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Now Isaac Newton and/or Joseph Raphson figured out how to do this kind of thing for all differentiable functions.
既然牛顿和拉复生已经,指数了如何解这种可导函数,因此我们就不用太担心了。
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But then having found one answer, you can add to it anything that gets killed by the act of taking derivatives.
一旦你算出了一个答案,你就可以往式子里加入任何,在求完导后消失的项
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Let's listen to Luciano Pavarotti sing a leading tone, and as by coincidence, it happens to be the one-year anniversary of Pavarotti's death this week.
让我们听一下,卢西亚诺·帕瓦罗蒂演唱的导音,巧合的是,这周恰好是,帕瓦罗蒂逝世一周年
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All right, what we gonna do, we want to take a derivative of this thing.
接下来我们要求导了
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If you took the two derivatives of this guy, In other words, Same thing there.
对这一项求两次导,也就是说,这里也是一样
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Okay. So this is a leading tone going in to that particular note.
好吧,这是一个引出那个特定音符的导音
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Okay, so we've got across the idea of tonics and leading tones.
好了,我们了解了主音和导音的概念
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To do that, let's take one more derivative.
要得到结果,我们再求一次导
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They love to kind of luxuriate in the opportunity to show off their voice on a leading tone, build up expectation, make you wait even more for that tonic.
他们总是喜欢借机,在导音部分卖弄下自己的歌喉,以增加听众的期望,让大家更渴望听到主音
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At any point on the graph you can take the derivative, which will be tangent to the curve at each point, and its numerical value will be what you can call the instantaneous velocity of that point and you can take the derivative over the derivative and call it the acceleration.
在图上的任意一点,你可以进行求导,得到曲线上每一点的切线斜率,所得到的数值,即为该点处的瞬时速度,然后你再求一次导,得出它的加速度
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Okay.This is the intro.
好，这一段是导奏。
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You know that if you took a derivative of this, you will find v of t is v0+at.
如果你对这个式子求一次导,你将会得到v=v0+at
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You can easily check by taking two derivatives that this particle does have the acceleration a.
你可以很容易地通过求两次导来验证,这个质点的加速度确实为 a
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You can take a derivative of the derivative and you can get the acceleration vector, will be d^2r over dt^2, and you can also write it as dv over dt.
你可以对导数再求一次导,你就可以得到加速度矢量,也就是 d^2 r / dt^2,你也可以写成 dv / dt
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take the derivative of this, get the velocity vector and you notice his magnitude is a constant Whichever way you do it, you can then rewrite this as v square over r.
对这个式子求一次导,就能得到速度矢量,你会发现其模长是常数,不管用什么方法,加速度也可以写成 v^2 / r
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That means, when I take two derivatives, I want to get a, then you should know enough calculus to know it has to be something like at^, and half comes from taking two derivatives.
也就是说,当我想求二阶导时,得到了a,你应该有足够的微积分知识,才能知道必须有类似at^的项,而这个1/2则是因为求了两次导数
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The most important result from last time was that if you took this r, and you took two derivatives of this to find the acceleration, d^2 r over dt^2, try to do this in your head.
上节课最重要的结论,就是如果你把 r 写成这样,对 r 求两次导就能得到加速度,d^2 r / dt^2,心算一下
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I would say, okay, this guy wants me to find a function which reduces to the number a when I take two derivatives, and I know somewhere here, this result, which says that when I take a derivative, I lose a power of t.
出题者想要我找出一个函数,它在经过两次求导后得到数字a,我知道这里的某个地方,这个结论告诉我,我每求一次导,t就降一次幂
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Then there's one other that we have to know about, it's called the leading tone.
还有一个音符我们需要了解,我们称之为导音
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Okay, and you can hear him sort of-- he didn't go-- You could hear him slide that leading tone up into the tonic in that case.
你可以听到,他并没有,大家可以听出,他是从导音,滑到了主音
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The tempo here is very slow and he slows it down even more and then it just sits out there, this huge wonderful voice that he had, on the leading tone.
速度非常缓慢,经他处理后就更慢了,余音缭绕,他在导音上展现了洪亮,精妙的歌喉
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So, here's Luciano Pavarotti singing a leading tone.
卢西亚诺·帕瓦罗蒂演唱的导音
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The leading tone is always the seventh degree.
导音总是第七级音
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