考前速通

各 Topic 核心公式

\cosh x = \frac{e^x + e^{-x}}{2}, \quad \sinh x = \frac{e^x - e^{-x}}{2}

\tanh x = \frac{\sinh x}{\cosh x}, \quad \operatorname{coth} x = \frac{\cosh x}{\sinh x}

\cosh^2 x - \sinh^2 x = 1, \quad 1 - \tanh^2 x = \operatorname{sech}^2 x

\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)

\cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right), \quad x \geq 1

\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right), \quad |x| < 1

z = re^{i\theta}, \quad z^n = r^n e^{in\theta}

\text{Roots: } z^{1/n} = r^{1/n} e^{i(\theta + 2k\pi)/n}, \quad k = 0, 1, \dots, n-1

\cos n\theta = \operatorname{Re}(e^{in\theta}), \quad \sin n\theta = \operatorname{Im}(e^{in\theta})

f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots

\det(A - \lambda I) = 0 \quad \text{(特征方程)}

A^n = PD^nP^{-1} \quad \text{(对角化)}

\frac{dy}{dx} + P(x)y = Q(x), \quad \text{IF} = e^{\int P\,dx}

\frac{d}{dx}(y \cdot \text{IF}) = Q \cdot \text{IF}

a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)

辅助方程： $am^2 + bm + c = 0$

根的情况	CF
实根 $m_1 \neq m_2$	$Ae^{m_1x} + Be^{m_2x}$
重根 $m_1 = m_2$	$(Ax + B)e^{m_1x}$
复根 $m = \alpha \pm i\beta$	$e^{\alpha x}(A\cos\beta x + B\sin\beta x)$

\sum_{r=1}^{n} r = \frac{n(n+1)}{2}, \quad \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}

每分约 1.6 分钟。10 分的题最多花 16 分钟。