Interpreting the Derivative in Context
AP Calculus BC Topic 4 8:26 English narration · English + 中文 subtitles burned in
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A tank is draining.
一个水箱正在放水。
A cup of coffee is cooling.
一杯咖啡正在变凉。
A city is growing.
一座城市正在扩张。
Every one of those sentences hides the same phrase. The rate at which something changes.
这三句话里都藏着同一个短语： 某个东西变化的速率。
And every time you meet that phrase in a word problem, that phrase is a derivative.
而每当你在应用题里遇到这个短语时，它就是一个导数。
This unit is about turning derivatives into sentences about the real world.
本单元讲的就是如何把导数变成关于真实世界的句子。
So the first habit to build is simple. Read the units, and they will tell you which quantity's rate you are holding.
所以要养成的第一个习惯很简单： 读单位，单位会告诉你手上拿的是哪个量的变化率。
Once you can compute derivatives, you use them to describe the world.
一旦你会求导，接下来就是用导数来描述这个世界。
Let's begin.
让我们开始吧。
Units matter. The unit of a derivative is the unit of the output divided by the unit of the input.
导数的单位，是输出的单位除以输入的单位。
That single rule settles every units question.
这一条规则就能解决所有的单位问题。
Suppose a function gives an area in acres, and the input is time in weeks. Then its derivative is in acres per week.
假设一个函数给出的是以英亩为单位的面积，输入是以周为单位的时间。
Now, the exam does not want a number.
那么它的导数的单位就是英亩每周。
It says, using correct units, interpret the meaning.
而考试要的不是一个数字。
That asks for a full sentence, and the sentence needs four things. The value.
它会说： 使用正确的单位，解释其含义。
The quantity that is changing. The word per, with the units. And the moment it happens.
这要求你写出一个完整的句子，而这个句子需要四样东西： 数值；正在变化的量；带单位的"每"字；以及它发生的时刻。
The classic context is a particle moving along a line.
最经典的情境是一个质点沿直线运动。
Three functions of time are linked by differentiation.
三个关于时间的函数由求导联系在一起。
Position.
位置。
Velocity, which is the derivative of position.
速度，它是位置的导数。
And acceleration, which is the derivative of velocity.
加速度，它是速度的导数。
Two readings come up on almost every paper.
有两条读法几乎每张卷子上都会考。
The particle is at rest when the velocity is zero.
速度为零时，质点静止。
And the sign of velocity gives the direction.
速度的符号给出运动方向。
Positive means moving right, negative means moving left.
正表示向右运动，负表示向左运动。
The particle changes direction wherever velocity changes sign, not merely where it touches zero.
质点改变方向的地方，是速度变号的地方，而不只是速度碰到零的地方。
Now the distinction the exam loves.
现在讲考试最喜欢考的那个区别。
Velocity has direction. Speed does not.
速度有方向，速率没有。
Speed is the size of velocity, with the sign thrown away.
速率是速度的大小，把符号丢掉。
So how do you tell whether a particle is speeding up?
那么怎么判断质点是在加速呢？
Compare the signs of velocity and acceleration.
比较速度和加速度的符号。
When they share a sign, the particle is speeding up.
两者符号相同时，质点在加速。
When they have opposite signs, it is slowing down.
两者符号相反时，质点在减速。
Notice what this means.
注意这意味着什么。
A negative acceleration does not always mean slowing down.
加速度为负并不总是表示在减速。
If the velocity is also negative, the particle is speeding up while travelling left.
如果速度也是负的，那么质点是在向左运动的同时加速。
Work one through.
我们完整做一道。
A particle has position t cubed, minus six t squared, plus nine t.
一个质点的位置是 t 的三次方，减 6t 平方，加 9t。
Differentiate, then factor the velocity.
求导，然后把速度因式分解。
It becomes three, times t minus one, times t minus three.
它变成 3 乘以 (t 减 1) 乘以 (t 减 3)。
So the particle is at rest at time one and at time three.
所以质点在时刻 1 和时刻 3 静止。
Draw a sign chart.
画一张符号表。
Velocity is positive, then negative, then positive again, so the particle changes direction at both of those times.
速度先正，再负，然后又变正，所以质点在这两个时刻都改变了方向。
Now look at time two.
现在看时刻 2。
Velocity is minus three, so the particle is moving left.
速度是负 3，所以质点在向左运动。
Acceleration is zero right at that instant, but just after it is positive.
就在那一瞬间加速度为零，但紧接着它变成正的。
Velocity is negative and acceleration is positive, so the signs differ, and the particle is slowing down there.
速度为负而加速度为正，符号相反，所以质点在那里正在减速。
Motion is only one story.
运动只是其中一个故事。
The same derivative idea models any changing quantity.
同样的导数思想可以描述任何变化的量。
A draining tank, where the rate is litres per minute.
一个正在放水的水箱，变化率的单位是升每分钟。
A spreading population, where the rate is people per year.
一个正在增长的人口，单位是人每年。
A cooling cup, where the rate is degrees per minute.
一杯正在变凉的咖啡，单位是度每分钟。
The mathematics never changes, only the words.
数学从来没变，变的只是文字。
So read the units first.
所以先读单位。
They tell you which quantity is on top and which is on the bottom, and that settles what the number actually means.
单位会告诉你哪个量在上面、哪个量在下面，这就确定了这个数到底是什么意思。
Next, related rates.
接下来是相关变化率。
Here several quantities change together over time.
这里有好几个量随着时间一起变化。
You know some of the rates, and you want another one.
你知道其中一些变化率， 但想求另一个。
The engine is the chain rule.
核心工具是链式法则。
You take a relationship between the quantities, and you differentiate it with respect to time.
你取一个联系这些量的关系式，然后对时间求导。
Every variable in that relationship is secretly a function of time, so each one picks up a rate factor as you differentiate.
关系式里的每一个变量其实都是时间的函数，所以求导时每一个都会带出一个变化率因子。
Product and quotient rules often appear too, because the relationship is rarely a simple sum.
乘积法则和商法则也经常出现，因为关系式很少是简单的加法。
There is a five step template, and following it earns full credit.
有一个五步模板，照着做就能拿满分。
One. Name the variables, and write down which rates you are given and which one you want. Two.
第一，给变量命名，写下题目给了哪些变化率、 要求的是哪一个。
Write an equation relating the quantities, usually a geometry or volume formula. Three.
第二，写出一个联系这些量的方程，通常是几何公式或体积公式。
Differentiate both sides with respect to time.
第三，两边同时对时间求导。
Four. Substitute the known values, but only at this stage, at the instant of interest. Five.
第四，代入已知数值，但只能到这一步才代， 而且是代所关心那一瞬间的值。
State the answer with units and the correct sign, because a shrinking quantity has a negative rate.
第五，写出答案，带上单位和正确的符号， 因为一个在减少的量，它的变化率是负的。
Step three before step four is the classic error.
第三步要排在第四步之前，这正是最经典的错误。
Differentiate the general relationship first, then put the numbers in.
先对一般的关系式求导，再代数字。
Here is the standard one.
这是最标准的一道。
Air is pumped into a spherical balloon, and the volume grows at one hundred cubic centimetres per second.
往一个球形气球里打气，体积以每秒 100 立方厘米的速度增长。
How fast is the radius growing when the radius is five centimetres?
当半径为 5 厘米时，半径增长得有多快？
Step one and two.
第一、二步。
The volume of a sphere is four thirds pi r cubed.
球的体积是三分之四 π r 的三次方。
Step three.
第三步。
Differentiate with respect to time.
对时间求导。
The volume rate equals four pi r squared, times the radius rate.
体积的变化率等于 4π r 平方，乘以半径的变化率。
Notice the radius rate appearing, exactly as the chain rule promised.
注意半径的变化率出现了，正如链式法则所预示的那样。
Step four.
第四步。
Now substitute.
现在代入。
One hundred equals four pi times twenty five, times the radius rate.
100 等于 4π 乘以 25，再乘以半径的变化率。
So the radius rate is one over pi, which is about zero point three two centimetres per second.
所以半径的变化率是 π 分之一， 大约是每秒 0.32 厘米。
Now a different use of the tangent line.
现在讲切线的另一种用法。
Take any smooth curve and zoom in far enough at one point, and it becomes indistinguishable from its tangent.
取任何一条光滑曲线，在某一点处放得足够大， 它就会变得和它的切线无法区分。
That is called local linearity.
这叫做局部线性。
So near the point of tangency, we can replace the function by the tangent line formula.
所以在切点附近， 我们可以用切线公式来代替这个函数。
The value at the point, plus the derivative at the point, times how far you moved.
切点处的函数值，加上切点处的导数， 乘以你移动了多远。
That gives a linear approximation.
这就给出一个线性近似。
It is quick, and it is accurate, but only close by.
它很快，也很准，但只在很近的范围内准。
Move far from the point and the curve peels away from the line.
离开切点太远，曲线就会从直线上剥离开去。
Exams then ask whether your estimate is too big or too small.
接着考试会问：你的估计值是偏大还是偏小？
Concavity answers it, and the picture makes it obvious.
凹凸性给出答案，而且图一画就很明显。
If the curve is concave up near the point, it bends upward away from its tangent, so the curve sits above the line, and the tangent value is an underestimate.
如果曲线在该点附近是凹向上的，它会向上弯离切线，所以曲线在直线上方， 切线给出的值就是低估。
If the curve is concave down, it sits below the line, and the tangent value is an overestimate.
如果曲线是凹向下的，它在直线下方，切线给出的值就是高估。
When you write this up, justify it with the sign of the second derivative, not just with a drawing.
写答案的时候，要用二阶导数的符号来说明理由，而不只是画一张图。
Last idea.
最后一个概念。
Sometimes direct substitution gets stuck.
有时候直接代入会卡住。
You put the value in and you get zero over zero, or infinity over infinity.
你把值代进去，得到零比零，或者无穷比无穷。
Those are indeterminate forms.
这些叫未定式。
They do not mean the limit fails. They mean you cannot see it yet. Here is L'Hospital's Rule itself.
它们并不表示极限不存在，只表示你还看不出来。
The limit of the quotient equals the limit of the derivative of the top over the derivative of the bottom.
法则本身是这样的： 这个商的极限，等于分子的导数比分母的导数的极限。
Read that carefully, because it is not the quotient rule.
要仔细读，因为这不是商法则。
You differentiate the top and the bottom separately, and you never combine them.
你要分别对分子和分母求导，绝不把它们合起来算。
And you must check the form first.
而且你必须先检查形式。
Applying this rule to anything other than those two forms is simply wrong.
把这条法则用在那两种形式之外的任何情况上，都是错的。
Two quick examples.
两个快速的例子。
Take the limit as x goes to zero of the sine of x over x.
求 x 趋于 0 时，x 的正弦比 x 的极限。
Substituting gives zero over zero, so the rule applies.
代入得到零比零，所以法则适用。
Differentiate the top to get cosine of x.
对分子求导得到 x 的余弦。
Differentiate the bottom to get one.
对分母求导得到 1。
Now substitute again, and the answer is one.
再代入一次，答案是 1。
Here is the second one.
第二个例子。
The limit as x goes to zero of e to the two x, minus one, all over x.
求 x 趋于 0 时，e 的 2x 次方减 1，再除以 x 的极限。
That is zero over zero as well.
这也是零比零。
The top differentiates to two e to the two x, and the bottom to one.
分子求导得到 2 乘以 e 的 2x 次方，分母求导得到 1。
Substitute, and the answer is two.
代入，答案是 2。
Three marks students lose.
学生最常丢分的三个地方。
First, in motion problems: speed increases when velocity and acceleration share a sign.
第一，在运动问题中，只有当速度和加速度符号相同时，质点才在加速。
Second, in related rates, differentiate first and substitute last.
第二，在相关变化率中，先求导，最后才代入。
Third, when a question says interpret with units, answer in a sentence with the units, not with a bare number.
第三，当题目要求带单位解释含义时，要用一个带单位的句子来回答，而不是只写一个数。