微积分基础 / Calculus Fundamentals

本页结构

核心概念

求导与积分规则 Differentiation and integration rules
泰勒展开作为局部近似工具 Taylor expansion as local approximation
期望与级数题中常用的求和恒等式 Summation identities used in expectation and series problems

学习顺序

用泰勒展开估计非线性收益变化。 Use Taylor expansion to estimate nonlinear payoff changes.
注意近似阶数和余项。 Keep track of approximation order and remainder terms.
只有在近似条件合理时才把求和转成积分。 Convert sums to integrals only when the approximation is justified.

At all points $x$ where the functions and the derivatives are defined,

在函数与导数均有定义的所有点 $x$ 上，

\frac{d}{dx}(x^n) = nx^{n-1} \quad \frac{d}{dx}\sin(x) = \cos(x) \quad \frac{d}{dx}\cos(x) = -\sin(x) \quad \frac{d}{dx}\tan(x) = \sec^2(x)

\frac{d}{dx}\sec(x) = \sec(x)\tan(x) \quad \frac{d}{dx}\csc(x) = -\csc(x)\cot(x) \quad \frac{d}{dx}\cot(x) = -\csc^2(x)

\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}} \quad \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2} \quad \frac{d}{dx}\text{arcsec}(x) = \frac{1}{|x|\sqrt{1-x^2}}

\frac{d}{dx}(e^x) = e^x \quad \frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x) \quad \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + g'(x)f(x)

\frac{d}{dx}(\ln(x)) = \frac{1}{x} \quad \frac{d}{dx}f(g(x)) = f'(g(x))g'(x) \quad \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

\frac{d}{dx}(f(x)^{g(x)}) = f(x)^{g(x)} \left[g'(x)\ln(f(x)) + g(x) \cdot \frac{f'(x)}{f(x)}\right] \quad \frac{d}{dx}(x^x) = x^x(\ln(x) + 1)

Disregarding the $+C$ on all the integrals,

以下积分均省略常数项 $+C$ ：

\int x^n dx = \frac{x^{n+1}}{n+1}, n \neq -1 \quad \int \sin(x) dx = -\cos(x) \quad \int \cos(x) dx = \sin(x) \quad \int \tan(x) dx = -\ln|\cos(x)|

\int \sec(x) dx = \ln|\sec(x) + \tan(x)| \quad \int \csc(x) dx = \ln|\csc(x) - \cot(x)| \quad \int \cot(x) dx = \ln|\sin(x)|

\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x) \quad \int \frac{1}{1+x^2} dx = \arctan(x) \quad \int \frac{1}{|x|\sqrt{1-x^2}} dx = \text{arcsec}(x)

\int e^x dx = e^x \quad \int \frac{1}{x} dx = \ln|x| \quad \int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx

\int u(x)v'(x) dx = u(x)v(x) - \int v(x)u'(x) dx \quad \int f'(g(x))g'(x) dx = f(g(x))

Select some point $x = x_0$ . If $x_0 = 0$ , we have the Maclaurin series. Generally, $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n$ . Common Maclaurin series expansions:

选取某一点 $x = x_0$ 。当 $x_0 = 0$ 时，对应麦克劳林级数。一般地， $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n$ 。常见麦克劳林级数展开为：

e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots

\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots

\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots

\sum_{k=1}^n k = \frac{n(n+1)}{2} \quad \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} \quad \sum_{k=s}^\infty a \cdot r^k = a \cdot \frac{r^s}{1-r} \quad \sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}

补充讲解

导数是局部敏感度

Derivatives are local sensitivity

在优化和风险中，导数衡量目标量对小扰动的局部变化。梯度和金融 Greeks 都共享这套语言。

In optimization and risk, a derivative measures how a quantity changes under a small perturbation. This is the common language behind gradients and Greeks.

泰勒展开用于控制近似误差

Taylor expansion is approximation control

一阶和二阶泰勒展开解释 delta、gamma、凸性调整，以及为什么大幅波动下线性近似会失效。

First- and second-order Taylor expansions explain delta, gamma, convexity adjustments, and why linear approximations fail under large moves.

积分用于聚合密度

Integrals aggregate density

期望、期权收益定价和连续概率都依赖对收益函数或统计量按密度积分，这能把局部密度转换成整体数量。

Expected value, option payoff valuation, and continuous probability all rely on integrating a payoff or statistic against a density.

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