📚 10A Fundamentals of calculus | 微积分基础
Calculus is the mathematical study of continuous change. Built on the twin pillars of differentiation and integration, it provides the tools to analyse slopes of curves, rates of change, and the accumulation of quantities. In IB Mathematics, understanding these fundamentals opens the door to modelling everything from population growth to the motion of projectiles.
微积分是研究连续变化的数学分支。它以微分和积分为两大支柱，为分析曲线斜率、变化率以及量的累积提供了工具。在 IB 数学中，掌握这些基础知识便打开了为人口增长、抛体运动等现实问题建模的大门。
1. The Two Big Ideas of Calculus | 微积分的两大核心思想
Calculus revolves around two inverse operations: differentiation, which finds the instantaneous rate of change of a function, and integration, which accumulates an infinite sum of tiny pieces to compute area, volume, or total change. Together they are linked by the Fundamental Theorem of Calculus.
微积分围绕两个互逆的运算展开：微分——求函数在某一点的瞬时变化率；积分——将无穷多个微小片段累加起来计算面积、体积或总变化量。微积分基本定理将两者紧密联系起来。
2. Understanding Limits | 理解极限
A limit describes the value a function approaches as the input gets arbitrarily close to a given point. We write limx→a e as close as we wish to L by taking x sufficiently near a (but not necessarily equal to a).
极限描述的是当自变量无限趋近于某一点时函数值的趋向。如果通过让 x 充分靠近 a（但不必等于 a），f(x) 的值能任意接近 L，我们就记作 limx→a f(x) = L。
The limit does not depend on the function's value at a; it describes the trend from both sides. A limit exists only when the left-hand and right-hand limits match.
极限与该点处的函数值无关，它描述的是双侧趋势。只有当左极限与右极限相等时，极限才存在。
3. Evaluating Limits Algebraically | 代数方法求极限
Direct substitution works when the function is continuous at a. For indeterminate forms like 0/0, we can simplify by factoring, rationalising, or using known special limits such as limx→0 (sin x)/x = 1 and limx→0 (1 − cos x)/x = 0.
当函数在 a 处连续时，直接代入即可。遇到 0/0 这样的不定式时，可先通过分解因式、有理化或利用重要极限来化简，比如 limx→0 (sin x)/x = 1 以及 limx→0 (1 − cos x)/x = 0。
- limx→2 (x² − 4)/(x − 2) = limx→2 (x + 2) = 4
- limx→2 (x² − 4)/(x − 2) = limx→2 (x + 2) = 4
For limits at infinity, divide numerator and denominator by the highest power of x to see the horizontal asymptote.
处理无穷远处的极限时，可将分子分母同时除以 x 的最高次幂，从而看出水平渐近线。
4. Continuity and Differentiability | 连续性与可微性
A function is continuous at x = a if f(a) is defined, limx→a f(x) exists, and the limit equals f(a). Graphically, there are no breaks, jumps, or holes.
函数在 x = a 处连续，需要满足：f(a) 有定义，limx→a f(x) 存在，且极限值等于 f(a)。从图像上看，曲线在该点无断裂、跳跃或空洞。
Differentiability is stronger: a function must be continuous and have a well-defined tangent at every point in the domain. Corners, cusps, and vertical tangents break differentiability.
可微性要求更强：函数必须连续，且在定义域内每一点都有确定的切线。尖点、折点或垂直切线都会破坏可微性。
5. The Derivative as a Limit | 导数定义为极限
The derivative of f at x is the limit of the difference quotient:
f 在 x 处的导数定义为差商的极限：
f'(x) = limh→0 (f(x+h) − f(x)) / h
e y = f(x) at the point (x, f(x)). If the limit exists, f is differentiable at that point.
该表达式给出曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。如果极限存在，则 f 在该点可微。
6. Notation for Derivatives | 导数的符号
Several notations are commonly used: f'(x) (Lagrange), dy/dx (Leibniz), and ẏ (Newton for time derivatives). Leibniz notation is especially useful when applying the chain rule or solving differential equations.
常用符号有：f'(x)（拉格朗日记法）、dy/dx（莱布尼茨记法）以及 ẏ（牛顿对时间导数的记法）。在应用链式法则或求解微分方程时，莱布尼茨符号尤为方便。
Second derivatives are written as f''(x) or d²y/dx², representing the rate of change of the rate of change.
二阶导数写作 f''(x) 或 d²y/dx²，表示变化率的变化率。
7. Power Rule and Basic Rules | 幂法则与基本求导规则
The power rule states that if f(x) = xⁿ, then f' er with constant multiple and sum rules, we can differentiate any polynomial term by term.
幂法则指出：若 f(x) = xⁿ，则 f'(x) = n·xⁿ⁻¹，其中 n 为任意实数。结合常数倍法则与和差法则，即可对多项式逐项求导。
| Rule / 法则 | Formula / 公式 |
| Constant / 常数 | d/dx [c] = 0 |
| Constant multiple / 常数倍 | d/dx [c·f(x)] = c·f'(x) |
| Sum/Difference / 和差 | d/dx [f(x) ± g(x)] = f'(x) ± g'(x) |
| Power / 幂 | d/dx [xⁿ] = n·xⁿ⁻¹ |
8. Product and Quotient Rules | 乘积法则与商法则
When differentiating a product of two functions, the product rule must be used: (f·g)' = f'·g + f·g'. For a quotient, the quotient rule gives (f/g)' = (f'·g − f·g') / g².
求两个函数乘积的导数时，必须使用乘积法则：(f·g)' = f'·g + f·g'。对于商，商法则为 (f/g)' = (f'·g − f·g') / g²。
These rules prevent common mistakes like differentiating the product as the product of derivatives. Always apply them step by step, and simplify algebraically at the end.
这些法则能避免常见错误，比如将乘积的导数误当导数之积。应始终按步骤应用，最后再作代数化简。
9. The Chain Rule | 链式法则
functions: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. In function notation, (f ∘ g)'(x) = f'(g(x))·g'(x).
链式法则用于复合函数求导：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。用函数记号表示为 (f ∘ g)'(x) = f'(g(x))·g'(x)。
cos(2x)·2 = 2cos(2x). The rule is essential for handling functions inside other functions.
例如，对 sin(2x) 求导得 cos(2x)·2 = 2cos(2x)。这一法则对于处理嵌套函数至关重要。
10. Higher-Order Derivatives | 高阶导数
Differentiating a derivative yields a second derivative, denoted f''(x) or d²y/dx². This measures the concavity of the graph and the acceleration in motion problems. We can continue to find third, fourth, and higher derivatives as long as the function remains differentiable.
对导数再次求导就得到二阶导数，记作 f''(x) 或 d²y/dx²。它衡量函数图像的凹性和运动问题中的加速度。只要函数可微，就可以继续求出三阶、四阶甚至更高阶的导数。
In kinematics, if s(t) is displacement, then v(t) = s'(t) is velocity and a(t) = s''(t) is acceleration.
在运动学中，若 s(t) 表示位移，则 v(t) = s'(t) 为速度，a(t) = s''(t) 为加速度。
11. Antiderivatives and Indefinite Integrals | 反导数与不定积分
An antiderivative of f is a function F such that F' e derivative of a constant is zero. We write the indefinite integral ∫ f(x) dx = F(x) + C.
如果 F'(x) = f(x)，那么 F 就是 f 的一个反导数。由于常数导数为零，最一般的反导数包含任意常数 C。我们记作不定积分 ∫ f(x) dx = F(x) + C。
Basic formulas mirror differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ eˣ dx = eˣ + C, ∫ (1/x) dx = ln|x| + C.
基本积分公式与导数公式对应：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ eˣ dx = eˣ + C，∫ (1/x) dx = ln|x| + C。
12. The Fundamental Theorem of Calculus (First Glimpse) | 微积分基本定理（初探）
This theorem connects differentiation and integration in two parts. Part 1 says that if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). Part 2 states that the function A(x) = ∫ₐˣ f(t) dt is differentiable and A'(x) = f(x).
该定理分两部分将微分与积分联系起来。第一部分指出，若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。第二部分则说明函数 A(x) = ∫ₐˣ f(t) dt 是可微的，且 A'(x) = f(x)。
This means integration is the inverse of differentiation, and evaluating a definite integral can be done by finding any antiderivative and subtracting its values at the limits.
这意味着积分是微分的逆运算，计算定积分只需找到一个反导数并求其在上下限处的差值即可。
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