Test for Critical Points

Calculus II

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Second Derivative Test does not always work

Calculus II

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Global or Absolute Extrema

Calculus II

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Global or Absolute Extrema

Calculus II

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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

Calculus II

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Constrained Optimization

Calculus II

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Constrained Optimization Finding Maximum Volume of a Case

Calculus II

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Constrained Optimization Example

Calculus II

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Lecture Review (I)

Calculus II

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Lecture Review (II)

Calculus II

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Lesson Review (III)

Calculus II

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Lecture Review (and IV)

Calculus II

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Homework

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2.1_2.2_Optimizacion_y_Puntos_Criticos  

Calculus II Topic 1: Introduction to Optimization Calculus II Santiago de Vicente 1 Santiago de Vicente 2 Calculus II Santiago de Vicente 3...

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