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 great illustrations , a fun little story, and lots of calculus.

One of the things I like about this book is that the math is made applicable to the real world through a variety of applications to business, economics, environmental science, and other sciences. This makes it easier to see how calculus benefits people in all sorts of different professions. Also, the math is quite solid, and you could probably learn a lot from this book alone.

I would recommend this book for those people who are interested in Manga, visual learners, and especially younger students learning calculus. It’s fun and easy to read, but there are lots of great calculus lessons inside.

On the other hand, this book will not replace your calculus textbook, and it’s not quite enough to learn calculus on your own. You should think of this book as a supplement to your calculus class, not a replacement.

The Manga Guide to Calculus is a fun read, and you’re sure to learn something inside. If you’re interested, Amazon has a great price. (The links above are my Amazon affiliate links, so if you order it from there I’ll get a few cents. Thanks!)

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 need to know are two rules: first, logs turn multiplication into addition:

.

Second, taking the log of a power turns into multiplication by the exponent:

.

From these, we can deduce a rule for quotients:

Here’s an exercise: how could I have shown the third rule if I only knew the first two?

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 having a limit of or . In that case, the limit exists, and is equal to plus or minus infinity. In the case above, the limit doesn’t even exist.

Here’s another example that sometimes trips students up: what is the following limit?

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:

.

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, so you should think about just buying one.

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 nicer.

The trick is to use integration by parts. Let’s look at it backwards:

If we were really smart or really lucky, we might approach this problem by saying

.

(Is this really true? Use the product rule to show it.) Then we can integrate both sides:

.

Now using the Fundamental theorem of calculus, the integral on the left is just equal to . Solving for , we get

.

Yay!

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 to say that

.

Since , this is in turn equal to

.

Now we can add to get

,

which is pretty much what we wanted. Cool, huh?

It turns out that you can figure out pretty much any of the identities you’re gonna see in calculus or differential equations by just using this technique.

Can you figure out how to show these identities?

• 
• 
• For practice, try to figure this one out without looking up what the right answer is.

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 for historical reasons.

Now what if we start squaring ? What is ? Computing, we get , so
.

Similarly, we can compute that . Check these facts to make sure you understand.

Next time, we’ll use them for something!

In the last post we talked about two important trig identities. I said you should memorize them (and you probably should), but what if you forget? I want to talk about a powerful way to derive trig identities by remembering just a couple of facts. This is great if you’re forgetful like me! To get started, though, we need to look at a wonderful formula discovered by Leonhard Euler. It’s called, appropriately enough, Euler’s Formula:

Here is a thing whose square is negative one: .

If this formula doesn’t seem weird, there’s probably something wrong with you. What the heck does it mean to raise to an imaginary power? That’s bizarre. But it works, as Euler found, and using this formula you can deduce all sorts of interesting things.

One of the most beautiful formulas of mathematics follows from putting in in Euler’s formula. In that case, after adding one to both sides we get

,

a formula that relates five of the most important mathematical constants to one another.

Let’s remember a couple of trigonometric identities that are useful in integrating. How about the first one:

This is useful whenever we need to integrate . Duh. The second, a close cousin, is

.

These are very similar, so you should learn them together. How are the different? (The plus/minus sign.) How can you tell which one is which? (Set and see which is equal to which.)

These two trig identities come up over and over in calculus and differential equations, so do yourself a favor and learn them today.