Overview
As we continue our studies in Chapter 3, which is focused on how we can use the derivative of a function to tell us key information about the function itself, we are at present emphasizing the concept of optimization. That is, we are interested in determining where the value of a function is greatest or least, and the input value(s) at which such extremes occur. As we move from Section 3.3 into Section 3.4, we start to emphasize problems that occur in a more applied setting, ones that are based in some sort of physical reality. Here, you will be challenged to read carefully and interpret different possible scenarios, identify variables and determine functions, and to use calculus to justify the reasoning behind your ultimate answers.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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(Review) Understand where we look for extreme values of a function on a closed interval, and what we need to do when the interval is open.
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Identify and classify all critical values of a function (within any given domain of interest).
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Through problems such as Example 3.4, recognize how we often need to introduce a function in order to optimize some quantity of interest:
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Identify the variables in the problem as well as the constraints on the variables.
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Identify the “target quantity”, i.e. the quantity to be optimized.
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Set up an equation to relate the target quantity to the other variables.
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Use a constraint in the problem to reduce the number of other variables in the equation to one, and solve for the target quantity to obtain a function of one variable.
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Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- In a wide range of contexts, be able to identify and formulate a function of interest and then use Calculus to accurately justify where the function is optimized on an appropriate domain.
To prepare for class
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Read the introduction to section 3.4, making sure to click on the mentioned examples and reading through their solutions.
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Do the Preview Activity for section 3.4 (on WeBWorK if required by your teacher).
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Read subsection 3.4.1, which lists the different steps for solving an optimization problem.
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Watch the following video, making yourself a list of at least 3 things you learned from this and the following videos or your reading - and include this as part of your feedback:
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Watch the following video where we solve a fencing problem. Note that the optimum is found using a sign chart since we are minimizing over an open interval. (video created at CCSL):
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Watch the following video where we solve a geometry problem. Note that the optimum is found using the closed interval method of section 3.3.
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Do the following optimization problem: Find two positive numbers \(x\) and \(y\) whose product is 10 such that \(2x+y\) is minimal. Make sure that you use a sign chart (of the first or the second derivative of the target function) or the closed-in interval method, whichever is appropriate, in order to justify that you have reached a minimum. In other words, it is not sufficient to make the derivative equal to 0 to justify that a minimum value has been found. You should find \(x=\sqrt{5}\) and \(y=2\sqrt{5}\).
After class
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Watch the following video which shows how optimization can work when there are more than 1 variable (video created at CCSL):
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Do some experimentation with the following interactive applets (by Marc Renault) showing important examples of applied optimization problems (some of which might match WeBWorK problems you have been assigned!):
