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).

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

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.

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

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

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

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

Solution (PDF) Pages 9 to 11

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

Solution (PDF) Page 4

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

Solution (PDF) Page 2

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

Solution (PDF)

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.

Solution (PDF - 4.0 MB) Pages 23 to 27

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.

Solution (PDF)^# Page 28

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

Solution (PDF)^# Page 1

Minimizing the material required to make a popcorn container.

Solution (PDF)^# Page 1

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

Solution (PDF)^# Page 1

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

Solution (PDF)^# Page 1