Calculus 1, Math120
Mar
10,
2016

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...

Calculus 2, Math141
Apr
19,
2015

Calculus 2, Math141
Apr
19,
2015

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:

- There is a BIG difference between finite and infinite series. Look at the problem carefully before you decide how to approach it.
- Many tests are inconclusive about whether an infinite series converges/diverges. In that case, you must try another test.
- Just because we can prove that an infinite series converges, doesn't mea...

Calculus 3, Math241
Mar
29,
2015

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...

Calculus 2, Math141
Mar
27,
2015

If you're reading this post, you've probably arrived at the point in your calculus course where they try to teach you how to do integration by partial fraction expansion. And if you're anything like I was when I first learned calculus, you're probably scratching your head and going **whadafuhhhh**? **What is the point of all this**?

Here's what you need to know. *Partial fraction expansion is not an integration technique*. It's an algebraic technique. That being said, it's useful for making certain algebraic expressions (i.e. rational expressions) easier to integrate by breaking them into smaller, simpler chunks. Sound familiar? Its that same mode of thinking that all your other calculus classes are pushing at - taking a difficult problem, decomposing it into several easier problems, and then putting the pieces back together.

Got it? Let's get down to business. Here's our problem:

Rewrite the following expression as the sum of three partial fraction terms: $$\frac{1}{(x-1)(x^2+x+1)}$$

Just to quickly review the st...

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