4.1The Derivative

Thederivativeis the pattern of slices we get as we X-Ray a shape.It’s indicated by the black trend line, which might be flat, rising constantly, rising and falling, etc.Now here’s the trick: although the derivative generates the entire sequence of slices, we can also extract asingleslice.

Think about a function like\( f(x) = x^2 \).它描述了每一个可能的平方值(1、4、9、16、25等)，我们可以把它们都画在一个图表上。But, we can also ask for the value of\( f(x) \)at a specific value, such as at\( x = 3 \).

The derivative is similar.Officially, it’s the complete pattern of slices we get after X-Raying a shape.However, we can pull out an individual slice by asking for the derivativeata certain value.(The derivative is a function, just like\( f(x) = x^2 \), and mathematicians assume you’re talking about the entire function unless you ask for a specific slice.)

So, what do we need to find the derivative?Just the shape to split apart, and the path to follow as we cut it up (the orange arrow).术语是“对<某模式>对<某方向>求导”。For example:

• The derivative of a circlewith respect tothe radius creates rings (which always increase)
• The derivative of a circlewith respect tothe perimeter creates slices (which are equal-sized)
• The derivative of a circlewith respect tothe x-axis creates boards (which get larger, peak, and get smaller)

I agree that “with respect to” sounds formal:Honorable Grand Poombah radius, it is with respect to you that we take the derivative.Math is a gentleman’s game, I suppose.

Taking the derivative is also called “differentiating”, because we are finding the difference between successive positions as a shape grows.As we grow the radius of a circle, the outer ring is the difference between the size of the current disc and the next size up.

4.2The Integral, Arrows, and Slices

Theintegral是将一堆切片粘在一起(延时)，然后测量最终结果。For example, we glued together the rings (into a “ring triangle”) and saw it accumulated to\( \pi r^2 \)也就是圆的面积。

Here’s what we need to find the integral:

• Which direction are we gluing the steps together?Along the orange line (the radius, in this case)
• 我们什么时候开始和停止?At the start and end of the arrow (we start at 0, no radius, and move to r, the full radius)
• How big is each step?Well… each item is a “ring”.Isn’t that enough?

Nope!We need to be specific.We’ve been saying we cut a circle into “rings” or “pizza slices” or “boards”.但这还不够具体;就像烧烤食谱上写着“煮肉”。Flavor to taste.”

Maybe an expert knows what to do, but we need more specifics.How large, exactly, is each step (technically called the “integrand”)?

Ah.A few notes about the variables:

• If we are moving along the radius\( r \), then\( dr \)is the little chunk of radius in the current step
• The height of the ring is the circumference, or\( 2 \pi r \)

There’s several gotchas to keep in mind.

First,\( dr \)is its own variable, and not “d times r”.It represents the tiny section of the radius present in the current step.This symbol (\( dr \),\( dx \), etc.)is often separated from the integrand by just a space, and it’s assumed to be multiplied (written\( 2 \pi r \ dr \)).

Next, if\( r \)is the only variable used in the integral, then\( dr \)is assumed to be there.So if you see\( \int 2 \pi r \)this still implies we’re doing the full\( \int 2 \pi r \ dr \).(Again, if there are two variables involved, like radius and perimeter, you need to clarify which step we’re using:\( dr \)or\( dp \)?)

Last, remember that\( r \)(the radius) changes as we Time-Lapse, starting at 0 and eventually reaching its final value.When we see\( r \)in thecontext of a step, it means “the size of the radius at the current step” and not the final value it may ultimately have.

这些问题非常令人困惑。I’d prefer we use\( r_{dr} \)for “\( r \)at the current step” instead of a general-purpose\( r \)that’s easily confused with the max value of the radius.We can’t change the symbols at this point, unfortunately.

4.3Practicing The Lingo

Let’s learn to speak like calculus natives.Here’s how we can describe our X-Ray strategies:

Remember, the derivative just splits the shape into (hopefully) easy-to-measure steps, such as rings of size\( 2 \pi r \ dr \).我们把乐高积木拆开，碎片散落在地板上。We still need an integral to glue the parts together and measure the new size.The two commands are a tag team:

• 导数说:“好，我为你把这个形状分开。It looks like a bunch of pieces\( 2 \pi r \)tall and\( dr \)wide.”
• The integral says: “Oh, those pieces resemble a triangle – I can measure that!The total area of that triangle is\( \frac{1}{2} \textit{base} \cdot \textit{height} \), which works out to\( \pi r^2 \)在这种情况下。”

Here’s how we’d write the integrals to measure the steps we’ve made:

A few notes:

• Often, we write an integrand as an unspecified “pizza slice” or “board” (use a formal-sounding name like\( s(p) \)or\( b(x) \)if you like).首先，我们建立积分，然后我们考虑板或片的精确公式。
• Because each integral represents slices from our original circle, we know they will be the same.粘合任何一组切片都应该返回总面积，对吧?
• The integral is often described as “the area under the curve”.It’s accurate, but shortsighted.Yes, we are gluing together the rectangular slices under the curve.但这完全忽略了之前x射线和延时拍摄的思想。Why are we dealing with a set of slices vs.首先是曲线?Most likely, because those slices are easier than analyzing the shape itself (how do you “directly” measure a circle?).

4.4Questions

At a high level, can you find another activity madeeasierwith symbols, instead of using full English sentences?Would practitioners ever go back to written descriptions?

Math is just like that.Let’s try a few phrases, even if we aren’t fluent yet.

Question 1:Can you describe the integrals below in “Math English”?

Assume the arrow spans half the radius.The description should follow the format:

integrate [size of step] from [start] to [end] with respect to [path variable]

Have an idea?Here’s the answer for thefirst integral¹and thesecond integral².These links go to Wolfram Alpha, an online math solver, which we’ll learn to use.

Question 2:你能用“数学英语”来描述我们的披萨片创意吗?

The math description should be something like this:

integrate [size of step] from [start] to [end] with respect to [path variable]

记住，每一片基本上是一个三角形(那么面积是多少?)切片沿着周长移动(从哪里开始和停止?)Have a guess for the command?Here it is, theslice-by-slice description³.

Question 3:Can you describe how to move from volume to surface area?

Assume we know the volume of a sphere is4/3 * pi * r⌃3.Think about the instructions to separate that volume into a sequence of shells.我们要通过哪个变量?

take derivative of [equation] with respect to [path variable]

Have a guess?Great.Here’s the command to turnvolume into surface area⁴.