Optimization Flashcard

Front

What is the formula for the volume of a cylinder?

Back

V = πr²h, where V is volume, r is radius, and h is height.

Front

What is the relationship between the radius and height of a cylinder when minimizing surface area for a fixed volume?

Back

For a fixed volume, the surface area is minimized when the height equals twice the radius.

Front

What is a Norman window?

Back

A Norman window is a combination of a rectangle and a semicircle on top, often used in optimization problems to maximize area.

Front

What is the formula for the area of a rectangle?

Back

A = length × width.

Front

What is the formula for the area of a semicircle?

Back

A = (1/2)πr², where r is the radius.

Front

What is the perimeter of a rectangle?

Back

P = 2(length + width).

Front

How do you find the maximum area of a rectangle given a fixed perimeter?

Back

The maximum area occurs when the rectangle is a square.

Front

What is the profit function in terms of price and quantity?

Back

Profit = Revenue - Cost, where Revenue = price × quantity and Cost includes fixed and variable costs.

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex represents the maximum or minimum point of the parabola, which is crucial for optimization problems.

Front

What is the first derivative test in optimization?

Back

The first derivative test helps determine whether a critical point is a local maximum, local minimum, or neither.