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Typical phrases that indicate an Optimization problem include: Before you can look for that max/min value, you first have to develop the function that you’re going to optimize.There are thus two distinct Stages to completely solve these problems—something most students don’t initially realize [Ref]. Now maximize or minimize the function you just developed.Above, for instance, our equation for $A_\text$ has two variables, We can now make this substitution $h = \dfrac$ into the equation we developed earlier for the can’s total area: \[ \begin A_\text &= 2\pi r^2 2 \pi r h \[8px] &= 2\pi r^2 2 \pi r \left( \frac\right) \[8px] &= 2\pi r^2 2 \cancel \cancel \left(\frac\right) \[8px] &= 2\pi r^2 \frac \end \]We’re done with Step 3: we now have the function in terms of a single variable, , and we’ve dropped the subscript “total” from $A_\text$ since we no longer need it.
Your first job is to develop a function that represents the quantity you want to optimize. Campbell for his specific research into students’ learning of Optimization: “College Student Difficulties with Applied Optimization Problems in Introductory Calculus,” unpublished masters thesis, The University of Maine, 2013.] access to step-by-step solutions to most textbook problems, probably including yours; (2) answers from a math expert about specific questions you have; AND (3) 30 minutes of free online tutoring.
One common application of calculus is calculating the minimum or maximum value of a function.
We’ve labeled the can’s height Having drawn the picture, the next step is to write an equation for the quantity we want to optimize.
Most frequently you’ll use your everyday knowledge of geometry for this step.
Let’s break ’em down and develop a strategy that you can use to solve them routinely for yourself.
Optimization problems will always ask you to maximize or minimize some quantity, having described the situation using words (instead of immediately giving you a function to max/minimize).
In this problem, for instance, we want to minimize the cost of constructing the can, which means we want to use .
So let’s write an equation for that total surface area:\begin A_\text &= A_\text A_\text A_\text \[8px] &= \pi r^2 2\pi r h \pi r^2 \[8px] &= 2\pi r^2 2 \pi r h \end That’s it; you’re done with Step 2!
Notice, by the way, that so far in our solution we haven’t used any Calculus at all.
That will always be the case when you solve an Optimization problem: you don’t use Calculus until you come to Stage II.