全部书籍

1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

1 Primary & Middle School Physics1–9 年级在线 2 High School Physics10–12 年级在线 3 University Physics — Year 1本科一年级在线 4 University Physics — Year 2本科二年级在线 5 University Physics — Year 3本科三年级在线

1 School Chemistry — Grades 1 to 121–12 年级在线 2 University Chemistry — Year 1本科一年级在线 3 University Chemistry — Year 2本科二年级在线 4 University Chemistry — Year 3本科三年级在线

1 Primary & Middle School Biology1–9 年级在线 2 High School Biology10–12 年级在线 3 University Biology — Year 1本科一年级在线 4 University Biology — Year 2本科二年级在线 5 University Biology — Year 3本科三年级在线

1 Primary & Middle School Earth and Space Sciences1–9 年级在线 2 High School Earth and Space Sciences10–12 年级在线 3 University Earth and Space Sciences — Year 1本科一年级在线 4 University Earth and Space Sciences — Year 2本科二年级在线 5 University Earth and Space Sciences — Year 3本科三年级在线

1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

全部游戏 Math Fuse Word Reactor Atlas Dash

应用关于

Coach

English Français Español Nederlands Português हिन्दी العربية Bahasa Indonesia 简体中文 বাংলা

登录开始阅读

书籍暂未推出简体中文版——当前显示英文版。

Mathematics · 术语表

什么是 Increasing, decreasing？

又称： increasing function · decreasing function

Definition 3.8 High School Mathematics · 第 3 章 — Functions

Let $f$ be defined on an interval $I$ .

$f$ is increasing on $I$ when for all $u, v \in I$ , if $u < v$ then $f(u) \leq f(v)$ : the outputs grow with the inputs, and the graph climbs from left to right.
$f$ is decreasing on $I$ when $u < v$ implies $f(u) \geq f(v)$ : the graph falls from left to right.

With strict inequalities $f(u) < f(v)$ (resp. $>$ ) we say strictly increasing (resp. decreasing).

示例

Example 3.10

Consider the function graphed below on $\intcc{-2}{4}$ .

A function defined on [-2, 4], increasing then decreasing. — A function defined on $\intcc{-2}{4}$ , increasing then decreasing.

Reading the graph: $f$ increases from $f(-2) = -1.5$ up to its maximum $f(1) = 3$ , then decreases down to $f(4) = -1.5$ . Its variation table is

$x$	$-2$		$1$		$4$
$f$	$-1.5$	$\nearrow$	$3$	$\searrow$	$-1.5$

Example 3.11 (Proving a variation)

Show that $f(x) = 3x + 1$ is strictly increasing on $\R$ . Take any $u < v$ and compare the images:

f(v) - f(u) = (3v + 1) - (3u + 1) = 3(v - u) > 0,

because $v - u > 0$ . So $f(u) < f(v)$ : the function is strictly increasing. The same computation with a negative slope, e.g. $g(x) = -2x + 5$ , gives $g(v) - g(u) = -2(v-u) < 0$ : strictly decreasing.

在原文中阅读 →

Definition 11.4 High School Mathematics · 第 11 章 — Functions and Variations

Let $f$ be defined on an interval $I$ . The function $f$ is increasing on $I$ if it preserves order:

\text{for all } u, v \in I: \quad u < v \implies f(u) \leq f(v),

and decreasing if it reverses order ( $u < v \implies f(u) \geq f(v)$ ). With strict inequalities on the right, $f$ is strictly increasing (resp. decreasing). A function that is increasing or decreasing on $I$ is monotonic on $I$ .

示例

Example 11.10

Study $f(x) = \sqrt{x^2 + 1}$ on $\R$ . The inner function $u(x) = x^2 + 1$ is positive, decreasing on $\intoc{-\infty}{0}$ and increasing on $\intco{0}{+\infty}$ (a shifted square). By point 3, $f$ has the same variations: decreasing then increasing, with minimum $f(0) = 1$ .

在原文中阅读 →