What Optimization Problems Are Really Asking
Optimization problems ask you to find the largest or smallest possible value of some quantity. In AP Calculus, that quantity might be area, volume, distance, cost, revenue, or another measurable quantity.
The calculus itself is usually not the hardest part. The challenging step is translating the words of the problem into a mathematical function that you can differentiate.
For an area or volume problem, the basic strategy is:
- Identify what quantity you are trying to maximize or minimize.
- Write a formula for that quantity.
- Use the given information to express everything in terms of one variable.
- Find the critical points by setting the derivative equal to zero or identifying where it is undefined.
- Determine which critical point gives the maximum or minimum.
- Answer the question in the context of the original problem.
The Central Idea
Suppose the quantity you want to maximize is represented by . A maximum can occur at a critical point, where the derivative is zero or does not exist, or at an endpoint of the domain.
Geometrically, a maximum occurs when the graph of the objective function changes from increasing to decreasing. At a smooth interior maximum, the tangent line is horizontal.
The most important practical rule is this: do not differentiate until you have reduced the problem to one variable.
A Step-by-Step Optimization Method
Step 1: Define the variables
Give names to the unknown dimensions. For example, if a rectangle has length and width, let and .
Step 2: Identify the objective
Ask: What am I trying to maximize or minimize? That quantity becomes your objective function.
For a rectangle, area is . For a rectangular box, volume is .
Step 3: Find the constraint
The problem normally gives you some restriction connecting the variables. This is the constraint equation.
Step 4: Rewrite the objective using one variable
Use the constraint to solve for one variable and substitute it into the objective function.
Step 5: Differentiate
Find the derivative of the one-variable objective function.
Step 6: Find critical points
Solve and check any points where the derivative does not exist.
Step 7: Verify the maximum
A critical point is not automatically a maximum. Use the first derivative test, second derivative test, or compare the critical point with the endpoints of the domain.
Step 8: State the answer with units
If the question asks for dimensions, give dimensions. If it asks for maximum area, give square units. If it asks for maximum volume, give cubic units.
Worked Examples
A rectangle has a perimeter of 40 units. What dimensions maximize its area?
A farmer has 100 meters of fencing to enclose a rectangular field along a straight river. No fencing is needed along the river. What dimensions maximize the area?
Squares of side length are cut from each corner of a 20-by-12-inch sheet of cardboard. The sides are folded upward to form an open-top box. What value of maximizes the volume?
A cylinder has a fixed volume of . Find the relationship between its radius and height that minimizes its surface area.
How to Know You Found a Maximum
Finding only tells you that you found a critical point. You still need to establish whether that point produces a maximum.
One useful method is the first derivative test. If the derivative changes from positive to negative, the function changes from increasing to decreasing, so the critical point is a local maximum.
Another method is the second derivative test. If at a critical point, the graph is concave down there, indicating a local maximum.
For many AP Calculus optimization problems, you can also use the endpoints of the feasible interval. If the objective function has only one interior critical point and the endpoint values are smaller, you have established the absolute maximum.
Common Mistakes
An optimization problem may begin with two or more variables, but your objective function should normally be written in terms of one variable before you differentiate. Use the constraint first.
Read the geometry carefully. If a rectangle is next to a river, for example, the river may replace one side of the fencing. If a box is open on top, there is no top face in the surface-area constraint.
Algebra can produce critical points that make no physical sense. A length cannot be negative, and a cutout in a box cannot be larger than half of the corresponding sheet dimension.
A critical point could be a minimum or neither. Use the first derivative test, second derivative test, or compare the endpoints.
If the question asks for the dimensions, do not stop after finding the maximum area or volume. If it asks for the maximum area, report the area rather than only the dimensions.
Practice Problems
A rectangle has a perimeter of 60 units. What dimensions maximize its area?
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A farmer has 120 meters of fencing to enclose a rectangular field along a river. No fencing is required along the river. What dimensions maximize the area?
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A 30-by-18-inch sheet of cardboard is used to make an open-top box by cutting equal squares from each corner. What side length of the squares maximizes the volume?
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A rectangle is inscribed under the curve and above the x-axis, with its base on the x-axis. Find the dimensions that maximize its area.
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