Howdy folks, Alex here! I thought I'd start this blog off right, with one of the most popular (and head-spinning) problems that get thrown at my calculus students. The problem goes something like this:
Find the area of the region enclosed by the lines $y=2x$, $y=3x$, and $y=2$. You must use calculus or you will not receive any credit!
Whoa. Strong words from the guy with the gradebook. Alright, well hopefully you've already seen problems that ask for the area between two functions. If not, break out your textbook ;-) The 2-function area problems are solved by integrating the difference between the "top" and "bottom" functions, like so:
But wait a minute! Our problem is giving us THREE functions, not two. How in the name of Bieber can we apply the above formula to a region enclosed by three functions?
I'll tell you how. We're going to figure out a way to decompose, or break down, this difficult problem into several smaller, easier problems. Learning how to break down problems, by the way, is the reaso...
This part is optional, as students are unlikely to be tested on this material, but if you would like a better understanding of what the natural logarithm means conceptually, you may want to read this section.
Sometimes students will see $ ln(x) $ on a paper, refer to it as "el-en", but not know what it actually means. Perhaps the easiest way to understand it is to know its relation to the number, $ e $. Remember: $ e $ is just a constant, approximately equal to $ 2.71828 $. $ e^x $ and $ ln(x) $ are inverses.
What does this mean? Think about cubing a number and taking the cube root of a number. Imagine that we start out with the number $7$. If we cube it, we get $343$. If we take the cube root of this new number, $343$, we get $7$, which is the number we started with. Similarly, imagine that we start out with the number $125$. If we take the cube root, we get $5$. If we cube this new number, $5$, we get $125$, which is the number we started with...
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...