Category Archives: calculus
Absolute Maxima And Minima Problems
Finding the absolute maxima and minima of functions (the biggest and smallest values a function attains) is important for many real world calculus applications. For example, companies want to maximize their profits, engineers want to minimize the drag on an airplane, computer scientists want to get your youtube video from the server to your computer … Continue reading
What is a Bounded Function?
Often in calculus we want to use properties of functions in order to learn more about a specific problem. Functions can have lots of weird behavior, so it’s nice when we come across a function that has properties we can use to reason about it. One great type of function is a bounded function. Here … Continue reading
The Manga Guide to Calculus
A few weeks ago, I received a copy of The Manga Guide to Calculus. This is a translation of a Japanese book, published in the US by No Starch Press. They’ve translated quite a few Manga Guides (for example, to Physics, Statistics, Biology, and more) and this book is quite a good read. It’s got … Continue reading
Properties of Log
There are some properties of logarithms that you absolutely must know to get through calculus. Most students shriek or faint when they see a logarithm, but if you learn these simple rules they become much easier to work with. Get a little practice, and you’ll be on your way to mastering logarithms! All you really … Continue reading
Limits That Don’t Exist
Some functions don’t have a limit as the value of their parameter approaches . Here’s one example: just doesn’t exist. Why is that? Because as increases, keeps oscillating between 1 and -1. It never approaches any one value, and so we can’t assign a limit to it. Notice that this is different from a function … Continue reading
Integration by Parts
When we looked at integrating the natural logarithm, we used a little trick: we took the product rule , integrated it to get , and made a clever choice of and to find the integral we wanted. This is called integration by parts, and it’s usually written like this: .
Textbook Rental
There’s a new service for students to rent textbooks for the semester: chegg.com. If you’re not planning on using your books later, this might be a good option for you. Check it out! I should mention that if you’re in a science or engineering field, you will need your calculus book again at some point, … Continue reading
Integrating the Natural Logarithm
You’ll probably see this integral someday: It looks so simple, and you think, “gee, I probably was supposed to memorize that” or “oh I can do that, it looks so easy.” But then you don’t remember the antiderivative and get stuck. Cause, heck, what can you do with just anyway? It doesn’t decompose into anything … Continue reading
Remembering One Trig Identity
We’ve been talking about a way to figure out trig identities without having to just memorize them. Here is how I showed the identity , by just knowing Euler’s formula and some facts on complex numbers. I started off by using Euler’s formula to write . Then I used the fact from complex numbers that … Continue reading
Some Facts About Complex Numbers
Let’s look at some properties of complex numbers, numbers of the form where and are real numbers and . We’ll use these together with Euler’s formula to deduce some trig identities. First, if , I’m going to call the real part of and the imaginary part. We write and ; the big goofy symbols are … Continue reading