Key takeaway: Calculus optimization is the use of derivatives to find the input that makes a quantity as large or as small as possible while obeying a stated constraint.
Grade 12 – AP Calculus ABAP Calculus AB Unit 5Calculus
Definition of Optimization
Calculus optimization answers questions that ask for the most, the least, the cheapest or the largest. The quantity to improve is written as a formula; a constraint ties the variables together, so one variable can be eliminated.
Once the quantity depends on a single variable over a sensible domain, the extreme value theorem and the derivative tests locate the best input. The final answer must come back to the words of the problem, with units.
Formula and notation
The objective function is the quantity to improve, such as area \(A=xy\). A constraint, such as \(2x+4y=540\), lets you rewrite it as \(A(y)\). Then solve
\[ A^{\prime}(y)=0 \]
and compare the critical value with the endpoints of the domain.
Key facts
- Draw a picture and name every variable before writing any formula.
- State the domain from the context; lengths and counts cannot be negative.
- Confirm the answer is a maximum or minimum, using a sign check or the second derivative.
- Give the answer in the original units and answer the question that was asked.
Optimization example
A rancher has 540 feet of fencing for a rectangular field split into three equal pens by two interior fences parallel to the short sides. What dimensions give the greatest total area?
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Let the field be \(x\) feet wide and \(y\) feet tall. The fence used is \(2x+4y=540\), so \(x=270-2y\).
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Area: \(A(y)=y(270-2y)=270y-2y^2\), with \(0\lt y\lt 135\).
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Differentiate: \(A^{\prime}(y)=270-4y=0\), so \(y=67.5\), and then \(x=270-135=135\).
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Check: \(A^{\prime\prime}(y)=-4\) is negative, so this is a maximum. \(A=135\cdot 67.5=9{,}112.5\).
The field should be 135 feet by 67.5 feet, enclosing a maximum area of 9,112.5 square feet.
Common mistake
Watch out
A student solves \(A^{\prime}(y)=0\), gets \(y=67.5\), and reports \(67.5\) square feet as the largest area.
Why it is wrong
The number 67.5 is a length, the best value of \(y\), not the area. Calculus optimization asks for the optimal input first; the requested quantity comes from substituting it back into the area formula. Here that gives 9,112.5 square feet.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus