In this activity, we will look more closely at what we’ve already learned about how calculus can reveal much about the behavior of a function, whether it is represented with a formula or graphically. In probably the most famous application of derivatives, we will reveal to a waiting world a function’s greatest and lowest moments of its life, whether the function likes it or not.

We will be working with functions that are represented by its formula only. You may use the graph of a function to “check”, but a graphing calculator’s ability to always pinpoint a peak or valley of a function is not always foolproof.

In this activity, you will develop a process for identifying -values (if any) for which a given function has local maximum and minimal -values (i.e., peaks and valleys). You’ve already done most of the conceptual groundwork; it’s just a matter of putting together a step-by-step recipe for success!

What must happen when a function has a peak (local max) or valley (local min)? That is, what should we look for to identify -values for which a function is to even have a chance to have a peak or valley? This was something discovered in the first or second week of the course.
Assuming the criteria in the previous part has occurred at a value of , how can we tell whether or not (without looking at the graph of ) is a max, min, or neither? Hint: What must the function (and its derivative) be doing around (just before and after) the suspected -value if it gives a max? If it gives a min? If it gives neither? The answer to this question is the basis for what is called the “First Derivative Test”.
If the previous part’s result is too cumbersome to algebraically work out, is there another, easier way to find if our suspected -value give a max, min, or neither for ? In a previous homework, you dealt with a “Second Derivative Test”, in which you looked at the concavity around a suspected max or min. For example, if is our suspected , then if is positive, what does that say about at (and why?)? If is negative, what does that say about at (and why?)? If is zero, what does that say about at (and why? Be careful on this one)?
Write down a step-by-step process for finding value(s) of suspected ’s and determining if it gives a local max value of , a local min value of , or neither. Your process should start with this “precalculus” step: FIND A FORMULA FOR THE THING WE WANT TO MINIMIZE OR MAXIMIZE. This often is the hardest part before we let “calculus do the driving” on the formula to find the max/mins (e.g., recall the container on the final question in Conjunction Junction.)

The Special Case When We Only Care About Part of a Function

In several real-life situations, we only care about our beloved function for certain values of (usually a closed interval). Most of the time, the function in question will only make sense in the context of the situation at hand for only those values of (e.g., other values might make a physical measurement such as area a negative number). Make sure to ask if this is the case whenever you approach an optimization problem (or any problem!).

What should we do in these cases if we only have a formula? To develop a plan, take a look at these “partial graphs” and identify where the largest and smallest values of take place in the interval (these will be called absolute maxes and mins for that interval). After you do this, take the process you developed above for general functions and adapt it to this case.

 
 
 

Now practice your newly-developed processes on the following problems:

Leo Duh Vinci wants to build a rectangular piece of Florida land comprising square miles for his snowman collection. The piece of land will lie along a river, on which side there is no need for fencing (the snowmen are not afraid of alligators). What is the least amount of fencing he will need to buy to enclose the land? Do the same if he had square miles.
Elpo, a small dog food manufacturer makes pounds of dog food per year. Analysts say that the last couple years’ figures suggest that it costs the company dollars to produce pounds of dog food and that the company was charging dollars per pound of dog food. Assuming this is to hold true for this year, how many pounds of dog food should be manufactured to maximize the company’s profits?
A by inch rectangular piece of cardboard is to be used to construct an open-topped box. The box is to be constructed by cutting out a square (of the same size) from each corner of the rectangle and folding it up. What should be the size of a square to make the box of maximum volume? What will that volume be?
What are the dimensions of an aluminum can that holds cubic inches of juice and that uses the least amount of aluminum? What is that amount of aluminum? Assume that the can is cylindrical and is capped on both ends.
You have a piece of wire of length inches from which you construct a square and/or a circle. How should you do this so that you have the largest possible total area inside the figures? Do the same if you had a piece of wire of length .
2024-10-10 13:51:33