# Optimization: Absolute & Relative Extrema

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## Lecture Notes

#### Applied Maximum/Minimum Problems

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Section 2, Page 1 to page 3

Maximization and minimization problems are worked through step-by-step. Maximizing the area enclosed by a given fence length, and minimizing the travel time of a swimmer who has to get to a point on the shore (relates to Snell's law).

Instructor: Prof. Jason Starr
Prior Knowledge: First Derivative Test, Critical and Extremal Points (lecture 9)

#### Another Applied Max/Min Problem

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Section 3, Page 4 to page 5

Max/Min problem of maximizing area enclosed by a trapezoid inscribed in a semicircle.

Instructor: Prof. Jason Starr
Prior Knowledge: Max/Min Problems (section 2 of lecture 10)

#### Review Problems

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Section 3, Page 3 to page 4

Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.

Instructor: Prof. Jason Starr
Prior Knowledge: Tangent Lines (section 1 of lecture 2), Max/Min Problems (section 2 of lecture 10), Volume of Solids of Revolution (section 3 of lecture 19), Inverse Substitution (section 3 of lecture 25), Integration by Parts (section 1 of lecture 27)

## Online Textbook Chapters

#### Quadratic Behavior at Critical Points(18.013A, Spring 2005)

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Definition of a critical point and its use in finding maxima and minima of a function.

Instructor: Prof. Daniel J. Kleitman

#### General Conditions for Maximum or Minimum(18.013A, Spring 2005)

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Finding the extremal values of a function, including distinction between local and global maxima and minima.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None

#### Finding One Dimensional Extrema(18.013A, Spring 2005)

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Iterative divide and conquer method for finding a local maximum or minimum on a curve.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Conditions for Maximum or Minimum (OT14.1)

## Practice Problem

#### Maximization

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Problem 2 (page 2)

An optimization problem involving two fixed rays and a segment that is allowed to slide between them.

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### Optimization

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Problem 3 (page 6)

Finding the maximum volume of a box made from two square sheets of metal.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Optimization

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Problem 5 (page 1)

Finding the maximum volume for a trash can made from a cylinder and a hemisphere.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Optimization

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Problem 3.1 (page 2) to problem 3.2 (page 2)

Two problems which involve minimizing the cost of a sculpture and maximizing the area enclosed by a fence.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Max/Min Problems

PDF - 2.2 MB
Problem 2C-1 (page 13) to problem 2C-15 (page 15)

Fifteen optimization questions drawn from various applications including largest volume of a box, shortest length of fence for a barnyard, and the optimal fare for an airline.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### More Max/Min Problems

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Problem 2D-1 (page 15) to problem 2D-7 (page 16)

Seven optimization questions which include finding the optimum attack angle for a plane and the best moment to add milk to a cup of coffee to keep it hot.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Optimization

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Problem 3 (page 1) to problem 4 (page 1)

Two questions which involve minimizing the area of a triangle and minimizing the length of wire needed to brace the legs of a table.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Optimization

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Problem 2 (page 1)

Minimizing the material required to make a popcorn container.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Optimization: Triangular Fence

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Problem 3 (page 1)

Finding the maximum area of a triangular enclosure formed from two sides of fence and a wall for the third side.

Instructor: Prof. David Jerison
Prior Knowledge: None