Optimization: Absolute & Relative Extrema

 

Applied Maximum/Minimum Problems

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).

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Another Applied Max/Min Problem

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

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Review Problems

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.

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Quadratic Behavior at Critical Points

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

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General Conditions for Maximum or Minimum

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

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Finding One Dimensional Extrema

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

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Maximization

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

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Optimization

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

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

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

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

Minimizing the material required to make a popcorn container.

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Max/Min Problems

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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.

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.

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Optimization: Triangular Fence

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

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Optimization: Area of a Rectangle

Finding the largest possible area of a rectangle with two corners that lie on a given parabola.

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