补充笔记 ¶

常见放缩 ¶

1. \(\sin x < x < \tan x\quad(0 < x < \tfrac{\pi}{2})\) （当 \(-\tfrac{\pi}{2} < x < 0\) 时，\(\tan x < x < \sin x\)）

2. \(\displaystyle \sin x \ge x - \tfrac12 x^2\quad(x \ge 0)\) （当 \(x \le 0\) 时，\(\displaystyle \sin x \le x + \tfrac12 x^2\)）

3. \(\displaystyle \cos x \ge 1 - \tfrac12 x^2\)

4. \(x + 1 \le e^x \le \dfrac1{1 - x}\quad(x < 1)\)

5. \(\displaystyle e^x \ge ex\)

6. \(\displaystyle 1 - \tfrac1x \le \ln x \le x - 1\)

7. \(\displaystyle \ln x \le \tfrac{x}{e}\)

8. \(\displaystyle \ln x > \tfrac12\!\bigl(x - \tfrac1x\bigr)\quad(0 < x < 1)\) （当 \(x > 1\) 时，\(\displaystyle \ln x < \tfrac12\!\bigl(x - \tfrac1x\bigr)\)）

9. \(\displaystyle \ln x > \sqrt{x} - \tfrac1{\sqrt{x}}\quad(0 < x < 1)\) （当 \(x > 1\) 时，\(\displaystyle \ln x < \sqrt{x} - \tfrac1{\sqrt{x}}\)）

10. \(\displaystyle \sqrt{x} \le \tfrac{x + 1}{2}\)

11. \(\displaystyle e^x > \ln x + 2\)

12. \(\displaystyle e^x > \ln(x + 2)\)

13. \(\displaystyle \sin x \ge \tfrac{2}{\pi}x\quad(0 \le x \le \tfrac{\pi}{2})\) （当 \(-\tfrac{\pi}{2} \le x \le 0\) 时，\(\displaystyle \sin x \le \tfrac{2}{\pi}x\)）

14. \(\displaystyle \cos x \ge 1 - \tfrac{2}{\pi}x\quad(0 \le x \le \tfrac{\pi}{2})\)

15. \(\displaystyle \sin x \ge x\cos x\quad(0 \le x \le \tfrac{\pi}{2})\) （当 \(-\tfrac{\pi}{2} \le x \le 0\) 时，\(\displaystyle \sin x \le x\cos x\)）