Calculus for Smart Kids who are bored in Jordan's 5th grade Math Class

So, you want to learn to read "mathonese", eh? Do you have what it takes. Can you stare at the scribbles for hours, days, maybe years on end and ponder their secrets. Are you that CURIOUS?

"Welcome to my parlor, said the spider to the fly."

Chapter 1:

"I’ve Reached My Limit!"

Or

“How to Make your own rules, impress your teachers and meet cute girls.”


Pick a number, any number. Try again, but make it smaller, anything but zero. I said smaller, not vanishing altogether. Do it again. Again, Again. Keep going! Why'd ya stop? Are we there yet?

I told you that ε ("epsilon") is a Greek letter that represents a very peculiar number, 2.171228… . Well, Greek is a very versatile language. We sometimes use the same Greek letters to mean other things that have nothing to do with each other. Today, we’ll be generous and let you make ε any number you like. Hey, Perfesser Wizard is easy! Chill baby. In fact, ε has another Greek friend, ‘Δ’ ("delta", which means "different"). You can make ‘Δ’ be any number you like too. How about that!

Well, there might be SOME limits. HAR! HAR! That’s a pun! (This story is about “limits”, get it?)

You see you can either pick ε or Δ, but not both at the same time. Whatever you pick for ε will decide what Δ has to be. Or, if you prefer, whatever you pick for Δ will decide what ε has to be. Hey! I didn’t make the rules. It’s how stuff works in math. The Sun comes up. The Sun goes down. You can pick either ε or Δ. Live with it!

Let’s think about a typical expression in algebra. Say,

y = x + 1 .

We could compute a whole list of results, i.e, values of y for this little snippet of addition. Pick a value for x and the result is one more than that value. So, let’s say that we start with x equal to 100 and we calculate x+1, then make x = 99, then x=98, and on and on. The sum of x+1 keeps getting smaller every time we change x. How small will it get? What’s your limit?

What is the “limit” of the expression y = x+1 as we make x smaller and smaller?

If x is 100, then y becomes 101.
If x is 99, then y becomes 100.
Every time we make x a little “smaller”, or closer to zero, y follows.

Another way we can say that we are making x smaller is to ask how different it is from zero. That’s where ε comes in handy. We use ε to represent the difference between the value of x and zero. Sometimes, if we like, we can use ε to represent the difference between any two numbers we choose. Greek is versatile. Perfesser Wizard is easy! Life is good!

How small is ε? Not much, but it’s not 0!

ε = x – 0 (We can compare x to other numbers too. But this example happens to use 0.)

ε’s Greek pal Δ is sort of a copycat. He represents how close the sum of x+1 is to some number, we'll call it y, while we keep making ε smaller and smaller. How close do you want it to get? Pick a number? If you want the sum of x+1 to be Δ = ½ different than what it will be if x were 0, then you have to make ε be small enough to make that happen. Or said another way, the difference between x and the value that makes x+1 as close as Δ to the “limit” is ε.

As x – 0 gets smaller ( ε gets smaller), the difference between y and( x+1) gets smaller, i.e. Δ gets smaller.

y + Δ = (x + 1) + ε

As you make ε smaller, the difference between y and (x+1), which we call Δ, gets smaller. Or we can say,

(x + 1) becomes Δ close to y, as x becomes ε close to 0.

The fancy, schmancy, braniac way to write this so you look like you work for NASA (or came from Mars) is:

y = lim (x+1)
ε = (x-0)-> 0

"The limiting value of the sum (x+1) as ε get very small, i.e., very close to 0, is equal to a number represented by y.)

ε-> 0 is how we write “as the value of ε gets close t 0.” How close? Why ε close, of course!

How close does (x+1) get to y? Or asked another way, “what is the limit of (x+1) as x gets close to the value 0?

Well, (x+1) get Δ close to y. It will never get smaller than y because that is it’s “limit” as ε approaches 0.

But y = x +1 is kind of a lame example. Let’s try something tough like:

y = x² / x



What do you suppose the value of y will become if we make x be equal to 0? Hmmm! I think I heard somewhere that we’re not allowed to divide by zero. In fact, dividing by zero is not even defined in the official rules of Intergalactic Math. (Okay, I made some of that up. But you can’t divide by 0, okay?)

What to do? WHAT TO DO?

Well, let’s take it in baby steps. Let’s see what y looks like it’s going to become as we slowly make x smaller and smaller.

x = 10, x² = 100, x²/x = 10
x = 5, x² = 25, x²/x = 5
x = 2, x² = 4, x²/x = 2
x= 0.01, x² = .0001, x²/x = 0.01

Hmmm! It kind of looks like, y wants to be whatever x is. In fact if x were 0 …, no wait, we can’t go there, it’s against the intergalactic rules.

What about x = .00000000000001 ? :

x² = .000000000000000000000000001, y = x²/x = .00000000000001 (trust me, I’m a professional!)

Wow. X is getting pretty close to 0 and so is y!

Maybe the limit of y in this example is, GULP! YES IT IS! It’s 0!

Lim x²/x = Lim (x times x)/x = Lim x = 0 EUREKA!
ε = x->0 ε = x->0 ε = x->0

Notice how I got rid of one of the x’s in the numerator by dividing (also called canceling)? You can’t do that if you let x = 0 because YOU CAN’T DIVIDE BY ZERO!

But if you sneak up on it with a “limit”, you can see what will happen if you make x as close as you like to zero, you know, ε close. Go ahead, pick a number, any number!

Calculus is a very powerful method of mathematical calculation based on the properties of limits, It’s so powerful, it can break the Intergalactic Rules of Mathematics. Well, it can bend them anyway! But there's lots more to limits. The Intergalactic Congress has been very busy!

It’s all Greek to me!

By the way, about that promise of meeting cute girls, it helps if they are as smart as you are and just as bored with 5th grade math.