Optimization word problems use calculus to find the best possible value of a quantity, such as the largest area, smallest cost, or shortest time. They matter because many real decisions involve limits, like fixed material, fixed distance, or a budget. The main skill is translating words into a mathematical model with variables, equations, and a target quantity.
Once the model is built, derivatives help locate where the maximum or minimum occurs.
A typical problem has an objective function, which is the quantity to optimize, and one or more constraints, which describe the limits in the situation. The constraint is used to rewrite the objective function in terms of one variable, so ordinary single-variable calculus can be applied. Critical points are found by setting the derivative equal to zero, then checking endpoints or using a second derivative or sign chart.
The final answer must be interpreted in the original context with correct units.
Key Facts
- Objective function: the formula for the quantity being maximized or minimized.
- Constraint equation: a relationship that limits the variables, such as 2x + 2y = P.
- Use the constraint to write the objective as one variable: A(x) = x(P/2 - x).
- Critical points occur where f'(x) = 0 or where f'(x) is undefined.
- Second derivative test: if f''(c) > 0, f(c) is a local minimum; if f''(c) < 0, f(c) is a local maximum.
- Always check the domain and endpoints because absolute maxima and minima can occur at boundaries.
Vocabulary
- Optimization
- Optimization is the process of finding the maximum or minimum value of a quantity under given conditions.
- Objective function
- An objective function is the equation that represents the quantity you want to maximize or minimize.
- Constraint
- A constraint is an equation or inequality that describes a limit or required relationship in the problem.
- Critical point
- A critical point is an input where the derivative is zero or undefined and a maximum or minimum may occur.
- Domain
- The domain is the set of input values that make sense for the model and the real-world situation.
Common Mistakes to Avoid
- Optimizing the wrong quantity: students sometimes differentiate the constraint instead of the objective function. The derivative must be taken of the quantity being maximized or minimized.
- Keeping too many variables in the objective function: students may write A = xy but never use the constraint to eliminate one variable. For single-variable calculus, the objective should usually be rewritten in terms of one variable.
- Ignoring the realistic domain: students may allow negative lengths, times, or costs. The domain must match the physical meaning of the problem.
- Forgetting to check endpoints: students often stop after solving f'(x) = 0. Absolute maxima and minima on a closed interval can occur at critical points or endpoints.
Practice Questions
- 1 A farmer has 100 m of fencing to make a rectangular pen along a straight river, so only three sides need fencing. Let x be the side perpendicular to the river. Find the dimensions that maximize the area.
- 2 A closed rectangular box has a square base and volume 32 cubic centimeters. If the base side length is x and the height is h, find the dimensions that minimize the surface area.
- 3 A student says the largest area rectangle with perimeter 40 cm occurs when one side is 0 cm and the other is 20 cm because that uses all the perimeter. Explain why this reasoning is incorrect and describe how calculus identifies the true maximum.