Chapter 8Playing With Squares
We’ve seen how lines behave: they change the same amount with each step.Now let’s try a more complex function like\( f(x) = x^2 \).It’s a more detailed scenario, so let’s visualize it.
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.
Your plan is to build the garden incrementally, foot-by-foot, until it looks right.Let’s say you start from scratch and build up to a 10\( \times \)10 plot:
To the untrained eye, you have single a 10\( \times \)10 garden, which uses 40 feet of perimeter fencing (10\( \times \)4) and 100 square feet of topsoil (10\( \times \)10).(Assume topsoil is sold by the square foot, with a standard thickness.)
8.1Bring On The Calculus
就这些吗?The analysis just figures out 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:
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 add 4 corner pieces (in yellow):
The visual is helpful, but not required.After our exposure to how lines behave, we can glance at an equation like\( p = 4x \)and realize that\( p \)jumps by 4 whenever\( x \)跳1。
8.2Changing Area
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 (\( 1^2 - 0^2 = 1 \)) |
| 1 | 1 | 3 (\( 2^2 - 1^2 = 3 \)) |
| 2 | 4 | 5 (\(3^2 - 2^2 = 5 \)) |
| 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\( 5 \).以此类推——奇数被夹在正方形之间!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\( 5 \times 5 \)square, getting to a\( 6 \times 6 \)will be a jump of\( 2(5) + 1 = 11 \).And yep,\( 36 - 25 = 11 \).)
再一次，视觉效果很好，但是很费力气。Algebra can simplify the process.
In this setup, if we set our change to\( dx = 1 \), we get
Algebra predicts the size of the slices without a hitch.
8.3Integrals and the Veggie Mafia
导数有一个形状，一个切割方向，并找到切片的模式。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…) butincreasingorders of topsoil (1, 3, 5, 7, 9, 11…).他们能解决什么问题?
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.”
但这还不够!那笨蛋不知道你想做什么。He saw order after order go by without noticing the deeper pattern.
The crime boss is different: he wants theindefiniteintegral, 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 pattern!”
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?同样，通过回溯。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 a steady 4, he knows the perimeter must be\( 4x \).Ah!There aren’t too many shapes with both properties: a square is his guess.(通过足够的练习，你开始识别常见的模式;像Wolfram Alpha这样的工具可以提供帮助。)
Now suppose 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 arectangulargarden, with one side determined by\( x \), and the other side a fixed 20 feet, for a 20-by-x rectangle.
这个猜想可行吗?Assuming this is the pattern, let’s measure the perimeter, area, and how they change:
Wow, it checks out: the changes in perimeter and area match the patterns (2, 2, 2…) and (20, 20, 20…).难怪他是教父。
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 youstartedwith a\( 2 \times 2 \)garden.Your first area jump was by 5, which would have happened if\( x \)was already 2 (solve\( 2x + 1 = 5 \)and we see\( x = 2 \)).
黑帮老大是古董商高手:他能看到你带来的古董的图案，很快就能确定原来的形状(不定积分)。随从只能告诉你到目前为止的运行总数(定积分)。
8.4Wrapping It All Up
It looks like we’re ready for another rule, to explain how squares change.If we leave\( dx \)as it is, we can write:
Ok!这是“边长边长”的缩写。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.