Objectives
- Model real world questions as an optimization problem about a function \(f\text{.}\)
- Use calculus techniques to fully analyze a modeled optimization problem and provide an answer in the form of a complete sentence.
Section 26 Applied optimization
Example 26.1. Box of maximum volume.
A square piece of cardboard of dimension \(2\) meters is made into a box by cutting squares of equal dimension out of each corner and folding up the sides. (See Figure 26.2.) Find the dimension of the cutout squares that maximizes the volume of the resulting box.
Solution.
Procedure 26.3. Applied optimization.
The following steps are useful for modeling and solving a word problem involving optimization.
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Model the problem.
- Make a detailed, labeled diagram summarizing the situation described.
- Clearly define and name all important quantities in the problem.
- Write down any important equations or formulas involving the quantities at play.
- Identify the quantity \(q\) that is to be optimized.
- Look for language suggesting maximum or minimum values: e.g., “largest”, “most”, “greatest”, “smallest”, “least”, etc..
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Solve optimization problem.
- Express the quantity we wish to optimize as a function \(q=f(x)\) of exactly one variable.
- If the quantity appears to be given as a function of more than one independent variable, look for a constraint equation that allows us to reduce to exactly one independent variable.
- Make explicit what the domain \(D\) of this function is, using the context of the problem.
- Translate the given word problem as one of our types of optimization problems (e.g., EVT, find/classify critical points, curve sketching, etc.) for the function \(q=f(x)\text{.}\)
- Use an appropriate procedure (e.g., EVT procedure, find/classify critical points, curve sketching, etc.) to solve the given optimization problem.
- The domain \(D\) of \(f\) plays an important role in this step. In particular, pay attention to whether or not \(D\) is a closed finite interval.
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Summarize.
- Communicate the answer you derived in Step 2 in a plain English sentence that makes use of the language/context of the stated problem.
- Make sure that you do indeed answer the problem posed and include units details if applicable.
Example 26.4. Minimal fence perimeter.
Farmer Dudley is building a rectangular pen for his iguanas. He will use the 50 meter long side of his barn as one side of the pen, and will construct fencing for the remaining three sides of the pen. The pen must have a total area of 200 m\(^2\text{.}\) What is the minimum length of fencing Dudley must build to create an iguana pen matching these specifications.
Solution.
It is often the case that the domain \(D\) of the function we wish to optimize is not of the simple form \([a,b]\text{,}\) where Procedure 20.13 applies. The following generalization comes in handy in such situations.
Procedure 26.5. Generalized extreme value theorem.
Let \(I\) be an interval of any sort (i.e., \(I\) can be open, closed, or neither, and \(I\) can be finite or infinite), and let \(a\) and \(b\) be the (possibly infinite) left and right endpoints of \(I\text{.}\) Assume \(f\) is continuous on \(I\text{.}\) To investigate whether \(f\) attains a maximum or minimum value on \(I\text{,}\) proceed as follows.
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Identify candidate inputs.The candidate inputs where \(f\) potentially attains a local maximum or minimum value consist of the (i) any finite endpoints of \(I\) that are elements of \(I\text{,}\) and (ii) any critical points of \(f\) lying in \(I\text{.}\)
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Evaluate \(f\).Evaluate \(f\) at all candidates you found in Step 1. Let \(m\) be the minimum of these values, and let \(M\) be the maximum of these values.
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Evaluate endpoint limits.Evaluate the two endpoint limits \(\lim\limits_{x\to a^+}f(x)\) and \(\lim\limits_{x\to b^-}f(x)\text{.}\) (In the case where the given endpoint is infinite, the one-sided limit is understood simply as the corresponding limit at infinity.)
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Compare \(m\text{,}\) \(M\text{,}\) and endpoint limits.Assume that each limit in Step 3 either exists or is infinite.
- If neither of the endpoint limits from Step 3 is less than \(m\) or equal to \(-\infty\text{,}\) then \(m\) is the absolute minimum value of \(f\) on \(I\text{.}\) Otherwise, there is no absolute minimum value.
- If neither of the endpoint limits from Step 3 is greater than \(M\) or equal to \(\infty\text{,}\) then \(M\) is the absolute maximum value of \(f\) on \(I\text{.}\) Otherwise, there is no absolute maximum value.
Example 26.6. Closest point on parabola.
Find the \(Q\) on the parabola \(y=1-x^2\) whose distance to \(P=(-3,1)\) is the smallest possible. The distance between two points \(P_1=(x_1,y_1)\) and \(P_2=(x_2,y_2)\) in \(\R^2\) is defined as
\begin{equation*}
d(P_1,P_2)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\text{.}
\end{equation*}
