Real problems involving optimization in one variable are not unheard of, but they are atypical. Careful thought should be given to the ways in which the settings and techniques being taught are misleading.
With respect to modeling, insofar as an optimization problem comes up in one or two variables, an appropriate solution technique is graphical: Look at the graph and pick the optimum.
With respect to mathematics, we should use techniques that apply in multiple variables, with the one-variable setting being used to sharpen insight.
What are the important concepts and techniques that justify teaching about optimization in one variable?
You have a function \( f(x)\) given as an algebraic formula. Find the maximum. You're not allowed to graph the formula; just consider the derivative.
You have a function \( f(x)\) that you want to maximize. You don't have a formula for \( f(x)\), but you can evaluate it.
Soln: Graph the function and pull out the \( x\) that produces the maximum output.
There is a function \( f(x)\) that you want to maximize. You can evaluate \( f(x)\) at any point you choose; it costs $100,000.00 per evaluation.
What \( x\) will you choose? How will you know when you are done?
Insofar as we want to teach about algorithms, we should give students an easy way to generate objects on which to apply and test their algorithms. Random, smooth functions are nice for this.
As a classroom activity, construct a smooth function and let them guess:
f = rfun(~x, seed = 20120710) f(0)
##  -4.082
##  -4.563
Suppose that you can compute the first and second derivatives:
df = D(f(x) ~ x) ddf = D(f(x) ~ x & x)
Describe the search method you would use. How would you know when to stop searching?
What are the units of \( f'(x)/f'‘(x)\)? How is this quantity potentially relevant to a search?
Often, models are used in teaching optimization in order to provide an objective function, the surface area of a tin can, the best direction to swim across a river or cross a snowy field, etc.
There are much more fundamental modeling concepts that relate to optimization. 1. The idea of an objective function, and of “policy variables”. 2. The idea of multiple incommensurate objectives. The mathematical problem of defining “best” in such a setting. 3. Attempts to deal with incommensurability by arbitrary conversion, e.g., “the value of a life.” 4. The relationship between constraints and multiple objectives.
In dealing with multiple objectives, it's important to remember that there can be no best solution and therefore automatic solution techniques are not key. Instead, the purpose of a model is often to allow people to explore alternatives and the consequences of those alternatives.
You have a policy variable \( x\) and two objective functions \( f(x)\) and \( g(x)\). Find the best value of \( x\), the one that maximizes \( f(x)\) and \( g(x)\).
Set a constraint on one of the functions, for instance, \( g(x) \geq 20\). Find the best value of \( f(x)\) subject to this constraint.
Now vary the level of the constraint by a little and see how much the optimal \( f(x)\) can improve. Was it worth it?
“A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.” Stewart, Calculus: Concepts and Contexts 2/e, p. 309
We're going to sell oil in a can.
Prices of some containers … specialtybottle.com
Area = makeFun(2 * pi * r^2 + 2 * pi * r * h ~ r & h) Volume = makeFun(2 * pi * r^2 * h ~ r & h) plotFun(Area(r = r, h = h) ~ r & h, r.lim = c(3, 7), h.lim = c(5, 15))
plotFun(Volume(r = r, h = h) ~ r & h, add = TRUE, col = "red", filled = FALSE)
## 2-D adding temporarily discontinued.
What's the configuration that minimizes the area given a volume of 1000 cm3.
An olive-oil can will be made in the shape shown in the picture.
Notice that picture comes from an advertisement for a "2 pack.” There's a reason why they are selling two 1 L can instead of a single 2 L can.
For ergonomic reasons, the depth of the can should be 2.5 inches and the radius of the bend should be no less than 0.75 inches. Retail store shelves restrict the height to 12 inches. The shipping boxes must be 20 x 18 x 18 inches. Oil in contact with air will go rancid, so we want to make the top surface of the oil as small as possible. …
Construct a function giving the price of a can as a function of the policy variables. Then let the designers discuss what sorts of shapes are worthwhile. You will be able to tell them how much any given change will cost.
Math 135 Assignment
Model effort against velocity, for instance effort goes as the square of velocity. Energy is effort time time. Current is carrying you downstream. Is there an optimal effort to minimize total energy?
Problem 42 on pp. 316-7 of Stewart
Extract out the modeling issues in this problem from Stewart.
Why do we consider just the resistance along the path ABC. Don't we want the total resistance? Are there flow constraints? Why is the radius of the smaller vessel two-thirds that of the larger?