Herb Gross Uses Limit Concept to Resolve Zeno's Paradox

赫伯·格罗斯运用极限概念化解芝诺悖论 微积分化解了关于运动与面积的古老悖论，因为极限概念使数学家能够在不除以零的情况下计算瞬时变化率与无穷级数和。 赫伯·格罗斯教授在介绍《重温微积分》（Calculus Revisited）时概述了该课程的架构，将其划分为教学“硬件”（讲座、教材、补充讲义与学习指南）和概念“软件”。他将微积分阐述为融入了极限概念的高中数学，解释了微分学如何解决0/0的不定型问题以定义瞬时速度与切线斜率。随后，他通过古希腊的穷竭法探讨了积分学，并揭示了微积分基本定理如何将变化率与面积累积联系在一起。最后，格罗斯利用无穷级数的收敛性化解了芝诺悖论，并概述了从集合论到高等微积分应用的演进过程。 麻省理工学院的《重温微积分》结合了讲座、教材、补充讲义和结构化学习指南，将直观的概念概述与严谨的数学推导连接起来。 (0:00) 微积分通过极限概念拓展了高中数学，克服了0/0不定型问题以定义瞬时速度和连续变化率。 (3:00) 古希腊用于计算面积的穷竭法与牛顿用于计算变化率的微分学，通过微积分基本定理在本质上被统一为互逆运算。 (18:55) 芝诺的“龟兔赛跑”悖论展示了无穷级数的概念难题，而现代微积分通过证明递减项的无穷和可以收敛为有限量化解了这一悖论。 (26:27)

Professor Herb Gross introduces Calculus Revisited by characterizing calculus as elementary mathematics augmented by the limit concept, an analytical device that resolves foundational paradoxes of motion and geometry. He demonstrates that determining instantaneous rates of change and computing accumulated areas are inverse operations united by the Fundamental Theorem of Integral Calculus. Differential Rates and the Indeterminate Form 0/0 (3:00) Calculating instantaneous speed exposes a foundational arithmetic dilemma: at an exact point, an object is statically located, and collapsing two physical observers onto that single point forces the speed formula into the ratio 0/0. Gross explains that 0/0 is indeterminate rather than undefined because division is the inverse of multiplication; any number multiplied by zero yields zero, meaning any value satisfies 0/0. To avoid this arithmetic breakdown, calculus replaces infinite physical observers with continuous functions, such as Galileo’s falling-body law s = 16t^2, and evaluates the slopes of secant lines as time intervals shrink. Differential calculus formally defines instantaneous speed as the limit of the change in distance over the change in time as time elapsed approaches, but never reaches, zero. Gross emphasizes that this core definition remains permanent across the entire curriculum. While the arithmetic, geometric, and algebraic manipulation grows demanding for intricate physical relationships, the underlying mathematical mechanism of the derivative never changes. Area Accumulation and the Method of Exhaustion (13:42) Integral calculus originated around 600 BC with the ancient Greek method of exhaustion, designed to find the area under curves such as the parabola y = x^2 on the interval from 0 to 1. By partitioning the base interval into n segments, geometers bounded the true area between inscribed rectangular sums (an underestimate based on subinterval minimums) and circumscribed rectangular sums (an overestimate based on maximums). Increasing n squeezes the region from within and without until both bounds converge to an exact limit L. Gross translates this geometric technique into physical mechanics by relabeling the axes to represent velocity v = t^2 and time t. Because distance equals speed multiplied by time over discrete slices, the area under the velocity curve precisely measures the total distance traversed. Differential calculus defines instantaneous rates of change, whereas integral calculus determines total accumulation by calculating the area under a continuous rate curve. Unification Through the Fundamental Theorem (18:55) More than two millennia separated the Greek method of exhaustion from Isaac Newton’s formulation of differential calculus around 1690 AD. Gross stresses that the intellectual power of calculus does not rest merely on having separate procedures for slopes and areas, but on the discovery that differentiation and integration are exact inverse operations. Gross visualizes this union by tracking an area being swept out beneath a curve by an advancing vertical boundary. As this line moves along the horizontal axis, the instantaneous rate at which new area accumulates is determined directly by the vertical height of the curve at that boundary. This relationship, formalized as the Fundamental Theorem of Integral Calculus, links rates of change directly to cumulative enclosed area. Convergence and Resolving Zeno's Paradox (24:19) Both branches of calculus depend on evaluating infinite processes, raising the core question of how an infinite sum behaves. Gross demonstrates that a sum of shrinking quantities can diverge if grouped components repeatedly sum to 1 (such as adding 1/n, n times). Conversely, an infinite geometric series with a common ratio of 1/2 continually halves the remaining distance to 2, proving that an infinite number of additions can converge to a finite total. This convergence mathematically resolves Zeno’s paradox of the Tortoise and the Hare. Zeno asserted that a Hare running at 1 yard per second could never overtake a Tortoise running at 1/2 yard per second with a 1-yard head start, because the Hare must continually cross an infinite sequence of diminishing intervals (1, 1/2, 1/4 seconds). Gross resolves this with basic algebra: equating travel times via x / 1 = (x - 1) / (1/2) yields x = 2 yards in 2 seconds. The physical paradox dissolves because the infinite series of time increments sums to a finite duration. Gross outlines the structural organization of the course, which combines lectures, a textbook, supplementary notes, and a central study guide. The curriculum systematically advances from set theory and functions into limits, derivatives, integrals, and applied engineering problems, ultimately returning full circle to the formal analysis of infinite series.

MIT教授Herb Gross指出，微积分本质上是由“极限”思想强化的初等数学，其核心使命在于借助极限工具化解诸如“0/0”等未定式危机，并将瞬时变化率（微分）与累积总量（积分）统一为互逆的知识整体。 课程体系设计：教学“硬件”与概念“软件” (0:00) Herb Gross将本课程分解为教学工具（硬件）与概念基石（软件）两大维度。硬件系统由四个模块构成：讲座负责提供宏观视野与直观洞察；教材提供严格的理论推导；补充讲义弥补教材与讲座之间的衔接缝隙；学习指南则是全套体系的“骨干”，提供单元规划、诊断性预测试、模块期末测试与启发性练习题。在概念“软件”层面，Gross强调微积分的基石仍是中学初等数学，但引入了革命性的极限概念。课程将以现代数学的语言——“集合”为起点，逐步过渡至函数关系，确立导数与积分的普适定义，最终延伸至物理工程应用与无穷级数。 微分学与瞬时速度：化解“0/0”未定式危机 (3:00) 微分学源于非匀速运动中对“瞬时速度”的精确界定。经典公式“距离 = 速度 × 时间”仅适用于恒速运动；若速度实时变动，该公式便无法直接使用。若尝试通过缩短两名观察者O1与O2之间的间距来逼近某位置点P的瞬时速度，当观察者最终在P点重合时，距离差与时间差均变为0，计算将崩溃为0/0。Gross强调，0/0在数学上属于“未定式”而非单纯的无定义，因为根据除法作为逆乘法的定义，任何数值乘以0均得0，故0/0可对应任意结果。 为规避除以零，数学以解析“函数”（例如伽利略自由落体定律s = 16t^2）取代无限多对实体观察者。在几何上，平均速度对应连接两点的割线斜率；微分学则将时间增量Δt无限趋近于0但绝不等于0时的比值极限定义为瞬时变化率与切线斜率。Gross指出，无论函数形式多么复杂，瞬时变化率的核心定义在整个微积分中保持恒定，演变的仅是求极限时所需的代数或三角运算难度。 积分学与穷竭法：几何面积向物理位移的映射 (13:42) 积分学的源头可追溯至约公元前600年的古希腊几何学，其初衷是求解曲线下方的面积。Gross以区间[0, 1]上的抛物线y = x^2为例，阐明古希腊人提出的“穷竭法”：将底边划分为n个等分子区间，取各区间曲线最低点构建内接矩形族，其面积和构成下界An；取最高点构建外切矩形族得到上界An。随着划分数n趋于无穷，内接与外切矩形从内外两侧挤压目标区域，两界最终收敛至同一公共极限L，即为精确面积。 Gross强调，该几何构型通过重新定义坐标轴可直接解释物理现象。若将横轴设为时间t、纵轴设为速度v = t^2，则在各子区间内利用最小速度与最大速度计算的矩形面积，正好对应质点行进位移的下限与上限。当时间分片无限细化，矩形面积累加的极限即为质点经历的总位移。这表明微分学所关注的瞬时速度，与积分学所处理的在速度曲线下求累积位移，具有内在的对应逻辑。 微积分基本定理：统一变化率与累积量的桥梁 (18:55) Gross回顾指出，古希腊穷竭法（约公元前600年）与牛顿等人创立微分学（约公元1690年）之间相隔了两千三百年。微积分的核心成就不仅在于分别发明了求切线斜率与求面积的技术，更在于证明了两者互为逆运算。若设想一条垂直边界线沿水平轴匀速向右推移并扫出曲线下方的动态面积，面积累积的瞬时变化速率完全取决于该边界处曲线的垂直高度。这一发现即为“微积分基本定理”，它正式宣告了微分学与积分学的统一。 两大分支在底层均受极限概念支配。在微分几何中，切线并非如16世纪数学家所言“穿过曲线上相邻两点”——光滑曲线上不存在离散的相邻点，两点重合即无法确定直线；现代微积分通过割线随第二点无限逼近第一点时的极限斜率解决了这一悖论。在积分学中，极限则表现为无限项矩形面积和的收敛。极限概念由此成为弥合几何直观与严格算术的关键纽带。 级数收敛性与芝诺悖论：无限项求和的有限性 (24:19) 针对积分中“无限细分相加到底有多大”的疑问，Gross对比了发散与收敛两类无穷级数。若将3个1/3、4个1/4直至n个1/n相加，由于每个分组求和恒为1，整体将无界递增至无穷大；但在几何级数1 + 1/2 + 1/4 + 1/8 + ...中，每一项依次减半，如同在2英寸长的线段上每次跨越剩余距离的一半，步骤虽有无穷多个，总和却严格收敛于有限值2。 Gross以此引申至公元前芝诺提出的“乌龟与兔子”悖论：兔子以1码/秒追赶领先1码、速度为1/2码/秒的乌龟，追赶每个落后距离所需的时间恰好对应上述几何级数（1秒、1/2秒、1/4秒……）。初等代数通过方程x/1 = (x - 1)/(1/2)可轻易算出兔子在2秒后、位移2码处追上乌龟。Gross总结道，芝诺悖论的困惑不在于物理现实，而在于古典思维难以理解“无限多个动作如何累加出有限的总体结果”，而这正是微积分级数理论所解答的核心问题。 Gross总结表示，微积分的根本分析框架始终一致：无论面对多么复杂的函数关系，瞬时变化率与累积总量的基本定义绝不改变，改变的只是代数计算的复杂度。作为后续铺垫，课程将首先以集合论作为现代数学的底层语言展开学习，进而系统深入函数、极限与微积分的完整体系。

## Background 本内容由 MIT OpenCourseWare 于 2011 年 5 月 6 日在 YouTube 平台发布，主讲人为 Herb Gross。这是《重温微积分：单变量微积分》（Calculus Revisited: Single Variable Calculus）经典系列课程的先导序言讲座（Preface），旨在为整门课程的教学资源架构与核心数学思想奠定基础。 ## TL;DR Herb Gross 教授在开篇首先介绍了由视频讲座、主教材、补充讲义和学习指南构成的“硬件”学习闭环。随后，他将微积分概括为“高中数学加上额外的极限概念”，从匀速运动公式的失效切入，深入剖析了瞬时速度中 0/0 不定式的代数本质及几何图象中的割线斜率映射。最后，他系统梳理了源自古希腊穷竭法的曲线下面积问题，通过微积分基本定理将导数与积分融为一体，并借助两个对比级数化解芝诺悖论，清晰擘画了从集合论到无穷级数的完整认知蓝图。 ## Atmosphere & style 讲座洋溢着老派 MIT 课堂特有的质朴与智慧，Herb Gross 以幽默亲切的语调开场，举重若轻地消除学生对高等数学的畏难情绪。他的讲解兼具严密的代数推演与生动的直觉比喻，从黑板前设想的两名观测员到龟兔赛跑悖论，层层递进、环环相扣。整堂课既有拨云见日的顿悟感，又充盈着古典数学思想跨越两千余年交相辉映的深邃与宏大。 ## Bulletin - [03:00](https://www.youtube.com/watch?v=MFRWDuduuSw&t=180s) 微积分本质上是初等高中数学与额外引入的极限概念的结合，旨在精确解决非匀速运动中瞬时速度与曲边图形面积等经典算术无法处理的问题。 - [05:58](https://www.youtube.com/watch?v=MFRWDuduuSw&t=358s) 在计算瞬时速度时直接让两观测点重合会遭遇 0/0 不定式，其比值受分子分母相对微小程度支配，既可能极大也可能极小，必须借助极限过程来规避。 - [16:16](https://www.youtube.com/watch?v=MFRWDuduuSw&t=976s) 起源相隔 2200 多年的古希腊穷竭法（积分学雏形）与 17 世纪牛顿时代的微分学，通过微积分基本定理紧密结合，确立了变化率与面积累积之间的对偶互逆关系。 - [26:27](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1587s) 针对芝诺龟兔赛跑悖论，通过无穷几何级数求和证明无限多个递减的时间项可以收敛为一个有限数值（2 秒），从数学上化解了直观与逻辑的矛盾。 ## Key takeaways - [00:00](https://www.youtube.com/watch?v=MFRWDuduuSw&t=0s) 课程教学体系由视频讲座、主教材、补充讲义与学习指南四大部分构成，讲座侧重阐述宏观机理与应用直觉，教材与讲义负责深入理论细节。 - [03:00](https://www.youtube.com/watch?v=MFRWDuduuSw&t=180s) 初等数学公式“距离 = 速度 × 时间”仅适用于匀速运动，面对变化速度时传统算术失效，促使微分学诞生以解决瞬时速度问题。 - [05:58](https://www.youtube.com/watch?v=MFRWDuduuSw&t=358s) 通过不断缩短两名观测员间距逼近瞬时速度时，两点若完全重合将导致距离与时间均为零，陷入除法未定义的 0/0 不定式危机。 - [07:15](https://www.youtube.com/watch?v=MFRWDuduuSw&t=435s) 两个极微小量之比并不必然很小，其数值完全不可预估，因此代数上不能直接用 0 除以 0，而必须采用无限逼近但永不相等的极限机制。 - [09:10](https://www.youtube.com/watch?v=MFRWDuduuSw&t=550s) 数学函数相当于在每一个连续时间点安放一名虚拟观测员，从而在几何上将抽象运动速率映射为直角坐标系中割线的倾斜斜率。 - [12:19](https://www.youtube.com/watch?v=MFRWDuduuSw&t=739s) 微分学核心定义写为当时间改变量趋近于零时位移改变量与时间改变量比值的极限，该定义在整个微积分学习中保持恒定，变化仅为代数计算量。 - [13:42](https://www.youtube.com/watch?v=MFRWDuduuSw&t=822s) 古希腊穷竭法通过构建内接矩形和（下逼近）与外切矩形和（上逼近）进行两极夹逼，证明了曲边面积可以通过有限近似的极限来精确确定。 - [16:16](https://www.youtube.com/watch?v=MFRWDuduuSw&t=976s) 在速度-时间图（v-t 图）中，曲线下面积代表质点运动的物理距离，使得纯几何面积概念在工程与物理学中获得直观意义。 - [18:50](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1130s) 微积分基本定理揭示了微分与积分的深刻对偶：当垂直线扫过图形时，面积累积的瞬时变化率恰恰等于曲线上边界的高度。 - [21:02](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1262s) 几何学中曲线上不存在真正的“相邻两点”，切线并非穿过重合单点的直线，而是割线端点无限逼近时的极限状态。 - [24:19](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1459s) 无限多个逐渐趋向于零的项相加既可能发散到无穷，也可能收敛到有限常数，项数的无限性不能决定总和的大小。 - [26:27](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1587s) 芝诺悖论的实质在于人类直觉难以理解“无限多步操作可以在有限时间内完成”，公比为 1/2 的几何级数收敛为 2 解释了兔子必然追上乌龟。 - [29:48](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1788s) 整门微积分课程遵循以集合论为数学语言、函数为映射骨架，递进推导极限、导数、积分，最终收敛于无穷级数的认知阶梯。 ## Detailed walkthrough ### [00:00](https://www.youtube.com/watch?v=MFRWDuduuSw&t=0s) 课程硬件架构：从讲座、教材到学习指南 讲师 Herb Gross 以轻松幽默的语调开启了《重温微积分》（Calculus Revisited）的第一堂课。他调侃道，任何课程最难讲的往往都是第一讲，讲师总会忍不住看着台下的听众说：“你们一定很纳闷，我为什么把大家召集到这里。”为此，他将本讲定名为“序言”（Preface），旨在从“硬件”（课程配套构件）与“软件”（核心数学思想）两个维度，为整门课程提供全景式的概览。 ```mermaid flowchart TD accTitle: 课程硬件组成架构 accDescr: 展示讲座、教材、补充讲义与学习指南四者构成的学习闭环 subgraph Core ["核心教学资源"] L["讲座 (Lectures)<br/>建立全局概览与应用直觉"] T["主教材 (Textbook)<br/>提供严谨深入的知识体系"] end subgraph Support ["学习支持与评估"] N["补充讲义 (Supplementary Notes)<br/>填补教材缝隙与进阶细节"] G["学习指南 (Study Guide)<br/>单元划分、前测、终考与学习习题"] end L --> T T --> N N --> G G -->|"反馈驱动深入理解"| L ``` 在“硬件”层面，整套教学方案由四个紧密咬合的板块构成：首先是系列视频讲座，其核心目的并非死板推导，而是提供宏观视角、揭示计算背后的机理，并展现概念的具体应用方式；其次是作为核心的主教材，用来弥补讲座受时间限制而无法深入的理论细节；再次是补充讲义（supplementary notes），专门填补教材中可能遗留的知识空白；最后是作为全书骨架的学习指南（study guide）。针对学习指南的功用，Herb Gross 强调：“学习指南包含课程的结构拆解，它会明确各堂讲座与学习单元，提供诊断准备情况的前测（pretest），并在每个模块结束时设置期末测验；更重要的是，从工程应用的角度来看，指南中编排的‘学习习题’（learning exercises）能够激发应用欲望，并作为检验理论重要性的跳板。” - 讲座定位：阐明“为什么这样计算”（why）以及“如何落地应用”（how）。 - 学习指南功能：包含预备知识摸底的前测、知识模块结课考试（final examination），以及连接工程实战与数学理论的学习习题。 --- ### [03:00](https://www.youtube.com/watch?v=MFRWDuduuSw&t=180s) 从高中平均速度到微积分的瞬时速度 转入“软件”层面后，Herb Gross 抛出了核心设问：微积分到底是什么？他给出了一个极为平实质朴的界定——微积分本质上就是“高中数学加上一个额外的极限（limit）概念”。在初等数学中，人们习惯处理匀速运动，经典公式“距离 = 速度 × 时间”暗含了速度恒定不变的前提；然而一旦速度随时间变化，由于无法确定该用哪一个速度去乘时间，旧有的公式立刻失效。微分学（differential calculus）正是为了解决“瞬时速度”（instantaneous speed）而诞生的。 为了直观探讨瞬时速度，Herb Gross 设想了一个物体沿着直线运动并穿过点 P 的物理场景：当询问物体恰好在点 P 这一瞬间的速度有多快时，思维会陷入悖论——因为在位于点 P 的那一刹那，物体在某种意义上根本没有位移。为了把新问题转化为已知的初等问题，物理直觉促使人们在点 P 前后各安插一名观测员（$O_1$ 与 $O_2$）。测量出两名观测员之间的距离以及物体穿过该区间所需的时间后，二者相除即可得到高中意义上的平均速度。 面对可能出现的质疑，Herb Gross 模拟了一段对话：“有人可能会说，哎呀，这个答案很精彩，但算错问题了；我们问的不是从 $O_1$ 到 $O_2$ 的平均速度，而是那一瞬间的瞬时速度。”他解释道，只要两名观测员彼此足够靠近，区间内留给速度剧烈变化的距离就极短，此时平均速度就会成为瞬时速度极佳的近似；于是人们顺理成章地设想：不断缩短两位观测员的间距，当两名观测员重合在点 P 时，似乎就能迎来最完美的理想结果。 --- ### [05:58](https://www.youtube.com/watch?v=MFRWDuduuSw&t=358s) 0/0 不定式的数学本质与反直觉特性 然而，两名观测员彻底重合的瞬间，恰恰暴露了微积分诞生之初所面临的根本危机。Herb Gross 激动地指出：“一旦两名观测员重叠在一起，它们之间的距离变成了 0，从一个移动到另一个所需的时间也是 0；因此，如果将距离除以时间，我们将不可避免地遭遇 0/0。” 对于 $0/0$，Herb Gross 明确指出，比起“未定义”（undefined），用“不定式”（indeterminate）来形容更为确切。在日常算术中，小数的运算规律通常具有可预测性： 1. 两个极小数相加，结果依然是极小数； 2. 两个极小数相减或相乘，结果仍然极小； 3. 但两个极小数的**商**（比率）却完全不可捉摸。 为了具象化这种反直觉的特性，他列举了数值对比：若将 $10^{-6}$（即百万分之一，0.000001）除以更微小的 $10^{-12}$，由于分母相比分子实在太小，得到的结果竟是巨大的 $10^6$（一百万）；反过来，若颠倒分子分母，计算 $10^{-12} / 10^{-6}$，所得结果则是微小的 $10^{-6}$。这证明了小量之比既可能极大，也可能极小，或者处于两者之间的任何数值，其比值完全无法预先断定。 ```mermaid flowchart LR accTitle: 0除以0的不定性逻辑推导 accDescr: 从传统除法定义推导为什么0除以0可以等于任何数 A["传统除法定义<br/>a/b = c 意味着 c * b = a"] --> B["代入 a=0, b=0<br/>0/0 = c 意味着 c * 0 = 0"] B --> C{"寻找满足条件的数值 c"} C -->|"任何数乘0都得0"| D["c 可以是任意数 (Any Number)"] D --> E["结论：0/0 是不定式 (Indeterminate)，必须规避直接代入"] ``` 随后，Herb Gross 从严格的代数结构层面给出了形式化证明：依照经典除法定义，$a/b$ 表示满足“乘以 $b$ 等于 $a$”的那个数；当考察 $0/0$ 时，问题就变成了寻找一个数，使其乘以 $0$ 能够得到 $0$。针对这个逻辑推演，Herb Gross 总结道：“什么样的数具有‘乘以 0 等于 0’的性质？答案是任何数；这正是为什么 0/0 是不定式——如果你问一个人‘告诉我哪个数乘以 0 能得到 0’，回答只能是任意数。”因此，直接计算 $0/0$ 是一条死胡同，在微积分中必须不惜一切代价规避。 --- ### [09:10](https://www.youtube.com/watch?v=MFRWDuduuSw&t=550s) 函数模型与斜率的几何映射 为了绕开 $0/0$ 的陷阱，物理直觉方案必须调整为：让两名观测员无限靠近，但**永远不要真正重合**。这种妥协带来了一个新的数学挑战——只要两名观测员之间还存在微小缝隙，理论上其间就能再塞入一对更接近的观测员。这意味着物理世界里需要“无穷多对观测员”才能逼近极值，而实际操作中无法雇佣无限多的人。 微积分给出的精妙解法，正是引入“函数”（function）的概念。Herb Gross 以伽利略的自由落体运动公式 $s = 16t^2$（其中 $t$ 为秒，$s$ 为英尺）为例说明：这个看似平淡无奇的算式，实质上构建了一个完美的数学机器。在每一个给定的时刻 $t$，都能即时计算出对应的下落位移 $s$；也就是说，这条数学法则在物理上等效于在每一个连续的时间点上都布设了一名忠实的观测员。 ```mermaid flowchart TD accTitle: 函数对应与图象斜率映射 accDescr: 展示从函数输入输出到直角坐标系图象及割线斜率的映射过程 subgraph FunctionMachine ["函数机制：等效于无穷多对观测员"] In["输入: 时间 t (秒)"] --> Rule["法则: s = 16t²"] Rule --> Out["输出: 位移 s (英尺)"] end subgraph Geometry ["几何图象映射"] AxisX["横轴: 自变量 t"] AxisY["纵轴: 应变量 s"] Curve["抛物线函数图象"] Slope["割线斜率: Δs / Δt<br/>物理意义 = 平均速度<br/>几何意义 = 割线倾斜度"] end Out --> AxisY In --> AxisX AxisX & AxisY --> Curve Curve --> Slope ``` 函数的代数关系随即转化为几何图象（graph）：将时间 $t$ 视为输入排布在水平轴上，位移 $s$ 视为输出排布在与之垂直的垂直轴上，就形成了直观的可视化图线。在坐标系中，给定时间区间 $\Delta t$ 内产生的位移改变量 $\Delta s$，两者的比值 $\Delta s / \Delta t$ 便是该区间上的平均速度。Herb Gross 提炼了两者的对偶关系：“换句话说，平均速度之于函数，正如直线的斜率之于几何。”通过图象中的割线斜率，抽象的运动速率被赋予了清晰的几何形状，也为进一步利用切线斜率定义瞬时速度铺平了道路。 ### [12:19](https://www.youtube.com/watch?v=MFRWDuduuSw&t=739s) 微分学的基石：瞬时变化率与极限定义 Herb Gross 教授深入探讨了如何用数学语言精确表达“瞬时变化率”。我们考察距离改变量除以时间改变量的比值，并观察当时间间隔趋近于零时的表现，即写成极限形式：$\lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}$。这里的时间改变量 $\Delta t$ 变得任意小，但绝不能等于 0，以避免陷入无意义的 $\frac{0}{0}$ 形式。这正是整个微分学的核心工作定义。 这个定义的美妙之处在于其普遍性：它并不依赖于具体的运动方程（例如自由落体公式 $s = 16t^2$），距离 $s$ 可以是时间 $t$ 的任意函数。面对更加复杂的现实情境，底层的基本定义始终如一。改变的仅仅是处理 $s$ 与 $t$ 之间关系所需的运算量。Gross 强调，即使学到课程的最后阶段，瞬时变化率的这一定义也绝不会改变，变化的是为了从数值上计算出结果而必须运用的算术、代数、几何或三角函数技巧。 > “The important point is what? The basic definition stays the same. What changes is the amount of arithmetic that's necessary to handle the particular relationship between s and t.” —— Herb Gross --- ### [13:42](https://www.youtube.com/watch?v=MFRWDuduuSw&t=822s) 古希腊的穷竭法与积分学起源 微积分的第二大分支是积分学，其起源远早于微分学，早在公元前 600 年左右就由古希腊学者逐步奠定，旨在解决曲线下方所围图形的面积问题。Gross 以抛物线 $y = x^2$ 在区间 $[0, 1]$ 上构成的曲边三角形区域 $R$ 为例，介绍了古希腊人极具智慧的几何方案——“穷竭法”（method of exhaustion）。 穷竭法的核心操作机制可归纳如下： 1. **划分区间**：将基底区间划分为 $n$ 等份。 2. **内接矩形（下逼近）**：在每个子区间取函数最低点做内接矩形，由“底乘高”累加出总面积，记作 $\underline{A}_n$。该值完全落在区域 $R$ 内部，因此必然小于实际面积。 3. **外切矩形（上逼近）**：在每个子区间取函数最高点做矩形并累加，记作 $\overline{A}_n$。此值必然大于真实面积。 4. **两极夹逼**：区域 $R$ 的真实面积被严格夹在 $\underline{A}_n$ 和 $\overline{A}_n$ 之间。当划分份数 $n$ 不断增加时，下逼近从内部逐渐填满空隙而增大，上逼近从外部不断削去多余部分而减小。若上下界最终收敛于同一个极限值 $L$，则区域 $R$ 的面积必定就是 $L$。 ```mermaid flowchart LR accTitle: Method of Exhaustion Convergence accDescr: Diagram showing lower and upper Riemann-style bounds squeezing the true area under a curve. subgraph Bounds ["穷竭法逼近过程"] direction TB Lower["下逼近 A_n (内接矩形和，逐步增大)"] Upper["上逼近 Ā_n (外切矩形和，逐步减小)"] end Target["真实面积 R"] Limit["极限值 L"] Lower -->|"从下方夹逼"| Target Upper -->|"从上方夹逼"| Target Target -->|"收敛判定"| Limit ``` - 穷竭法是一门拥有约 2500 年历史的古老技术。 - 它在直观上展示了通过有限近似的序列极限来确定不规则几何图形大小的方法。 --- ### [16:16](https://www.youtube.com/watch?v=MFRWDuduuSw&t=976s) 从纯几何到物理世界：曲线下面积的物理诠释 面对许多工科和物理系学生的疑问——“我又不是几何学家，关心曲线下面积有什么用？”——Gross 指出，只要将坐标轴赋予具体的物理量，曲线下的面积便立刻拥有了鲜活的物理意义。将坐标从 $(x, y)$ 替换为时间与速度 $(t, v)$，考虑质点在 $t \in [0, 1]$ 期间以速度 $v = t^2$ 运动的情形。 在速度-时间图（$v$-$t$ 图）中，曲线下的面积恰好等于质点在该时间段内移动的距离。如果同样将时间区间划分为 $n$ 等份并绘制矩形，每个矩形的面积（速度 $\times$ 时间）就代表质点在某一恒定速度假设下走过的路程。由于实际运动速度在每个微区间内都高于左端点的瞬时速度，所有内接矩形面积之和必然小于实际行进路程。如同几何面积被上下界夹逼一样，质点所走过的物理路程也完全能通过这种极限方式被精准确定。 ```mermaid flowchart TD accTitle: Unified Calculus Architecture accDescr: Diagram illustrating how differential and integral calculus are linked by limits and the Fundamental Theorem. subgraph Branches ["微积分的两大支柱"] Diff["微分学<br>(牛顿/莱布尼茨，约公元 1690 年)<br>瞬时变化率 / 速度"] Integ["积分学<br>(古希腊人，约公元前 600 年)<br>曲线下面积 / 累积位移"] end Core["核心支撑：极限概念 (Limit Concept)"] FTC["微积分基本定理 (Fundamental Theorem of Calculus)"] Diff --> Core Integ --> Core Diff <-->|"面积变化率 = 曲线高度"| FTC Integ <-->|"互为逆运算"| FTC ``` 微积分真正的惊艳之处，并非仅仅在于分别发明了“瞬时速率”与“曲边面积”的解法，而在于将两门相隔了两千两百多年的学科熔铸在一起的深刻联系——微分学直到 1690 年代牛顿时代才确立，而它与古希腊积分学的融合构成了“微积分基本定理”。当一条坚线向右移动扫过曲面时，面积累积的快慢（面积的瞬时变化率）恰恰取决于该处曲线上边界的高度。 > “The true beauty lies in the fact that these apparently two different branches of calculus, one of which was invented by the Ancient Greeks as early as 600 BC, the other of which--differential calculus--was not known to man until the time of Isaac Newton in 1690 AD are related by a rather remarkable thing.” —— Herb Gross --- ### [21:02](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1262s) 两种极限形态：切线问题与无穷和 Gross 总结道，无论是微分学还是积分学，微积分通篇都依托于底层的“极限概念”。他从几何直观上剖析了极限的两种表现形态： 第一种形态是连续变化的割线斜率极限（切线问题）。若想求曲线在 $P$ 点的切线，可先在曲线上任取一点 $Q$，作出割线 $PQ$ 并求其斜率。随着点 $Q$ 沿着曲线一步步向 $P$ 逼近（经由 $Q_1, Q_2, \dots$），割线的斜率越来越接近切线斜率。16 世纪的人们曾给切线下过一个耐人寻味的定义：“切线是穿过曲线上两个‘相邻点’（consecutive points）的直线”。从严格的几何观点来看，平滑曲线上根本不存在所谓“相邻的两个点”。两点一旦完全重合便只剩下一个点，而单点无法确定一条直线；只要两点相异，就依然只是割线。古人用“两个相邻点”所极力描述的，实质上正是点 $Q$ 无限逼近点 $P$ 时的极限状态。 第二种形态是离散求和的无限项极限（面积与级数问题）。在利用矩形逼近面积时，我们将区间分成 $n$ 等份，然后让 $n$ 无界增加。此时需要将无限多个离散矩形的面积相加。这一问题引出了一个深刻的数学叩问：“一个无穷和到底有多大？”Gross 提醒大家，“无穷和”仅说明了相加的项数是无穷的，却完全没有说明每一项的大小，更无法直接断言相加后的总和会是多大。为了直观展现无穷求和的奥秘，他给出了一个具体的序列实验：从数字 1 开始，接着连续加上两次 $\frac{1}{2}$…… > “Now, obviously, a curve does not have two consecutive points. What they really meant was what? That as Q gets closer and closer to P, the secant line becomes a better and better approximation for the tangent line, and that in a way, if the two points were allowed to coincide, that should give us the perfect answer.” —— Herb Gross - 确定一条唯一平面直线必须依赖两个不同的点；一旦割线端点完全重叠成单点，斜率计算公式就会退化为无定义的 $\frac{0}{0}$。 - 无穷和（Infinite sum）刻画的是项数无止境累加的动态过程，其最终结果既可能发散到无穷，也可能收敛于有限值。 ### [24:19](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1459s) 两个对比级数：发散与收敛 赫伯·格罗斯（Herb Gross）首先通过构建一个具体的级数来说明无穷加和的性质。他展示了将 1/2 加两次、1/3 加三次、1/4 加四次、1/5 加五次、1/6 加六次以此类推的过程。在这一构造中，虽然每一项的数值都在逐渐变小，但每一组相同项相加的结果都恒等于 1。因此，只要不断往后延伸并累加足够多的 1，总和便会无界增长，变得想要多大就有多大。 紧接着，格罗斯给出了一个截然相反的例子：$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \dots$。在这个公比为 1/2 的几何级数（等比级数）中，每一项都是前一项的一半。格罗斯借助几何直观进行了解释：设想一条长度为 2 英寸的线段，先走一半（1 英寸），再走剩下的一半（1/2 英寸），接着再走剩余距离的一半（1/4 英寸），然后继续走剩余的一半（1/8 英寸、1/16 英寸……）。无论在何处停下，距离终点 2 英寸的位置都会越来越近，却永远不会在有限步内真正触达该点。 格罗斯总结了这一鲜明的反差：“在这里，有无穷多个项，其无穷级数的和为 2；而在前一个例子中，同样有无穷多个项，但其无穷级数的和却是无穷大，或者更确切地说，它无界增长。”（“In other words, here are infinitely many terms whose infinite sum is 2. Here are infinitely many terms whose infinite sum is infinity, we should say, because it increases without bound.”）这正是长期困扰古希腊学者的核心难题：人究竟如何在有限的时间内完成无穷多件事？ ### [26:27](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1587s) 芝诺悖论：阿喀琉斯与乌龟（龟兔赛跑） 古希腊人在探索积分学的同时，著名哲学家芝诺（Zeno）提出了著名的“芝诺悖论”。芝诺共有三个著名悖论，格罗斯在此专门引用了最能说明直观困境的“龟兔赛跑”（即阿喀琉斯与乌龟）问题。格罗斯为了贴合前述的数学模型，假设这是一只跑得较慢的兔子和一只跑得较快的乌龟，乌龟先获得 1 码的起跑优势，而兔子的奔跑速度恰好是乌龟的两倍。 依据芝诺的逻辑，兔子永远追不上乌龟。因为兔子要追上乌龟，必须先跑完乌龟原先领先的 1 码；但当兔子跑完这 1 码时，由于乌龟的速度是兔子的一半，乌龟又往前挪动了 1/2 码；当兔子补上这 1/2 码时，乌龟又前进了 1/4 码；等兔子再追完这 1/4 码，乌龟又走了 1/8 码。芝诺以此断定兔子只会无限逼近乌龟，却永远无法完成超越。格罗斯指出，悖论的本质在于“它看似言之成理，但结论显然是错误的”。芝诺本人也明知现实中兔子必然会超过乌龟，但他无法从数学和逻辑上清晰解释其中机理。 格罗斯用时间量度化解了这一悖论：假设兔子奔跑的速度是 1 码/秒（乌龟为 1/2 码/秒），那么追赶各阶段所消耗的时间依次是 1 秒、1/2 秒、1/4 秒、1/8 秒……这些时间间隔累加起来的总和恰好是 2 秒。如果改用初中八年级的代数方法，设追及点距离起点为 $x$ 码，根据“路程除以速度等于时间”列出方程： $$\frac{x}{1} = \frac{x - 1}{\frac{1}{2}}$$ 解此一元一次方程可平平凡凡地得出 $x = 2$ 码。在现实世界中，兔子在追上乌龟之前确实不会完成超车，这是显而易见的常识；格罗斯强调，真正深奥且不易直观察觉的本质在于：“这无穷多个项加起来竟然可以得到一个有限的数值。”（“That these infinitely many terms can add up to a finite sum.”） ### [29:48](https://www.youtube.com/watch?v=MFRWDuduuSw&t=1788s) 课程全景回顾与未来路线 在剖析完无穷级数与芝诺悖论后，格罗斯回顾了整门课程的宏观架构与认知阶梯。这门单变量微积分课程将从现代集合论视角下的函数概念讲起，因为函数本质上是不同对象集合之间的映射关系。随后，课程将以此为基石，依次建立极限、导数（变化率）以及积分（曲线下方围成的面积）的核心框架。 一旦这些基础构件成型，其内在逻辑便恒定不变。随后课程将进入关键的拓展与应用阶段——格罗斯特别提到，应用正是工程领域的灵魂所在。面对更复杂的函数关系，课程将引入更精妙的高阶数学技巧，并最终落脚在无穷级数这一压轴主题上，系统阐述如何严谨处理“无限多个逐渐趋向于零的微小量相加”的数学规律。 格罗斯在下课前向学生交代了接下来的学习节奏：接下来的几节课将作为前置铺垫，重点讲解集合论知识，学生可以结合补充讲义自主阅读。格罗斯指出：“请努力理解这些内容，因为集合的概念是构筑现代数学大厦的基石，也是现代数学的基础语言。”（“Learn to understand these because the concept of a set is the building block, the fundamental language of modern mathematics.”）在打牢集合论基础之后，课程将重新切回函数概念，循序渐进地以慢镜头方式，将本节先导课中勾勒的宏伟蓝图一一详尽拆解实现。 ```mermaid flowchart TD accTitle: 单变量微积分课程架构全景图 accDescr: 展示从集合与函数基础出发，经由核心概念到高级应用与级数的完整教学路径。 subgraph Phase1["第一阶段：核心理论基石"] A["集合理论<br/>(现代数学语言)"] --> B["函数映射"] B --> C["极限与导数<br/>(变化率)"] B --> D["定积分与微积分基本定理<br/>(曲边梯形面积)"] end subgraph Phase2["第二阶段：工程与高级技巧"] C & D --> E["复杂函数建模与高级分析技巧"] E --> F["工程实际应用<br/>(解决现实问题)"] end subgraph Phase3["第三阶段：无穷理论"] F --> G["无穷级数<br/>(无限多微小量累加)"] end ``` ## Notable quotes > [09:10](https://www.youtube.com/watch?v=MFRWDuduuSw&t=550s) 平均速度之于函数，正如直线的斜率之于几何。 — Herb Gross > [05:58](https://www.youtube.com/watch?v=MFRWDuduuSw&t=358s) 什么样的数具有‘乘以 0 等于 0’的性质？答案是任何数；这正是为什么 0/0 是不定式——如果你问一个人‘告诉我哪个数乘以 0 能得到 0’，回答只能是任意数。 — Herb Gross ## Mentioned Herb Gross, MIT OpenCourseWare, Calculus Revisited, 艾萨克·牛顿 (Isaac Newton), 戈特弗里德·威廉·莱布尼茨 (Gottfried Wilhelm Leibniz), 伽利略·伽利莱 (Galileo Galilei), 芝诺 (Zeno of Elea) ## Video description Preface Instructor: Herb Gross View the complete course: http://ocw.mit.edu/RES18-006F10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu [Source](https://www.youtube.com/watch?v=MFRWDuduuSw)

[9:10] Herb Gross: 平均速度之于函数，正如直线的斜率之于几何。 [5:58] Herb Gross: 什么样的数具有‘乘以 0 等于 0’的性质？答案是任何数；这正是为什么 0/0 是不定式——如果你问一个人‘告诉我哪个数乘以 0 能得到 0’，回答只能是任意数。

[9:53] Herb Gross: average speed is to functions what slope of a straight line is to geometry.

[0:00] NARRATOR: The following content is provided under a [0:01] Creative Commons license. [0:03] Your support will help MIT OpenCourseWare continue to [0:06] offer high-quality educational resources for free. [0:09] To make a donation or to view additional materials from [0:12] hundreds of MIT courses, visit MIT OpenCourseWare at [0:16] ocw.mit.edu. [0:39] PROFESSOR: Hi. [0:40] I'm Herb Gross, and welcome to Calculus Revisited. [0:44] I guess the most difficult lecture to give with any [0:46] course is probably the first one. [0:48] And you're sort of tempted to look at your audience and say [0:51] you're probably wondering why I called you all here. [0:54] And in this sense, I have elected to entitle our first [0:57] lecture simply Preface to give a double overview, an overview [1:03] both of the hardware and the software that will make up [1:07] this course. [1:08] To begin with, we will have a series of lectures of which [1:13] this is the first. [1:15] In our lectures, our main aim will be to give an overview of [1:19] the material being covered, an insight as to why various [1:22] computations are done, and insights as to how [1:26] applications of these concepts will be made. [1:29] The heart of our course will consist of a regular textbook. [1:35] You see, we have our lectures. [1:37] We have a textbook. [1:40] The textbook is designed to supply you with deeper [1:44] insights than what we can give in a lecture. [1:48] In addition, recognizing the fact that the textbook may [1:53] leave gaps, places where you may want some additional [1:56] knowledge, we also have supplementary notes. [2:00] And finally, at the backbone of our package is what we call [2:05] the study guide. [2:06] The study guide consists of a breakdown of the course. [2:10] It tells us what the various lectures will be, the units. [2:14] There are pretests to help you decide how well prepared you [2:19] are for the topic that's coming up. [2:21] There is a final examination at the end of [2:23] each block of material. [2:25] And perhaps most importantly, especially from an engineer's [2:28] point of view, in each unit that we study, the study guide [2:33] will consist of exercises primarily called learning [2:37] exercises, exercises which hopefully will turn you on [2:42] towards wanting to be able to apply the material, and at the [2:46] same time, serve as a springboard by which we can [2:50] highlight why the theory and many about our lecture points [2:54] are really as important as they are. [2:56] So much for the hardware of our course. [3:00] And now let's turn our attention to the software. [3:03] Just what is calculus? [3:05] In a manner of speaking, calculus can be viewed as [3:09] being high school mathematics with one additional concept [3:13] called the limit concept thrown in. [3:16] If you recall back to your high school days, remember [3:18] that we're always dealing with things like [3:21] average rate of speed. [3:23] Notice I say average or constant rate of speed. [3:26] The old recipe that distance equals rate times time [3:29] presupposes that the rate is constant, because if the rate [3:33] is varying, which rate is it that you use to multiply the [3:36] time by to find the distance? [3:39] You see, in other words, roughly speaking, we can say [3:43] that at least one branch of calculus known as differential [3:46] calculus deals with the subject of [3:49] instantaneous speed. [3:50] And instantaneous speed is a rather easy thing to talk [3:53] about intuitively. [3:55] Imagine an object moving along this line and passing the [3:59] point P. And we say to ourselves how fast was the [4:03] object moving at the instant that we're at the point P? [4:06] Now, you see, this is some sort of a problem. [4:09] Because at the instant that you're at P, you're not in a [4:13] sense moving at all because you're at P. [4:17] Of course, what we do to reduce this problem to an old [4:20] one is we say, well, suppose we have a couple of observers. [4:24] Let's call them O1 and O2. [4:26] Let them be stationed, one on each side of P. Now, certainly [4:30] what we could do physically here is we can measure the [4:33] distance between O1 and O2. [4:37] And we can also measure the time that it takes to [4:43] go from O1 to O2. [4:46] And what we can do is divide that distance by the time, and [4:50] that, you see, is our old high school concept of the average [4:54] speed of the particle as it moves from O1 to O2. [4:58] Now, you see, the question is, somebody says gee, that's a [5:01] great answer, but it's the wrong problem. [5:03] We didn't ask what was the average speed as we [5:05] went from O1 to O2. [5:07] We asked what was the instantaneous speed. [5:09] And the idea is we say, well, lookit. [5:11] The average speed and the instantaneous speed, it seems, [5:14] should be pretty much the same if the observers were [5:17] relatively close together. [5:19] The next observation is it seems that if we were to move [5:22] the observers in even closer, there would be less of a [5:26] discrepancy between O1 and O2 in the sense that-- not a [5:31] discrepancy, but in the sense that the average speed would [5:34] now seem like a better approximation to the [5:36] instantaneous speed because there was less distance for [5:39] something to go wrong in. [5:41] And so we get the idea that maybe what we should do is [5:45] make the observers gets closer and closer together. [5:48] That would minimize the difference between the average [5:50] speed and the instantaneous rate of speed, and maybe the [5:54] optimal thing would happen when the two [5:56] observers were together. [5:58] But the strange part is-- [5:59] and this is where calculus really begins. [6:02] This is what calculus is all about. [6:04] As soon as the observers come together, notice that what you [6:07] have is that the distance between them is 0. [6:10] The time that it takes to get from one to the other is 0. [6:13] And therefore, it appears that if we divide distance by time, [6:17] we are going to wind up with 0/0. [6:21] Now, my claim is that 0/0 should be called-- [6:25] well, I'll call it undefined, but actually, I think [6:30] indeterminate would be a better word. [6:36] Why do I say that? [6:38] Well, here's an interesting thing. [6:40] When we do arithmetic with small numbers, observe that if [6:43] you add two small numbers, you expect the result to be a [6:46] small number. [6:47] If you multiply two small numbers, you expect the result [6:51] to be a small number. [6:52] Similarly, for division, for subtraction, the difference of [6:57] two small numbers is a small number. [6:59] On the other hand, the quotient of two small numbers [7:02] is rather deceptive. [7:04] Because it's a ratio, if one of the very small numbers [7:08] happens to be very much larger compared with the other small [7:12] number, the ratio might be quite large. [7:15] Well, for example, visualize, say, 10 to the minus 6, [7:19] 1/1,000,000, 0.000001, which is a pretty small number. [7:27] Now, divide that by 10 to the minus 12th. [7:32] Well, you see, 10 to the minus 12th is a small number, so [7:35] small that it makes 10 to the minus sixth appear large. [7:38] In fact, the quotient is 10 to the sixth, which is 1,000,000. [7:43] And here we see that when you're dealing with the ratio [7:46] of small numbers, you're a little bit in trouble, because [7:49] we can't tell whether the ratio will be small, or large, [7:53] or somewhere in between. [7:54] For example, if we reverse the role of numerator and [7:57] denominator here, we would still have the quotient of two [8:00] small numbers, but 10 to the minus 12th divided by 10 to [8:03] the minus sixth is a relatively small number, 10 to [8:11] the minus 6. [8:13] Of course, this is the physical way of looking at it. [8:16] Small divided by small is indeterminate. [8:18] We have a more rigorous way of looking at this if you want to [8:21] see it from a mathematical structure point of view. [8:24] Namely, suppose we define a/b in the traditional way. [8:28] Namely, a/b is that number such that when we multiply it [8:33] by b we get a. [8:36] Well, what would that say as far as 0/0 was concerned? [8:40] It would say what? [8:41] That 0/0 is that number such that when we multiply [8:45] it by 0 we get 0. [8:48] Now, what number has the property that when we multiply [8:51] it by 0 we get 0? [8:53] And the answer is any number. [8:55] This is why 0/0 is indeterminate. [8:58] If we say to a person, tell me the number I must multiply by [9:01] 0 to get 0, the answer is any number. [9:05] Well, the idea then is that we must avoid the expression 0/0 [9:08] at all costs. [9:10] What this means then is that we say OK, let the observers [9:14] get closer to closer together, but never touch. [9:17] Now, the point is that as long as the observers get closer [9:20] and closer together and never touch, let's ask the question [9:23] how many pairs of observers do we need? [9:26] And the answer is that theoretically we need [9:28] infinitely many pairs of observers. [9:31] Well, why is that? [9:32] Because as long as there's a distance between a pair of [9:35] observers, we can theoretically fit in another [9:38] pair of observers. [9:40] This is why in our course we do not begin with this idea, [9:44] but looking backwards now, we say ah, we had better find [9:48] some way of giving us the equivalent of having [9:51] infinitely many pairs of observers. [9:53] And to do this, the idea that we come up with is the concept [9:59] called a function. [10:01] Consider the old Galileo freely falling body problem, [10:05] where the distance that the body falls s equals 16t [10:09] squared, where t is in seconds and s is in feet. [10:12] Notice that this apparently harmless recipe gives us a way [10:17] for finding s for each given t. [10:21] In other words, to all intents and purposes, this recipe [10:24] gives us an observer for each point of time. [10:28] For each time, we can find the distance, which is physically [10:32] equivalent to knowing an observer at every point. [10:36] In turn, the study of functions lends itself to a [10:40] study of graphs, a picture. [10:43] Namely, if we look at s equals 16t squared again, notice that [10:47] we visualize a recipe here. [10:49] t can be viewed as being an input, s as the output. [10:54] For a given input t, we can compute the output s. [10:58] In general, if we now elect to plot the input along a [11:03] horizontal line and the output at right angles to this, we [11:07] now have a picture of our relationship, a picture which [11:11] is called a graph. [11:14] You see, we can talk about this more explicitly as far as [11:17] this particular problem is concerned, just by taking a [11:20] look at a picture like this. [11:22] In other words, in this particular problem, the input [11:25] is time t, the output is distance s. [11:30] For each t, we locate a height called s by squaring t and [11:35] multiplying by 16. [11:37] And now, what average speed means in terms of this kind of [11:40] a diagram is the following. [11:42] To find the average speed, all we have to do is on a given [11:46] time interval find the distance traveled, which I [11:49] call delta s, the change in distance, and divide that by [11:53] the change in time. [11:54] That's the average speed, which, by the way, from a [11:58] geometrical point of view, becomes known as the slope of [12:02] this particular straight line. [12:03] In other words, average speed is to functions what slope of [12:08] a straight line is to geometry. [12:10] At any rate, knowing what the average rate of speed is, we [12:14] sort of say why couldn't we define the instantaneous speed [12:18] to be this. [12:19] We will take the change in distance divided by the change [12:24] in time and see what happens. [12:26] And we write this this way. [12:28] Limit as delta t approaches 0. [12:30] Let's see what happens as that change in time becomes [12:33] arbitrarily small, but never equaling 0 because we don't [12:37] want a 0/0 form here. [12:39] You see, this then becomes the working definition of what we [12:44] call differential calculus. [12:46] The point is that this particular definition does not [12:50] depend on s equaling 16t squared. [12:53] s could be any function of t whatsoever. [12:57] We could have a more elaborate type of situation. [13:00] The important point is what? [13:02] The basic definition stays the same. [13:05] What changes is the amount of arithmetic that's necessary to [13:09] handle the particular relationship between s and t. [13:13] This will be a major part of our course, the strange thing [13:18] being that even at the very end of our course when we've [13:20] gone through many, many things, our basic definition [13:24] of instantaneous rate of change will have never [13:26] changed from this. [13:27] It will always stay like this. [13:30] But what will change is how much arithmetic and algebra [13:34] and geometry and trigonometry, et cetera, we will have to do [13:38] in order to compute these things from a [13:40] numerical point of view. [13:42] Well, so much for the first phase of calculus called [13:45] differential calculus. [13:47] A second phase of calculus, one which was developed by the [13:50] Ancient Greeks by 600 BC, the subject that ultimately [13:54] becomes known as integral calculus, concerns problem of [13:58] finding area under a curve. [14:00] Here, I've elected to draw the parabola y equals x squared on [14:07] the interval from 0, 0 to 1, 0. [14:10] And the question basically is what is the area bounded by [14:17] this sort of triangular region? [14:19] Let's call that region R, and what we would like to find is [14:24] the area of the region R. [14:26] And the Ancient Greeks had a rather interesting title for [14:30] this type of approach for finding the area. [14:32] It is both figurative and literal, I guess. [14:35] It's called the method of exhaustion. [14:40] What they did was to -- [14:40] They would divide the interval, say, [14:42] into n equal parts. [14:44] And picking the lowest point in each interval, they would [14:47] inscribe a rectangle. [14:50] Knowing that the area of the rectangle was the base times [14:53] the height, they would add up the area of each of these [14:57] rectangles, and know that whatever that area was, that [15:00] would have to be too small to be the right answer because [15:04] that region was contained in R. And that would be [15:07] labeled A sub n-- [15:08] lower bar, say-- [15:10] to indicate that this was a sum of rectangles which was [15:13] too small to be the right answer. [15:15] Similarly, they would then find the highest point in each [15:20] rectangle, get an overapproximation by adding up [15:24] the sum of those areas, which they would call A sub n upper [15:28] bar, and now know that the area of the regions they were [15:31] looking for was squeezed in between these two. [15:34] Then what they would do is make more and more divisions, [15:37] and hopefully, and I think you can see this sort of [15:40] intuitively happening here, each of the lower [15:42] approximations gets bigger and fills out [15:45] the space from inside. [15:47] Each of the upper approximations gets smaller [15:51] and chops off the space from outside here. [15:54] And hopefully, if both of these bounds sort of converge [16:01] to the same value L, we get the idea that the area of the [16:05] region R must be L. [16:08] This is not anything new. [16:09] In other words, this is a technique that is some 2,500 [16:12] years old, used by the Ancient Greeks. [16:16] Of course, what happens with engineering students in [16:19] general is that one frequently says, but I'm not interested [16:22] in studying area. [16:23] I am not a geometer. [16:25] I am a physicist. [16:26] I am an engineer. [16:28] What good is the area under a curve? [16:30] And the interesting point here becomes that if we label the [16:35] coordinate axis rather than x and y, give them physical [16:39] labels, it turns out that area under a curve has a physical [16:43] interpretation. [16:44] Consider the same problem. [16:46] Only now, instead of talking about y equals x squared, [16:50] let's talk about v, the velocity, equaling the square [16:53] of the time. [16:54] And say that the time goes to 0 to 1. [16:57] In other words, if we plot v versus t, we get a [17:00] picture like this. [17:02] And the question that comes up is what do we mean by the area [17:05] under the curve here? [17:07] And again, without belaboring this point, not because it's [17:09] not important, but because this is just an overview and [17:12] we'll come back to all of these topics later in our [17:15] course, the point I just want to bring out here is, notice [17:19] that the area under the curve here is the distance that this [17:23] particle would travel moving at this speed if the time goes [17:28] from 0 to 1. [17:30] And notice what we're saying here. [17:32] Again, suppose we divide this interval into n equal parts [17:36] and inscribe rectangles. [17:39] Notice that each of these rectangles [17:43] represents a distance. [17:44] Namely, if a particle moved at the speed over this length of [17:53] time, the area under the curve would be the distance that it [17:57] traveled during that time interval. [17:59] In other words, what we're saying is that if the particle [18:02] moved at this speed from this time to this time, then moved [18:05] at this speed from this time to this time, the sum of these [18:10] two areas would give the distance that the particle [18:12] traveled, which obviously is less than the distance that [18:17] the particle truly traveled, because notice that the [18:19] particle was moving at a speed which at every instance from [18:22] here to here was greater than this and at every instant from [18:26] here to here was greater than this. [18:28] In other words, in the same way as before, that area of [18:32] the region R was whittled in between A sub n upper bar and [18:39] A sub n lower bar, notice that the distance traveled by the [18:43] particle can now be limited or bounded in the same way. [18:48] And in the same way that we found area as a limit, we can [18:52] now find distance as a limit. [18:55] And these two things, namely, what? [18:58] Instantaneous speed and area under a curve are the two [19:02] essential branches of calculus, differential [19:05] calculus being concerned with instantaneous rate of speed, [19:10] integral calculus with area under a curve. [19:12] And the beauty of calculus, surprisingly enough, in a way [19:17] is only secondary as far as these [19:19] two topics are concerned. [19:21] The true beauty lies in the fact that these apparently two [19:24] different branches of calculus, one of which was [19:27] invented by the Ancient Greeks as early as 600 BC, [19:30] the other of which-- [19:32] differential calculus-- [19:33] was not known to man until the time of Isaac Newton in 1690 [19:37] AD are related by a rather remarkable thing. [19:41] That remarkable thing, which we will emphasize at great [19:44] length during our course, is that areas and rates of change [19:49] are related by area under a curve. [19:52] Now, I don't know how to draw this so that you see this [19:54] thing as vividly as possible, but the idea is this. [19:58] Think of area being swept out as we take a line and move it, [20:04] tracing out the curve this way towards the right. [20:07] Notice that if we have a certain amount of area, if we [20:13] now move a little bit further to the right, notice that the [20:18] new area somehow depends on what the height of this curve [20:24] is going to be. [20:26] That somehow or other, it seems that the area under the [20:30] curve must be related to how fast the height of this line [20:35] is changing. [20:36] Or to look at it inversely, how fast the area is changing [20:40] should somehow be related to the height of this line. [20:44] And just what that relationship is will be [20:47] explored also in great detail in the course. [20:49] And we will show the beautiful marriage between this [20:53] differential and integral calculus through this [20:56] relationship here, which becomes known as the [20:59] fundamental theorem of integral calculus. [21:02] At any rate then, what this should show us is that [21:06] calculus hinges-- [21:07] whether it's differential calculus or integral calculus, [21:12] that calculus hinges on something [21:13] called the limit concept. [21:15] Again, by way of a very quick review, one [21:18] of the limit concepts-- [21:19] and I think it's easy to see geometrically rather than [21:22] analytically. [21:23] Imagine that we have a curve, and we want to find the [21:26] tangent of the curve at the point P. What we can do is [21:30] take a point Q and draw the straight line that joins P to [21:34] Q. We could then find the slope of the line PQ. [21:38] The trouble is that PQ does not look very much like the [21:41] tangent line. [21:42] So we say OK, let Q move down so it comes closer to P. We [21:48] can then find the slopes of PQ1. [21:50] We could find the slope of PQ2. [21:53] But in each case, we still do not have the slope of the line [21:57] tangent to the curve at P. But we get the idea that as Q gets [22:01] closer and closer to P, the slope, or the secant line that [22:05] joins P to Q, becomes a better and better approximation to [22:10] the line that would be tangent to the curve at P. [22:13] In fact, it's rather interesting that in the 16th [22:16] century, the definition that was given of a tangent line [22:20] was that a tangent line is a line which passes through two [22:25] consecutive points on a curve. [22:27] Now, obviously, a curve does not have [22:29] two consecutive points. [22:31] What they really meant was what? [22:32] That as Q gets closer and closer to P, the secant line [22:37] becomes a better and better approximation for the tangent [22:40] line, and that in a way, if the two points were allowed to [22:43] coincide, that should give us the perfect answer. [22:47] The trouble is, just like you can't divide 0 by 0, if P and [22:51] Q coincide, how many points do you have? [22:54] Just one point. [22:55] And it takes two points to determine a straight line. [22:59] No matter how close Q is to P, we have two distinct points. [23:03] As soon as Q touches P, we lose this. [23:06] And this is what was meant by ancient man or medieval man by [23:10] his notion of two consecutive points. [23:12] And I should put this in double quotes because I think [23:16] you can see what he's begging to try to say with the word [23:19] "consecutive," even though from a purely rigorous point [23:22] of view, this has no geometric meaning. [23:26] Now, the other form of limit has to do with adding up areas [23:30] of rectangles under curves. [23:32] Namely, we divided the curve up into n parts. [23:35] We inscribed n rectangles, and then we let n [23:38] increase without bound. [23:40] In other words, this is sort of a discrete type of limit. [23:44] Namely, we must add up a whole number of areas, but the sum [23:49] is endless in the sense that the number of rectangles [23:52] becomes greater than any number we want to preassign. [23:56] And the basic question that we must contend with here is how [24:02] big is an infinite sum? [24:04] You see, when we say infinite sum, that just tells you how [24:07] many terms you're combining. [24:08] It doesn't tell you how big each term, how big [24:11] the sum will be. [24:12] For example, look at the following sum. [24:15] I will start with 1. [24:17] Then I'll add 1/2 on twice. [24:19] Then I'll add 1/3 on three times. [24:23] And without belaboring this point, let me then say I'll [24:25] had on 1/4 four times, 1/5 five times, 1/6 [24:30] six times, et cetera. [24:33] Notice as I do this that each time the terms gets smaller, [24:37] yet the sum increases without any bound. [24:41] Namely, notice that this adds up to 1. [24:43] This adds up to 1. [24:45] The next four terms will add up to 1. [24:47] And as I go out further and further, notice that this sum [24:51] can become as great is I want, just by me [24:53] adding on enough 1's. [24:55] On the other hand, let's look at this one. [24:57] 1 plus 1/2 plus 1/4 plus 1/8 plus 1/16 plus 1/32. [25:05] In other words, I start with 1 and each time add on half the [25:09] previous number. [25:10] See, 1 plus 1/2 plus 1/4 plus 1/8. [25:13] You may remember this as being the geometric series whose [25:17] ratio is 1/2. [25:20] The interesting thing is that now this sum gets as close to [25:25] 2 as you want without ever getting there. [25:28] And rather than prove this right now, let's just look at [25:30] the geometric interpretation here. [25:33] Take a line which is 2 inches long. [25:37] Suppose you first go halfway. [25:39] You're now here. [25:40] Now go half the remaining distance. [25:42] That's what? [25:43] 1 plus 1/2. [25:44] That puts you over here. [25:45] Now go half the remaining distance. [25:47] That means add on 1/4. [25:49] Now go half the remaining distance. [25:51] That means add on on 1/8. [25:53] Now go half the remaining distance. [25:55] Add up this on 1/16, you see. [25:57] And ultimately, what happens? [25:59] Well, no matter where you stop, you've become closer and [26:02] closer to 2 without ever getting there. [26:04] And as you go further and further, you can get as close [26:06] to 2 as you want. [26:08] In other words, here are infinitely many terms whose [26:11] infinite sum is 2. [26:13] Here are infinitely many terms whose infinite sum is [26:18] infinity, we should say, because it [26:19] increases without bound. [26:20] And this was the problem that hung up the Ancient Greek. [26:23] How could you do infinitely many things in a [26:26] finite amount of time? [26:27] In fact, at the same time that the Greek was developing [26:31] integral calculus, the famous greek philosopher Zeno was [26:36] working on things called Zeno's paradoxes. [26:39] And Zeno's paradoxes are three in number, of which I only [26:42] want to quote one here. [26:44] But it's a paradox which shows how Zeno could not visualize [26:47] quite what was happening. [26:49] You see, it's called the Tortoise and the Hare problem. [26:52] Suppose that you give the Tortoise a 1 yard head start [26:56] on the Hare. [26:58] And suppose for the sake of argument, just to mimic the [27:00] problem that we were doing before, suppose it's a slow [27:03] Hare and a fast Tortoise so that the Hare only runs twice [27:06] as fast as the Tortoise. [27:08] You see, Zeno's paradox says that the Hare can never catch [27:11] the Tortoise. [27:12] Why? [27:13] Because to catch the Tortoise, the Hare must first go the 1 [27:16] yard head start that the Tortoise had. [27:18] Well, by the time the Hare gets here, the Tortoise has [27:22] gone 1/2 yard because the Tortoise travels half as fast. [27:25] Now, the Hare must make up the 1/2 yard. [27:27] But while the Hare makes up the 1/2 yard, the Tortoise [27:30] goes 1/4 of a yard. [27:32] When the Hare makes up the 1/4 of a yard, the Tortoise goes [27:36] 1/8 of a yard. [27:37] And so, Zeno argues, the Hare gets closer and closer to the [27:40] Tortoise but can't catch him. [27:43] And this, of course, is a rather strange thing because [27:46] Zeno knew that the Tortoise would catch the Hare. [27:49] That's it's called a paradox. [27:50] A paradox means something which appears to be true yet [27:54] is obviously false. [27:56] Now, notice that we can resolve Zeno's paradox into [27:59] the example we were just talking about. [28:01] For the sake of argument, notice what's happening here [28:03] with the time. [28:04] For the sake of argument, let's suppose that the [28:07] Tortoise travels at 1 yard per second. [28:09] Then what you're saying is-- [28:10] I mean, the Hare travels at 1 yard per second. [28:12] What you're saying is it takes the Hare 1 [28:15] second to go this distance. [28:17] Then it takes him 1/2 a second to go this distance, then 1/4 [28:22] of a second to go this distance. [28:24] And what you're saying is that as he's gaining on the [28:27] Tortoise, these are the time intervals which are [28:29] transpiring. [28:30] And this sum turns out to be 2. [28:34] Now, of course, those of us who had eighth grade algebra [28:38] know an easier way of solving this problem. [28:40] We say lookit, let's solve this problem algebraically. [28:44] Namely, we say give the Tortoise a 1 yard head start. [28:49] Now call x the distance of a point at which the Hare [28:55] catches the Tortoise. [28:56] Now, the Hare is traveling 1 yard per second. [28:59] The Tortoise is traveling 1/2 yard per second, OK? [29:05] So if we take the distance traveled and divided by the [29:12] rate, that should be the time. [29:16] And since they both are at this point at the same time, [29:19] we get what? x/1 equals x minus 1 divided by 1/2. [29:24] And assuming as a prerequisite that we have had algebra, it [29:28] follows almost trivially that x equals 2. [29:32] In other words, what this says is, in reality, that the Hare [29:36] will not overtake the Tortoise until he catches [29:39] him, which is obvious. [29:41] But what's not so obvious is what? [29:43] That these infinitely many terms can add [29:46] up to a finite sum. [29:48] Well, at any rate, this complete the overview of what [29:52] our course will be like. [29:53] And to help you focus your attention on what our course [29:58] really says, what we shall do computationally is this. [30:03] In review, we shall start with functions, and functions [30:06] involve the modern concept of sets because they're [30:09] relationships between sets of objects. [30:12] We'll talk about limits, derivatives, rate of change, [30:16] integrals, area under curves. [30:18] This will be our fundamental building block. [30:22] Once this is done, these things will never change. [30:25] But the remainder of our course will be to talk about [30:28] applications, which is the name of the game as far as [30:31] engineering is concerned. [30:33] More elaborate functions, namely, how do we handle [30:36] tougher relationships. [30:37] Related to the tougher relationships will come more [30:41] sophisticated techniques. [30:43] And finally, we will conclude our course with the topic that [30:46] we were just talking about: infinite series, how do we get [30:50] a hold of what happens when you add up infinitely many [30:53] things, each of which gets small. [30:56] At any rate, that concludes our lecture for today. [31:00] We will have a digression in the sense that the next few [31:04] lessons will consist of sets, things that you can read about [31:09] at your leisure in our supplementary notes. [31:12] Learn to understand these because the concept of a set [31:15] is the building block, the fundamental language of modern [31:18] mathematics. [31:20] And then we will return, once we have sets underway, to talk [31:24] about functions. [31:26] And then we will build gradually from there. [31:29] Hopefully, when our course ends, we will have in slow [31:31] motion gone through today's lesson. [31:35] This completes our presentation for today. [31:37] And until next time, goodbye. [31:45] NARRATOR: Funding for the publication of this video is [31:48] provided by the Gabriella and Paul Rosenbaum Foundation. [31:52] Help OCW continue to provide free and open access to MIT [31:56] courses by making a donation at ocw.mit.edu/donate.