10.1Analogy: Measuring Heart Rates

Imagine you’re a doctor trying to measure a patient’s heart rate while exercising.You put a guy on a treadmill, strap on the electrodes, and get him running.The machine spits out 180 beats per minute.那一定是他的心率，对吧?

Nope.That’s his heart ratewhen observed by doctors and covered in electrodes.Wouldn’t that scenario be stressful?And what if your Nixon-era electrodes get tangled on themselves, and tug on his legs while running?

Ah.We need the electrodes to getsomemeasurement.But, right afterwards, we need to remove the effect of the electrodes themselves.For example, if we measure 180 bpm, and knew the electrodes added 5 bpm of stress, we’d know the true heart rate was 175.

The key is making the knowingly-flawed measurement, getting a reading, then correcting it as if the instrument was never there.

10.2Measuring the Derivative

Measuring the derivative is just like putting electrodes on a function and making it run.For\( f(x) = x^2 \), we stick an electrode of\( +1 \)onto it, to see how it reacted:

水平条纹是沿形状顶部应用的更改的结果。The vertical stripe is our change moving along the side.And what’s the corner?

这是水平变化与垂直变化相互作用的一部分!This is an electrode getting tangled in its own wires, a measurement artifact that needs to go.

10.3Throwing Away Artificial Results

The founders of calculus intuitively recognized which components of change were “artificial” and just threw them away.They saw that the corner piece was the result of our test measurement interacting with itself, and shouldn’t be included.

In modern times, we created official theories about how this is done:

• 限制:我们让测量工件变得越来越小，直到它们有效地消失(无法从零区分)。
• Infinitesimals: Create a new type of number that lets us try infinitely-small change on a separate, tiny number system.When we bring the result back to our regular number system, the artificial elements are removed.

There are entire classes that explore these theories.The practical upshot is realizinghow进行测量，然后扔掉我们不需要的部分。

Here’s how the derivative is defined using limits:

Step	Example
Start with function to study	\( f(x) = x^2 \)
1.Increase the input by\( dx \), a sample change	\( f(x + dx) = (x + dx)^2 = x^2 + 2x\cdot dx + (dx)^2 \)
2.Find the resulting increase in output,\( df \)	\( df = f(x + dx) - f(x) = 2x\cdot dx + (dx)^2 \)
3.Find the ratio of output change to input change	\( \frac{df}{dx} = \frac{2x\cdot dx + (dx)^2}{dx} = 2x + dx \)
4.Throw away any measurement artifacts	\( 2x + dx \overset{dx \ = \ 0} \Longrightarrow 2x \)

Wow!We found the official derivative for\( \frac{d}{dx} x^2 \)on our own:

Now, a few questions:

• Why do we measure\( \frac{df}{dx} \), and not the actual change\( df \)?Think of\( df \)整个变化发生在我们迈出一步的时候。For easy comparison to other functions, we typically want the “per step” change\( \frac{df}{dx} \).(This is like comparing jobs by dollars/hour instead of by salary, or cars by miles-per-gallon instead of gallons used.)Sometimes the total change is helpful to consider, and we can rewrite\( \frac{df}{dx} = 2x \)as\( df = 2x \cdot dx \).
• How can we just set\( dx \)最后变成零?I see\( dx \)as the size of the instrument used to measure the change in a function.After we have the measurement with a real instrument (\( \frac{df}{dx} = 2x + dx \)), we figure out what the measurement would be if the instrument were perfect and did not interfere (\( \frac{df}{dx} = 2x + 0 = 2x \)).
• But isn’t the\( 2x + 1 \)pattern correct?The whole numbers (integers) are separated by an interval of 1, so assuming\( dx = 1 \)(and not letting it disappear) is accurate:\( 2x + 1 \)correctly predicts the gap of 5 between\( 2^2 \)and\( 3^2 \).然而，小数点(实数)之间没有固定的间隔。\( 2x \)is the ideal estimate for the rate of change between\( 2^2 \)and the infinitely-close number that follows – not 2.0001, or 2.0000000001, but whatever unnamed number comes next.Said another way, if\( dx \)doesn’t disappear, we’re saying the real numbers have a fixed interval between them, like the integers.
• If there’s no ‘+1’, when is the corner filled in?考虑面积的变化，而不是图上的细节。The corner overestimates how much growth happens onthis step(i.e., the radar clocked us at\( 2x + 1 \)but we’re only growing by\( 2x \)).But we’re still moving and make progress.

  I imagine a square that grows by bulging out its sides (\( x + x = 2x \)), then absorbing the new area to make a larger square.The new size is larger, but notquitebig enough to fill in the corner exactly.这是可以的，因为随着时间的推移，这个过程仍然会包含必要的领域。\( 2x + 1 \)overestimates our growth because it assumes the horizontal and vertical slices interact to create the corner piece.

Practical conclusion:We can start with a knowingly-flawed measurement,\( f'(x) \sim 2x + dx \), and deduce the perfect result it points to:\( f'(x) = 2x \).The theories of exactlyhowwe throw away\( dx \)今天不需要掌握。The key is realizing there are measurement artifacts – the shadow of the camera in the photo – that must be removed to accurately describe the true behavior.

(Still shaky about exactly how\( dx \)can appear and disappear?有很多人和你一样。This question took top mathematicians decades to resolve.Here’sa deeper discussion of how the theory works.)