📚 AP Calculus: Ultimate Review Summary | AP 微积分：复习要点总结
Preparing for the AP Calculus exam requires a solid grasp of limits, derivatives, integrals, and the fundamental theorem that ties everything together. Whether you are taking AB or BC, this revision guide walks you through the most critical concepts, formulas, and problem-solving strategies you need to master before test day. Each section pairs a concise English explanation with its Chinese equivalent to support bilingual learners and reinforce understanding.
备战 AP 微积分考试，需要牢牢掌握极限、导数、积分以及将它们贯穿起来的微积分基本定理。无论你参加的是 AB 还是 BC，这份复习指南都会带你回顾最核心的概念、公式和解题策略。每个要点都提供中英双语对照讲解，帮助双语学习者加深理解。
1. Limits and Continuity | 极限与连续性
The limit of a function describes the value that f(x) approaches as x gets arbitrarily close to a given point. One-sided limits from the left (x→a⁻) and right (x→a⁺) must be equal for the two-sided limit to exist. A function is continuous at x = a if limx→a f(x) = f(a), meaning there is no hole, jump, or vertical asymptote at that point.
函数的极限描述的是当 x 无限接近某一点时 f(x) 所趋近的值。只有当左极限 (x→a⁻) 和右极限 (x→a⁺) 相等时，双侧极限才存在。如果 limx→a f(x) = f(a)，则函数在 x = a 处连续，也就是说在该点没有洞、跳跃或垂直渐近线。
Three common types of discontinuity are removable (hole), jump, and infinite (vertical asymptote). The Intermediate Value Theorem states that if f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in [a,b] such that f(c) = k. This is often used to show the existence of roots.
三种常见的间断类型是可去间断点（洞）、跳跃间断点和无穷间断点（垂直渐近线）。介值定理指出，如果 f 在闭区间 [a,b] 上连续，k 介于 f(a) 与 f(b) 之间，那么在 [a,b] 内存在 c 使得 f(c) = k。这条定理常用来证明方程根的存在性。
2. Definition of the Derivative | 导数定义
The derivative of a function at a point is the instantaneous rate of change, defined by the limit of the difference quotient: f ‘(x) = limh→0 [f(x+h) – f(x)] / h, provided this limit exists. The alternate form f ‘(a) = limx→a [f(x) – f(a)] / (x – a) is equally important for calculating derivatives at a specific point.
函数在某点的导数是该点处的瞬时变化率，由差商的极限定义：f ‘(x) = limh→0 [f(x+h) – f(x)] / h，前提是该极限存在。另一种等价形式 f ‘(a) = limx→a [f(x) – f(a)] / (x – a) 在求具体点的导数值时同样常用。
A function is differentiable at a point if the derivative exists there, which requires the function to be continuous, but continuity alone does not guarantee differentiability. Corners, cusps, vertical tangents, and discontinuities all break differentiability. Notation for derivatives includes f ‘(x), dy/dx, and y’.
若函数在某点可导，则该点必然连续，但连续不一定可导。尖点、角点、垂直切线和间断点都会破坏可导性。导数的记号包括 f ‘(x)、dy/dx 和 y’。
3. Differentiation Rules | 求导法则
Memorising derivative rules is essential for speed and accuracy. The power rule: d/dx[xⁿ] = n xⁿ⁻¹. The product rule: (uv)’ = u’v + uv’. The quotient rule: (u/v)’ = (u’v – uv’) / v². The chain rule: d/dx[ f(g(x)) ] = f ‘(g(x)) · g'(x).
熟记求导法则对快速准确解题至关重要。幂函数法则：d/dx[xⁿ] = n xⁿ⁻¹。乘法法则：(uv)’ = u’v + uv’。除法法则：(u/v)’ = (u’v – uv’) / v²。链式法则：d/dx[ f(g(x)) ] = f ‘(g(x)) · g'(x)。
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
| arcsin x | 1/√(1 – x²) |
For inverse trigonometric functions, the derivatives often involve square roots. The derivative of arcsin x is 1/√(1 – x²), and the derivative of arctan x is 1/(1 + x²). These appear frequently in both differentiation and integration problems.
反三角函数的导数常常带有根号。arcsin x 的导数是 1/√(1 – x²)，arctan x 的导数是 1/(1 + x²)。这些在微分和积分题中经常出现。
4. Applications of Derivatives | 导数的应用
Derivatives unlock a wealth of information about a function’s behaviour. The first derivative f ‘(x) tells us where the function is increasing (f ‘ > 0) or decreasing (f ‘ < 0), and its sign changes identify local maxima and minima. Critical points occur where f '(x) = 0 or f '(x) does not exist.
导数揭示出关于函数性态的大量信息。一阶导数 f ‘(x) 告诉我们函数在何处递增 (f ‘ > 0) 或递减 (f ‘ < 0)，其符号变化可用于识别局部极大值和极小值。临界点出现在 f '(x) = 0 或 f '(x) 不存在的位置。
The second derivative f ”(x) describes concavity: f ” > 0 means concave up, f ” < 0 means concave down. Points of inflection occur where concavity changes. The second derivative test states that if f '(c) = 0 and f ''(c) > 0, there is a local minimum at c; if f ”(c) < 0, a local maximum.
二阶导数 f ”(x) 描述函数的凹凸性：f ” > 0 表示凹向上，f ” < 0 表示凹向下。拐点出现在凹凸性改变的地方。二阶导数判别法指出，若 f '(c) = 0 且 f ''(c) > 0，则 c 处有局部极小值；若 f ”(c) < 0，则有局部极大值。
Related rates use implicit differentiation with respect to time t to connect the rates at which related quantities change. Optimization problems apply derivative tests to find absolute maxima and minima on a closed interval. Mean Value Theorem guarantees at least one point where the instantaneous rate of change equals the average rate of change over the interval.
相关变化率利用对时间 t 的隐函数求导，把相互关联的量变化率联系起来。最优化问题应用导数判别法在闭区间上寻找绝对最大值和最小值。中值定理保证区间内至少存在一点，该点的瞬时变化率等于区间上的平均变化率。
5. The Definite Integral | 定积分
The definite integral ∫ab f(x) dx represents the net signed area between the curve and the x-axis from x = a to x = b. It is defined as the limit of Riemann sums: limn→∞ Σi=1n f(xi*) Δx, where Δx = (b – a)/n. The choice of sample points (left, right, midpoint) affects the accuracy of approximations.
定积分 ∫ab f(x) dx 表示从 x = a 到 x = b 曲线与 x 轴之间的带符号净面积。它被定义为黎曼和的极限：limn→∞ Σi=1n f(xi*) Δx，其中 Δx = (b – a)/n。取样点的选择（左端点、右端点、中点）会影响近似值的精度。
Basic properties of definite integrals include additivity over intervals, reversal of limits (∫ab = -∫ba), and linearity (∫ (cf + g) = c ∫ f + ∫ g). If f(x) ≥ 0 on [a,b], the integral gives the actual area under the curve. Understanding the integral as an accumulation function is key to the Fundamental Theorem.
定积分的基本性质包括区间上的可加性、上下限互换变号（∫ab = -∫ba）以及线性性质（∫ (cf + g) = c ∫ f + ∫ g）。如果 f(x) ≥ 0 在 [a,b] 上成立，积分就给出曲线下的实际面积。把积分理解为累积函数是理解微积分基本定理的关键。
6. Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem of Calculus (FTC) links differentiation and integration. Part 1: If g(x) = ∫ax f(t) dt, then g'(x) = f(x). This shows that the integral function is an antiderivative of f. If the upper limit is a function of x, use the chain rule: d/dx ∫au(x) f(t) dt = f(u(x)) · u'(x).
微积分基本定理（FTC）将微分与积分联系起来。第一部分：若 g(x) = ∫ax f(t) dt，则 g'(x) = f(x)。这表明积分函数是 f 的一个原函数。如果积分上限是 x 的函数，则需要使用链式法则：d/dx ∫au(x) f(t) dt = f(u(x)) · u'(x)。
Part 2: ∫ab f(x) dx = F(b) – F(a), where F is any antiderivative of f. This provides the primary tool for evaluating definite integrals exactly. You must ensure f is continuous on [a,b] for the theorem to apply. This is also the reason why indefinite integrals are written as F(x) + C.
第二部分：∫ab f(x) dx = F(b) – F(a)，其中 F 是 f 的任意一个原函数。这为精确计算定积分提供了主要工具。必须保证 f 在 [a,b] 上连续才能应用该定理。这也是不定积分写作 F(x) + C 的原因。
7. Integration Techniques | 积分技巧
Reverse power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. The integral of 1/x is ln|x| + C. Substitution (u-substitution) is the reverse of the chain rule: choose u = g(x), replace dx with du/g'(x), and integrate with respect to u. Always change limits when evaluating definite integrals.
幂函数反求导法则：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ -1。1/x 的积分是 ln|x| + C。换元积分法（u-代换）是链式法则的逆运算：选取 u = g(x)，将 dx 替换为 du/g'(x)，然后关于 u 积分。计算定积分时务必同时更换积分上下限。
Integration by parts: ∫ u dv = uv – ∫ v du, derived from the product rule for derivatives. It is useful for products of polynomials and exponentials, logs, or trig functions. The LIATE rule helps choose u. Partial fractions decompose rational functions into simpler fractions for integration. These techniques are heavily tested, especially in BC.
分部积分法：∫ u dv = uv – ∫ v du，来源于微分的乘法法则。它适用于多项式与指数函数、对数函数或三角函数的乘积。LIATE 法则可以帮助选取 u。部分分式分解可将有理函数拆分成更简单的分式再积分。这些技巧在考试中，尤其是在 BC 考试中占比很大。
8. Applications of Integrals | 积分的应用
The area between two curves y = f(x) and y = g(x) from a to b is ∫ab |f(x) – g(x)| dx, which simplifies to ∫ab (top – bottom) dx if the curves do not cross. Always sketch the region to identify the correct limits and the “top” function. For regions with y as the independent variable, use ∫ (right – left) dy.
两条曲线 y = f(x) 与 y = g(x) 从 a 到 b 之间的面积是 ∫ab |f(x) – g(x)| dx，如果两曲线不相交，可以简化为 ∫ab (上曲线 – 下曲线) dx。一定要画草图来确定正确的积分限和“上”函数。对于以 y 为自变量的区域，使用 ∫ (右曲线 – 左曲线) dy。
Volume by cross-sections: V = ∫ab A(x) dx where A(x) is the area of the cross-section perpendicular to the x-axis. For solids of revolution, the disc method (V = π ∫ [R(x)]² dx) and washer method (V = π ∫ ([R(x)]² – [r(x)]²) dx) are standard. The shell method (V = 2π ∫ r h dy) is sometimes required.
横截面积法求体积：V = ∫ab A(x) dx，其中 A(x) 是垂直于 x 轴的截面面积。对于旋转体，圆盘法（V = π ∫ [R(x)]² dx）和垫圈法（V = π ∫ ([R(x)]² – [r(x)]²) dx）是标准方法。有时也需要用到柱壳法（V = 2π ∫ r h dy）。
Average value of a function on [a,b] is (1/(b-a)) ∫ab f(x) dx. Position, velocity, and acceleration are connected through integrals: s(t) = ∫ v(t) dt, v(t) = ∫ a(t) dt. Total distance travelled is ∫ |v(t)| dt.
函数在 [a,b] 上的平均值是 (1/(b-a)) ∫ab f(x) dx。位置、速度和加速度通过积分相互关联：s(t) = ∫ v(t) dt，v(t) = ∫ a(t) dt。总运动路程是 ∫ |v(t)| dt。
9. Differential Equations | 微分方程
A differential equation relates a function and its derivatives. The simplest type is dy/dx = f(x), solved by direct integration: y = ∫ f(x) dx + C. Separable equations allow variables to be separated: dy/dx = g(x)h(y) can be rewritten as (1/h(y)) dy = g(x) dx and integrated. Always include the constant of integration.
微分方程将函数与其导数联系起来。最简单的一类是 dy/dx = f(x)，可通过直接积分求解：y = ∫ f(x) dx + C。可分离变量的微分方程可以将变量分离开：dy/dx = g(x)h(y) 可以改写成 (1/h(y)) dy = g(x) dx 然后积分。永远不要忘记积分常数。
Slope fields provide a graphical view of a differential equation. At each point (x,y), a short segment with slope dy/dx is drawn. Solution curves follow these slopes. Euler’s method approximates a solution using small steps: yn+1 = yn + h f(xn, yn), where h is the step size.
斜率场从图形角度呈现微分方程。在每一点 (x,y) 处画一条斜率为 dy/dx 的短线段。解曲线会沿着这些斜率延伸。欧拉方法通过小步长逼近解：yn+1 = yn + h f(xn, yn)，其中 h 为步长。
Exponential growth and decay models follow dy/dt = k y, with solution y = C ekt. Logistic growth (BC) involves the equation dy/dt = k y (1 – y/L), where L is the carrying capacity. These models appear in both multiple-choice and free-response questions.
指数增长与衰减模型遵循 dy/dt = k y，解为 y = C ekt。逻辑斯蒂增长（BC）涉及的方程是 dy/dt = k y (1 – y/L)，其中 L 是环境承载力。这些模型会出现在选择题和简答题中。
10. Parametric, Polar, and Vector Functions (BC) | 参数方程、极坐标与向量函数（BC）
Parametric equations define x and y in terms of a third variable t. The derivative dy/dx = (dy/dt) / (dx/dt) requires dx/dt ≠ 0. The second derivative is d²y/dx² = (d/dt [dy/dx]) / (dx/dt). Arc length for a parametric curve is ∫ √( (dx/dt)² + (dy/dt)² ) dt.
参数方程用第三个变量 t 来定义 x 和 y。导数 dy/dx = (dy/dt) / (dx/dt)，要求 dx/dt ≠ 0。二阶导数为 d²y/dx² = (d/dt [dy/dx]) / (dx/dt)。参数曲线的弧长公式为 ∫ √( (dx/dt)² + (dy/dt)² ) dt。
Polar coordinates describe points by (r, θ). The area enclosed by a polar curve r = f(θ) from θ = α to β is ½ ∫αβ [f(θ)]² dθ. To find the slope of a polar curve, convert to parametric: x = r cos θ, y = r sin θ. Vector-valued functions give position as
极坐标用 (r, θ) 描述点。极坐标曲线 r = f(θ) 从 θ = α 到 β 围成的面积是 ½ ∫αβ [f(θ)]² dθ。要求极坐标曲线的斜率，可将其转化为参数方程：x = r cos θ, y = r sin θ。向量值函数用
11. Infinite Sequences and Series (BC) | 无穷数列与级数（BC）
A sequence {an} converges if limn→∞ an exists and is finite. A series Σ an converges if the sequence of partial sums converges. The n-th term test: if lim an ≠ 0, the series diverges. Geometric series Σ arn converges to a/(1 – r) when |r| < 1.
如果 limn→∞ an 存在且有限，则数列 {an} 收敛。如果部分和数列收敛，则级数 Σ an 收敛。第 n 项检验法：若 lim an ≠ 0，级数发散。几何级数 Σ arn 当 |r| < 1 时收敛，和为 a/(1 - r)。
Common convergence tests: Integral test, p-series test (Σ 1/np converges for p > 1), comparison test, limit comparison test, alternating series test, and ratio test. Taylor and Maclaurin series represent functions as power series: f(x) = Σ f(n)(a)(x – a)n/n!. The radius and interval of convergence must be determined using the ratio test.
常见收敛性判别法：积分判别法、p 级数判别法（Σ 1/np 当 p > 1 时收敛）、比较判别法、极限比较判别法、交错级数判别法和比值判别法。泰勒级数和麦克劳林级数将函数表示为幂级数：f(x) = Σ f(n)(a)(x – a)n/n!。必须用比值判别法求出收敛半径和收敛区间。
Lagrange error bound gives the maximum error when a Taylor polynomial is used to approximate a function. The bound is |Rn(x)| ≤ M|x – a|n+1/(n+1)!, where M is the maximum value of |f(n+1)(z)| on the interval between a and x.
拉格朗日误差界给出了用泰勒多项式逼近函数时的最大误差。误差界为 |Rn(x)| ≤ M|x – a|n+1/(n+1)!，其中 M 是 |f(n+1)(z)| 在 a 与 x 之间区间上的最大值。
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