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For example, in Example \(\Page Index\), we are interested in maximizing the area of a rectangular garden.Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger.
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.
first have to develop the function you’re going to maximize or minimize, as we did in Stage I above.
Having done that, the remaining steps are exactly the same as they are for the max/min problems you recently learned how to solve.
Hence, you already know how to do all of the following steps; the only new part to maximization problems is what we did in Stage I above. We want to minimize the function $$ A(r) = 2\pi r^2 \frac$$ and so of course we must take the derivative, and then find the critical points. \[ \begin A'(r) &= \dfrac\left(2\pi r^2 \frac \right) \\[8px] &= \dfrac\left(2\pi r^2 \right) \dfrac\left(\frac \right) \\[8px] &= 2\pi \dfrac\left(r^2 \right) 2V \dfrac\left(r^ \right) \\[8px] &= 2\pi(2r) 2V \left((-1)r^ \right)\\[8px] &= 4 \pi r\, – \frac \end \]The critical points occur when $A'(r) = 0$: \[ \begin A'(r) = 0 &= 4 \pi r\, – \frac \\[8px] \frac &= 4 \pi r \\[8px] \frac &= r^3 \\[8px] r^3 &= \frac \\[8px] r &= \sqrt \end \] We thus have only one critical point to examine, at $r = \sqrt\,.$Step 5.
Next we must justify that the critical point we’ve found represents a minimum for the can’s surface area (as opposed to a maximum, or a saddle point).
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learning of Optimization: “College Student Difficulties with Applied Optimization Problems in Introductory Calculus,” unpublished masters thesis, The University of Maine, 2013.]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.
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).