Optimization, or finding the maximums or minimums of a function, is one of the first applications of the derivative you'll learn in college calculus.
In this video, we'll go over an example where we find the dimensions of a corral (animal pen) that maximizes its area, subject to a constraint on its perimeter. Other types of optimization problems that commonly come up in calculus are:
This video goes through the essential steps of identifying constrained optimization problems, setting up the equations, and using calculus to solve for the optimum points.
You're in charge of designing a custom fish tank. The tank needs to have a square bottom and an open top. You want to maximize the volume of the tank, but you...
A lot of the "word problems" that come up in calculus seem silly and contrived, because they are. The inventory cost problem, however, is something that comes up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back.
Hot Bod Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. They have an exclusive deal with Gallmart to supply the retail giant with 10,000 units over the next several years. The hot tub shells are made using injection-molding, in which molten plastic is squirted into metal molds at high pressure, and then allowed to cool. Once the shell has cooled, assembly workers finish the product by attaching the hoses and motors and installing insulation.
Being a small company, Hot Bod doesn't have their own factory - they will have to rent space from the Berry Plastics Corporation. Berry Plastics char...
The chapter on series is notorious for making calculus students want to hurl their textbooks into a fire, curl up into a fetal position, and sob uncontrollably the night before the exam (I've been there myself).
Honestly, its not that series are so difficult as much as it requires you to remember a lot of rules, how to apply them, and most importantly, in what order to apply them. And most calculus courses just don't do all that great a job of explaining this - they just hurl a bunch of theorems at you, a few tests, and tell you to sort it all out yourself. To help with this, I've created a flowchart that explains how to approach solving a mathematical series. The important things to keep in mind are:
Tonight, on the World's Most Extreme Values. One 2-variable function. One closed region. One shot at glory. Don't miss it!
...sorry, had to get that out of my system. The problem we're going to look at today goes like this:
Find the absolute minimum(s) and maximum(s) of the function $f(x,y)=xe^y-x^2-e^y$ on the rectangle with vertices $(0,0)$, $(0,1)$, $(2,0)$, and $(2,1)$.
Ok, we've seen extreme value (i.e., maximum and minimum) problems like this in Calculus 1. If you don't remember the gist of this, please go back and check your notes/textbook first. Just to review, the basic idea is that we find the derivative of a function, set it equal to zero, and solve the resulting equation. Together with the points where the the function is non-differentiable, these solutions give us a set of critical points where the function might have a maximum, minimum, or inflection point.
Our example has two new issues we must confront. First of all, we have a function of two variables, so what does it mean to "set the de...