We've seen how lines behave. Now let's hone our skills with a more complex function like $$$f(x) = x^2$$$.

This scenario is more complex, so let's visualize it and walk through the properties.

Imagine you're building a square garden, to plant veggies and enjoy cucumbers in a few months. You're not sure how large to make it. Too small, and there's not enough food, but too large, and you'll draw the attention of the veggie mafia.

You'll build the garden incrementally, foot-by-foot, until it looks right. Let's say you build it slowly, starting from scratch and growing to a 10×10 plot:

What do we have? To the untrained eye, we've built a 10×10 garden, which uses 40 feet of perimeter fencing (10 · 4) and 100 square feet of topsoil (10 · 10). (Pretend topsoil is sold by the square foot, assuming a standard thickness.)

That's it? The analysis is just the current perimeter and square footage? No way.

By now, you should be clamoring to use X-Ray and Time-Lapse vision to see what's happening under the hood. Why settle for a static description when we can know the step-by-step description too?

We can analyze the behavior of the perimeter pretty easily:

$$Perimeter = 4 \cdot x $$

$$\frac{d}{dx} Perimeter = 4 $$

The change in perimeter ($$$\frac{dP}{dx}$$$) is a constant 4. For every 1-foot increase in x, we have a 4-foot jump in the perimeter.

We can visualize this process. As the square grows, we push out the existing sides, and just add 4 corner pieces (in yellow):

The visual is nice, but not required. After our exposure to lines, we should glance at an equation like $$$p = 4x$$$ and realize that p jumps by 4 when x jumps by 1.

Now, how does area change? Since squares are fairly new, let's X-Ray the shape as it grows:

We can write out the size of each jump, like so:

x | x^2 | Jump to next square --|-----|-------------------- 0 | 0 | 1 1 | 1 | 3 2 | 4 | 5 3 | 9 | 7 4 | 16 | 9 5 | 25 | 11 6 | 36 | 13 7 | 49 | 15

Now that's interesting. The gap from 0^{2} to 1^{2} is 1. The gap from 1^{2} to 2^{2} is 3. The gap from 2^{2} to 3^{2} is 5. And so on -- the odd numbers are sandwiched between the squares! What's going on?

Ah! Growing to the next-sized square means we've added a horizontal and vertical strip (x + x) and a corner piece (1). If we currently have a square with side x, the jump to the next square is 2x + 1. (If we have a 5×5 square, getting to a 6×6 will be a jump of 2·5 + 1 = 11. And yep, 36 - 25 = 11.)

Again, the visualization was nice, but it took effort. Algebra can simplify the process.

In this setup, if we set our change to dx = 1, we get

$$df = f(x + 1) - f(x) = (x + 1)^2 - x^2 = (x^2 + 2x + 1) - x^2 = 2x + 1$$

Algebra predicted the size of the slices without a hitch.

The derivative takes a shape and get the slices. Can we work backwards, from the slices to the shape? Let's see.

Suppose the veggie mafia spies on your topsoil and fencing orders. They can't see your garden directly, but what can they deduce from your purchases?

Let's say they observe a constant amount of fencing being delivered (4, 4, 4, 4…) but *increasing* orders of topsoil (1, 3, 5, 7, 9, 11…). What can they work out?

A low-level goon might just add up the total amount accumulated (the definite integral): "Heya boss, looks like they've built some garden with a total perimeter of 40-feet, and total area of 100 square feet."

But that's not good enough! The goon doesn't know the shape you're trying to build. He saw order after order go by without noticing the deeper pattern.

The crime boss is different: he wants the *indefinite* integral, the pattern you are following. He's savvy enough to track the pattern as the orders come in: "The area is increasing 1, 3, 5, 7… that's following a 2x + 1 area increase!"

Now, there are likely many shapes that could grow their area by $$$2x + 1$$$. But, combined with a constant perimeter increase of 4, he suspects you're making a square garden after a few deliveries.

How does the godfather do it? Again, by working backwards. He's split apart enough shapes (triangles, squares, rectangles, etc.) that he has a large table of before-and-afters, just like the antiques dealer.

When he sees a change of $$$2x + 1$$$, a square ($$$x^2$$$) is a strong candidate. Another option might be a right triangle with sides $$$x$$$ and $$$2x$$$. Its area equation is $$$\frac{1}{2} x \cdot 2x = x^2$$$, so the area would change the same as a square.

And when he sees a perimeter change of 4, he knows the perimeter must be $$$4x$$$. Ah! There aren't too many shapes with both properties: a square is his guess. Part of learning calculus is getting familiar with the typical origins of the patterns you see, so you can quickly reverse-engineer them.

Suppose they see your orders change: your fencing deliveries drop to (2, 2, 2, 2…) and your topsoil orders change to (20, 20, 20, 20). What's going on? Make a guess if you like.

Ready?

The veggie boss figures you've moved to a *rectangular* garden, with one side determined by x, and the other side a fixed 20-feet. Does this guess work?

$$ f(x) = 20x $$ $$ Perimeter = 20 + 20 + x + x = 40 + 2x$$ $$ Area = 20x$$ $$ \frac{d}{dx} Perimeter = 2 $$ $$ \frac{d}{dx} Area = 20$$

Wow, it checks out. No wonder he's the godfather.

Lastly, what if the godfather saw topsoil orders of (5, 7, 9, 11, 13)? He might assume you're still building a square ($$$2x + 1$$$ pattern), but you *started* with a 2×2 garden already made. Your first jump was 5, which would have happened if x was already 2 ($$$2x + 1 = 5$$$).

The mob boss is a master antiques dealer: he sees the pattern of shards you're bringing in and quickly determines the original shape (indefinite integral). The henchman can only tell you the running totals so far (definite integral).

It looks like we're ready for another rule, to explain how squares change. If we leave $$$dx$$$ as it is, we can write:

$$\frac{d}{dx} x^2 = \frac{f(x + dx) - f(x)}{dx} = \frac{(x + dx)^2 - (x)^2}{dx} $$ $$=\frac{x^2 + 2x\cdot dx + (dx)^2 - x^2}{dx} = \frac{2x\cdot dx + (dx)^2}{dx} = 2x + dx$$

Ok! That's the abbreviated way of saying "Grow by two sides and the corner". Let's plug this into the computer to check:

Uh oh! We hand-computed the derivative of $$$x^2$$$ as $$$2x + dx$$$ (which is usually $$$2x + 1$$$), but the computer says it's just $$$2x$$$.

But isn't the difference from 4^{2} to 5^{2} exactly 25 - 16 = 9, and not 8? What happened to that corner piece? *The mystery continues in the next episode.*

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