Interpreting the Derivative in Context
AP Calculus AB Topic 4 6:54 English narration · English + 中文 subtitles burned in
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Now that we can compute derivatives, we use them to describe the real world.
现在我们能求导数了，就用它们来描述真实世界。
A derivative is a rate of change.
导数是一种变化率。
How fast is a population growing?
人口增长有多快？
How fast is a tank draining?
水箱排水有多快？
How fast is a car moving right now?
汽车此刻开得有多快？
Every time a problem asks the rate at which something changes, it is asking for a derivative.
每当一道题问"某个量变化的速率"时，它问的就是导数。
And the units tell you exactly which rate you have.
而单位会准确告诉你，这是哪个量的变化率。
Today we put derivatives to work — motion, related rates, linear approximation, and a shortcut for tricky limits.
今天我们让导数派上用场——运动、相关变化率、线性近似， 以及一个处理棘手极限的捷径。
Let's begin.
让我们开始吧。
Units matter. First, interpreting a derivative in words.
首先，用文字解读导数。
The unit of f prime is the unit of f, divided by the unit of the input.
f 撇 的单位，是 f 的单位除以输入的单位。
If a quantity is measured in acres, and time in weeks, then its derivative is in acres per week.
如果一个量以英亩计量，时间以周计量，那么它的导数就以每周多少英亩计量。
When the exam says interpret the meaning, write a full sentence: the value, the quantity, the word per, and the moment.
当考试要求解释含义时，写一个完整的句子：数值、量、"每"这个词、以及那个时刻。
For example: at that time, the quantity is increasing at about two point three units per week.
例如：在那个时刻，这个量正以每周大约二点三个单位的速度增加。
The most common context is straight-line motion.
最常见的情境是直线运动。
Position, velocity, and acceleration are linked by differentiation.
位置、速度和加速度，通过求导彼此相连。
Velocity is the derivative of position.
速度是位置的导数。
Acceleration is the derivative of velocity.
加速度是速度的导数。
Velocity is signed — its sign gives the direction: right or up when positive, left or down when negative.
速度是有符号的——它的符号给出方向： 为正时向右或向上，为负时向左或向下。
The particle is at rest when the velocity is zero, and it changes direction wherever the velocity changes sign.
当速度为零时物体静止，而只要速度改变符号，它就改变运动方向。
Now, speed versus velocity — the exam loves this difference.
现在讲速率与速度——考试特别爱考这个区别。
Speed is the size of the velocity, ignoring direction.
速率是速度的大小，不看方向。
Here is the key rule: the particle is speeding up when velocity and acceleration have the same sign, and slowing down when they have opposite signs.
关键法则是：当速度和加速度符号相同时，物体在加速；符号相反时，物体在减速。
So a negative velocity with a negative acceleration means speeding up, not slowing.
所以负的速度配上负的加速度，意味着在加速，而不是减速。
Watch the signs, not just the numbers.
要看符号，不只是看数字。
Apply all of that to one particle, because the exam asks it every year.
把这些全部用到一个质点上，因为考试每年都问。
Take a particle whose position is t cubed, minus six t squared, plus nine t.
设一个质点的位置是 t 的三次方，减六 t 平方，加九 t。
Differentiate, then factorise — that second step is the one that pays.
先求导，再因式分解——第二步才是关键。
The velocity is three t squared minus twelve t plus nine, which factors into three, times t minus one, times t minus three.
速度是三 t 平方减十二 t 加九，可以分解成三，乘 t 减一，乘 t 减三。
So it is at rest at t equals one and at t equals three, and because the velocity changes sign at each, it turns around at each.
所以它在 t 等于一和 t 等于三时静止； 而且速度在这两处都变号，所以它在这两处都掉头。
Now look at t equals two, between the two turning points.
现在看 t 等于二，在两个掉头点之间。
The velocity there is three, times one, times minus one — that is minus three, so it is moving left.
那里的速度是三，乘一，乘负一——等于负三，所以它在向左运动。
Acceleration is six t minus twelve, which is zero exactly at t equals two and positive just after.
加速度是六 t 减十二，在 t 等于二时恰好为零，随后变正。
So just after t equals two the velocity is negative while the acceleration is positive: opposite signs, so the particle is slowing down, even though it is still moving.
所以刚过 t 等于二时，速度为负而加速度为正： 符号相反，所以质点在减速，尽管它仍在运动。
Speeding up and moving forward are different questions, and this is where the marks go.
"在加速"和"在前进"是两个不同的问题，分数就丢在这里。
Sometimes several quantities change together over time.
有时几个量会随时间一起变化。
A ladder slides down a wall: as the top falls, the bottom slides out — and their rates are linked.
一架梯子沿墙下滑：顶端下落时，底端向外滑出—— 它们的变化率彼此相连。
This is a related rates problem.
这就是相关变化率问题。
The engine is the chain rule: we differentiate a relationship with respect to time.
它的引擎是链式法则： 我们对一个关系式关于时间求导。
Every variable becomes a function of time, so each one picks up a rate — a per-second factor.
每个变量都成为时间的函数， 于是每一个都带上一个变化率——一个每秒的因子。
Here is the reliable procedure — and the full-credit template.
这是可靠的步骤——也是拿满分的模板。
One: name the variables, and write the rate you know and the rate you want.
第一：给变量命名，写下你已知的变化率和想求的变化率。
Two: write an equation relating the quantities, often a geometry or volume formula.
第二：写一个把这些量联系起来的方程，常常是几何或体积公式。
Three: differentiate both sides with respect to time — before you put any numbers in.
第三：对两边关于时间求导——在代入任何数字之前。
Four: now substitute the known values at that instant, and solve.
第四：现在代入那一时刻的已知值，并求解。
Five: state the answer with units and the right sign.
第五：给出答案，带上单位和正确的符号。
Let's use it.
我们来用一用。
Air fills a spherical balloon, so its volume grows at one hundred cubic centimetres per second.
空气充入一个球形气球，使它的体积以每秒一百立方厘米增长。
How fast is the radius growing when the radius is five?
当半径为五时，半径增长得有多快？
Start from the volume formula, and differentiate with respect to time first: the rate of the volume equals four pi r squared, times the rate of the radius.
从体积公式出发，先对时间求导： 体积的变化率，等于四派 r 平方，乘以半径的变化率。
Now put in the numbers: one hundred equals four pi times twenty-five, times the rate.
现在代入数字： 一百等于四派乘以二十五，再乘以那个变化率。
Solve, and the radius grows at one over pi — about zero point three two centimetres per second.
解出来，半径以每秒派分之一增长—— 大约每秒零点三二厘米。
Substituting the radius too early is the classic mistake.
太早代入半径，是最典型的错误。
Zoom in on a smooth curve near a point, and it looks almost straight — like its tangent line.
把一条光滑曲线在某点附近放大，它看起来几乎是直的——就像它的切线。
This is called local linearity.
这叫做局部线性。
So the tangent line gives a linear approximation of the function near that point.
所以切线给出了函数在该点附近的线性近似。
Close to the point, the tangent and the curve nearly agree, so we can use the simple line to estimate the complicated function.
在该点附近，切线和曲线几乎重合，于是我们可以用这条简单的直线来估计那个复杂的函数。
Let's estimate the square root of four point one.
我们来估计四点一的平方根。
Take the square-root function, at the nearby point four, where the value is two.
取平方根函数，在附近的点四处，那里的值是二。
The derivative there is one quarter.
那里的导数是四分之一。
So the linear approximation is two, plus one quarter, times zero point one — which gives two point zero two five.
所以线性近似是二，加上四分之一，乘以零点一—— 得到二点零二五。
That matches the true value to three places.
这与真值精确到三位小数一致。
And because the square-root curve is concave down, the tangent sits above it, so our estimate is a slight overestimate.
而因为平方根曲线向下弯， 切线位于它上方，所以我们的估计略微偏高。
One more tool — L'Hospital's Rule, for tricky limits.
还有一个工具——洛必达法则，用来处理棘手的极限。
When direct substitution gives an indeterminate form — zero over zero, or infinity over infinity — you may differentiate the top and the bottom separately — this is not the quotient rule — and then try the limit again.
当直接代入给出零比零， 或无穷比无穷时，你可以分别对分子和分母求导——这不是商法则——然后再试一次极限。
But first, always confirm the form really is one of those two.
但首先，永远要确认这个形式确实是这两种之一。
For example, the sine of x over x gives zero over zero.
例如，x 的正弦除以 x，给出零比零。
Differentiate: cosine of x over one, which at zero is one — confirming our famous limit from Unit One.
求导：x 的余弦除以一，在零处等于一——这印证了我们在第一单元里的著名极限。
Before you go, three marks to keep.
结束之前，三个要守住的分。
First, in motion: an object speeds up only when velocity and acceleration share the same sign.
第一，运动问题里：只有当速度和加速度符号相同时，物体才加速。
Second, in related rates, differentiate with respect to time first, and substitute the numbers last.
第二，相关变化率里，先对时间求导，最后再代入数字。
Third, the tangent-line approximation is only accurate close to the point — and whether it is over or under comes from the concavity.
第三，切线近似只在该点附近才准确—— 而它是偏高还是偏低，取决于凹凸性。
Get these right, and this topic is yours.
把这些做对，这个专题就是你的了。