AP Calculus——考前的最后复习

不太常用的 导数表 & 积分表

• \(\tan'x = \sec^2 x\)
• \(\cot'x = - \csc^2 x\)
• \(\sec'x = \sec x \tan x\)
• \(\csc'x = - \csc x \cot x\)
• \(\text{arcsec}'x = \frac 1 {|x| \sqrt{x^2 - 1}}\)
• \(\text{arccsc}'x = - \frac 1 {|x| \sqrt{x^2 - 1}}\)

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为方便，以下省略 \(+C\)。

• \(\int \tan x dx = -\ln |\cos x| = \ln |\sec x|\)
• \(\int \cot x dx = -\ln |\sin x|\)
• \(\int \sec^2 x dx = \tan x\)
• \(\int \csc^2 x dx = - \cot x\)
• \(\int \sec x \tan x dx = \sec x\)
• \(\int \csc x \cot x dx = - \csc x\)
• \(\int \frac 1 {x \sqrt{x^2 - 1}} dx = \text{arcsec} |x|\)

Critical / Extremum / Inflection Point

	为 \(0\) 或不存在	两侧邻域内异号
一阶导	Critical Point	Extremum Point
二阶导	(Candidate Inflection Point)	Inflection Point

"异号" 是比 "为 \(0\) 或不存在" 更强的条件，所以右侧必然属于左侧。

IVT EVT MVT

IVT - Intermediate Value Theorem

• 条件：\(f\) 在 \([l,r]\) 上连续。
• 结论：对于任意 \(f(l)\) 和 \(f(r)\) 之间的 \(k\)，存在 \(c \in (l,r)\) 使得 \(f(c) = k\)。

EVT - Extreme Value Theorem

• 条件：\(f\) 在 \([l,r]\) 上连续。
• 结论：\(f\) 在 \([l,r]\) 上存在全局最大值和全局最小值。

MVT - Mean Value Theorem

• 条件：\(f\) 在 \([l,r]\) 上连续，\((l,r)\) 上可导。（闭区间连续，开区间可导）
• 结论：存在 \(c \in (l,r)\) 使得 \(f'(c) = \frac {f(r) - f(l)} {r - l}\)。（切线斜率等于割线斜率）

ODE

Exponential Growth & Decay

\[y' = ky \]

\[\boxed{ y = C e^{kt} }\]

Restricted Growth & Decay

\[y' = k (A - y) \]

\[\boxed{ y = A - C e^{-kt} }\]

Logistic Growth

\[y' = k y (A - y) \]

\[\boxed{ y = \frac A {1 + C e^{-Akt}} }\]

When applied to populations, \(A\) is called the carrying capacity or the maximum sustainable population.

• \(y < \frac A 2 \implies y'' > 0\)
• \(y = \frac A 2 \implies y'' = 0\)
• \(y > \frac A 2 \implies y'' < 0\)

Series - 牢九门

复制自《数列敛散性牢九门》

Geometric Series Test

\[|r| < 1 \iff \sum a r^n \text{ converges} \]

\(p\)-Series Test

\[p > 1 \iff \sum \frac 1 {n^p} \text{ converges} \]

\(n\)th Term Test

\[\lim\limits_{n \to \infty} a_n \ne 0 \implies \sum a_n \text{ diverges} \]

Integral Test

(Nonnegative series)

If \(f(x)\) is a continuous, positive, decreasing function and \(f(n) = a_n\), then

\[\int_1^\infty f(x) dx \text{ and } \sum a_n \text{ both converge or diverge} \]

Comparison Test

(Nonnegative series)

\[\sum b_n \text{ converges} \land a_n \le b_n \implies \sum a_n \text{ converges} \]

\[\sum b_n \text{ diverges} \land a_n \ge b_n \implies \sum a_n \text{ diverges} \]

Limit Comparison Test

(Nonnegative series)

• Case 1: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = L\) where \(0 < L < \infty\)

\[\sum a_n \text{ and } \sum b_n \text{ both converge or diverge} \]

• Case 2: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = 0\)

\[\sum b_n \text{ converges} \implies \sum a_n \text{ converges} \]

• Case 3: \(\lim\limits_{n \to \infty} \frac {a_n} {b_n} = \infty\)

\[\sum b_n \text{ diverges} \implies \sum a_n \text{ diverges} \]

Ratio Test

(Nonnegative series)

Let \(\lim\limits_{n \to \infty} \frac {a_{n+1}} {a_n} = L\) if exists.

\[L < 1 \implies \sum a_n \text{ converges} \]

\[L > 1 \implies \sum a_n \text{ diverges} \]

Root Test

(Nonnegative series)

Let \(\lim\limits_{n \to \infty} \sqrt[n]{a_n} = L\) if exists.

\[L < 1 \implies \sum a_n \text{ converges} \]

\[L > 1 \implies \sum a_n \text{ diverges} \]

Alternating Series Test

(\(a_n \ge 0\))

\[\begin{cases} a_{n+1} < a_n \\ \lim\limits_{n \to \infty} a_n = 0 \\ \end{cases} \implies \sum (-1)^n a_n \text{ converges}\]