📚 AP Calculus: Core Pre-Exam Knowledge Review | AP微积分：考前核心知识点复习
As the AP Calculus exam approaches, a structured review of the most essential topics is crucial for success. This guide covers the core concepts from limits and derivatives to integrals and their applications, zeroing in on what matters most for both AB and BC candidates. Each section is designed to refresh your memory, highlight common pitfalls, and provide you with the key formulas in a clear, accessible format.
随着AP微积分考试临近，系统梳理核心知识点至关重要。本指南涵盖从极限、导数到积分及其应用的必考内容，精准聚焦AB与BC考生共同需要掌握的重点。每一部分都旨在唤醒你的记忆、点明常见误区，并以清晰易懂的形式呈现关键公式。
1. Limits and Continuity | 极限与连续性
The limit of a function f(x) as x approaches a is L if the values of f(x) get arbitrarily close to L from both sides. We write limx→a f(x) = L. A function is continuous at a point if the limit exists, the function is defined, and the limit equals the function value.
如果当x趋近于a时，f(x)的值从左右两侧无限接近L，则极限limx→a f(x) = L。函数在该点连续的条件是：极限存在、函数有定义且极限值等于函数值。
One-sided limits (limx→a⁻ f(x) and limx→a⁺ f(x)) must be equal for a two-sided limit to exist. For rational functions, indeterminate forms like 0/0 can often be resolved by factoring or rationalizing. The limit at infinity describes end behavior, often leading to horizontal asymptotes.
单侧极限（limx→a⁻ f(x) 与 limx→a⁺ f(x)）相等时，双侧极限才存在。对于有理函数，0/0型不定式通常可通过因式分解或有理化求解。无穷远处的极限描述函数远端行为，常与水平渐近线相关。
limx→c f(x) = L ⇔ limx→c⁻ f(x) = limx→c⁺ f(x) = L
连续 ⇔ limx→c f(x) = f(c)
2. Definition of Derivative and Basic Rules | 导数定义与基本法则
The derivative of f at x is the limit of the difference quotient: f'(x) = limh→0 [f(x+h) – f(x)] / h. Geometrically, this gives the slope of the tangent line. The alternative form f'(a) = limx→a [f(x) – f(a)] / (x – a) is often used at a specific point.
函数f在x处的导数是差商的极限：f'(x) = limh→0 [f(x+h) – f(x)] / h。从几何上看，它给出了切线的斜率。另一种形式f'(a) = limx→a [f(x) – f(a)] / (x – a)常用于求某点导数。
Power rule (d/dx xⁿ = n xⁿ⁻¹), sum/difference rules, and constant multiple rules form the foundation. The derivatives of sine and cosine are cos x and –sin x, respectively. Memorize the derivatives of eˣ, aˣ, ln x, and the six basic trig functions.
幂法则(d/dx xⁿ = n xⁿ⁻¹)、和差法则与常数倍法则是基础。正弦与余弦的导数分别为cos x和–sin x。熟记eˣ, aˣ, ln x以及六个基本三角函数的导数。
| Function | Derivative |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| sin x | cos x |
| cos x | -sin x |
| eˣ | eˣ |
| ln x | 1/x |
3. Advanced Differentiation and Implicit Differentiation | 高阶导数与隐函数求导
The product rule: d/dx [u·v] = u’v + uv’. The quotient rule: d/dx [u/v] = (u’v – uv’) / v². The chain rule: d/dx f(g(x)) = f'(g(x))·g'(x). These allow us to differentiate composite, product, and rational functions efficiently.
乘积法则：d/dx [u·v] = u’v + uv’；商法则：d/dx [u/v] = (u’v – uv’) / v²；链式法则：d/dx f(g(x)) = f'(g(x))·g'(x)。这些法则使我们能高效地对复合函数、乘积和有理函数求导。
Implicit differentiation is used when y is not explicitly a function of x. Differentiate both sides with respect to x, treating y as a function of x and multiplying by dy/dx. Higher-order derivatives (f”, f”’, etc.) are obtained by differentiating the previous derivative, with notation d²y/dx².
当y不能显式表示为x的函数时，使用隐函数求导。方程两边对x求导，将y视为x的函数并乘以dy/dx。高阶导数（f”, f”’等）由前一级导数再次求导得到，记法为d²y/dx²。
d/dx [sin(x²)] = cos(x²)·2x
4. Applications of Differentiation: Related Rates and Optimization | 导数应用：相关变化率与最优化
Related rates problems involve finding the rate at which one quantity changes by relating it to other quantities whose rates are known. Write an equation linking the variables, differentiate with respect to time t, and solve for the unknown rate.
相关变化率问题通过将未知变化率的量与已知变化率的量关联，来求出该量的变化率。写出变量间的关系式，对时间t求导，然后解出未知变化率。
Optimization uses the first and second derivatives to find absolute maxima and minima on a closed interval. Set f'(x) = 0 to find critical points, evaluate f at critical points and endpoints. The second derivative test confirms if f”(c) > 0 (min) or f”(c) < 0 (max).
最优化利用一阶和二阶导数求闭区间上的绝对最大值与最小值。令f'(x)=0求临界点，计算f在临界点和端点的值。二阶导数检验确认f”(c)>0为极小，f”(c)<0为极大。
f'(c)=0 and f”(c) > 0 ⇒ local minimum at c
5. Theorems: Mean Value, Extreme Value, and L’Hôpital | 定理：中值定理、极值定理与洛必达法则
The Extreme Value Theorem guarantees that a continuous function on a closed interval [a,b] attains an absolute max and min. The Mean Value Theorem states there exists c in (a,b) such that f'(c) = [f(b) – f(a)] / (b – a). These theorems justify many derivative applications.
极值定理保证闭区间[a,b]上的连续函数必取得绝对最大值和最小值。中值定理指出在(a,b)内存在c使得f'(c) = [f(b) – f(a)] / (b – a)。这些定理为许多导数应用提供了理论依据。
L’Hôpital’s Rule helps evaluate limits that yield indeterminate forms 0/0 or ∞/∞. If lim f(x)/g(x) is indeterminate, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the limit of the derivatives exists. Repeated application may be necessary.
洛必达法则用于计算产生0/0或∞/∞不定式的极限。若lim f(x)/g(x)为不定式，则lim f(x)/g(x) = lim f'(x)/g'(x)，前提是导数之比极限存在。可能需要多次使用该法则。
limx→0 (sin x)/x = limx→0 (cos x)/1 = 1
6. Antiderivatives and Indefinite Integrals | 原函数与不定积分
An antiderivative of f is a function F such that F'(x) = f(x). The indefinite integral ∫ f(x) dx = F(x) + C represents the family of all antiderivatives. Basic integration rules mirror derivative rules: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1), ∫ cos x dx = sin x + C, ∫ 1/x dx = ln|x| + C.
若F'(x)=f(x)，则F是f的一个原函数。不定积分∫ f(x) dx = F(x) + C表示全体原函数族。基本积分法则与导数法则对应：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1)，∫ cos x dx = sin x + C，∫ 1/x dx = ln|x| + C。
The integral of eˣ is eˣ + C; for aˣ, use (aˣ)/(ln a) + C. Remember the constant of integration C, especially when given an initial condition to find a particular solution.
eˣ的积分为eˣ + C；aˣ的积分为(aˣ)/(ln a) + C。务必记住积分常数C，尤其是在给定初始条件求特解时。
7. Riemann Sums and Definite Integrals | 黎曼和与定积分
The definite integral ∫ab f(x) dx represents the signed area under the curve from a to b. It is defined as the limit of Riemann sums: limn→∞ Σ f(xᵢ* ) Δx, where Δx = (b–a)/n. Left, right, midpoint, and trapezoidal sums are common approximations.
定积分∫ab f(x) dx表示从a到b曲线下的带符号面积。它定义为黎曼和的极限：limn→∞ Σ f(xᵢ* ) Δx，其中Δx = (b–a)/n。左和、右和、中点及梯形和是常见近似。
Properties include ∫ab f(x)dx + ∫bc f(x)dx = ∫ac f(x)dx, and flipping limits changes the sign: ∫ab f(x)dx = –∫ba f(x)dx. The average value of f on [a,b] is (1/(b–a)) ∫ab f(x) dx.
性质包括∫ab f(x)dx + ∫bc f(x)dx = ∫ac f(x)dx，颠倒积分上下限变号：∫ab f(x)dx = –∫ba f(x)dx。f在[a,b]上的平均值为(1/(b–a)) ∫ab f(x) dx。
8. Fundamental Theorem of Calculus and Accumulation Functions | 微积分基本定理与累积函数
FTC Part 1: If g(x) = ∫ax f(t) dt, then g'(x) = f(x). This reveals that differentiation and integration are inverse processes. Accumulation functions can have variable upper limits, requiring the chain rule if the upper limit is a function of x.
FTC第一部分：若g(x)=∫ax f(t) dt，则g'(x)=f(x)。这揭示了微分与积分互为逆运算。累积函数的上限可以是变量，若上限是x的函数，求导时需使用链式法则。
FTC Part 2: ∫ab f(x) dx = F(b) – F(a), where F is any antiderivative of f. This is the evaluation theorem, enabling exact calculation of definite integrals without limits of Riemann sums.
FTC第二部分：∫ab f(x) dx = F(b) – F(a)，其中F是f的任一原函数。此评价定理使我们无需依赖黎曼和极限即可精确计算定积分。
d/dx ∫1sin x cos(t) dt = cos(sin x)·cos x
9. Integration Techniques: Substitution and Parts (BC) | 积分技巧：换元法与分部积分
U-substitution is the reverse of the chain rule. For ∫ f(g(x))g'(x) dx, let u = g(x), then du = g'(x)dx, transforming the integral into ∫ f(u) du. Choose u to be the inner function whose derivative appears in the integrand.
换元积分法是链式法则的逆运算。对于∫ f(g(x))g'(x) dx，令u=g(x)，则du=g'(x)dx，将积分化为∫ f(u) du。选取u为内层函数，确保其导数出现在被积函数中。
Integration by parts (BC) derives from the product rule: ∫ u dv = uv – ∫ v du. Use the LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) rule to choose u. This technique handles integrals like x eˣ dx or ln x dx.
分部积分法（BC）源于乘积法则：∫ u dv = uv – ∫ v du。使用LIATE法则（对数、反三角、代数、三角、指数）选择u。此法可处理形如x eˣ dx或ln x dx的积分。
10. Area, Volume, and Arc Length | 面积、体积与弧长
Area between two curves y = f(x) and y = g(x) from a to b is A = ∫ab [f(x) – g(x)] dx, where f(x) ≥ g(x). If curves are expressed as functions of y, integrate with respect to y.
两条曲线y=f(x)与y=g(x)在a到b之间围成的面积为A = ∫ab [f(x) – g(x)] dx，其中f(x)≥g(x)。若曲线以y的函数给出，则对y积分。
Volume by disk method: when revolving a region about the x‑axis, V = π ∫ab [f(x)]² dx. Washer method for space between curves: V = π ∫ab ([R(x)]² – [r(x)]²) dx. Cross-sections perpendicular to an axis can also yield volume formulas.
圆盘法求体积：绕x轴旋转区域，V = π ∫ab [f(x)]² dx。垫圈法处理曲线间区域：V = π ∫ab ([R(x)]² – [r(x)]²) dx。垂直于轴的截面也可导出体积公式。
Arc length (BC) for y = f(x) on [a,b] is L = ∫ab √(1 + [f'(x)]²) dx. For parametric curves, L = ∫ √((dx/dt)² + (dy/dt)²) dt.
弧长（BC）：曲线y=f(x)在[a,b]上的弧长L = ∫ab √(1 + [f'(x)]²) dx。参数曲线弧长L = ∫ √((dx/dt)² + (dy/dt)²) dt。
11. Differential Equations and Slope Fields | 微分方程与斜率场
A separable differential equation can be written as dy/dx = g(x)h(y). Separate variables: ∫ (1/h(y)) dy = ∫ g(x) dx. Integrate both sides and solve for y, using an initial condition to find the particular solution.
可分离变量的微分方程可写为dy/dx = g(x)h(y)。分离变量：∫ (1/h(y)) dy = ∫ g(x) dx。两边积分并解出y，利用初始条件求特解。
Slope fields provide a graphical representation of differential equations. At each point (x,y), a small segment with slope f(x,y) is drawn. Solution curves follow these slope segments. Euler’s method (BC) approximates values using tangent line steps.
斜率场给出微分方程的图形表示。在每点(x,y)处画出斜率为f(x,y)的小线段，解曲线沿这些线段延伸。欧拉方法（BC）通过切线步进近似求解。
dy/dx = xy ⇒ ∫ 1/y dy = ∫ x dx ⇒ ln|y| = x²/2 + C
12. BC Topics: Parametric, Polar, and Series | BC专题：参数、极坐标与级数
For parametric equations x = f(t), y = g(t), the derivative dy/dx = (dy/dt)/(dx/dt). The second derivative requires differentiating dy/dx with respect to t and dividing by dx/dt. Velocity is (dx/dt, dy/dt); speed is √((dx/dt)² + (dy/dt)²).
对于参数方程x=f(t), y=g(t)，导数dy/dx = (dy/dt)/(dx/dt)。二阶导数需对dy/dx关于t求导，再除以dx/dt。速度为(dx/dt, dy/dt)，速率为√((dx/dt)² + (dy/dt)²)。
Polar coordinates (r,θ): x = r cosθ, y = r sinθ. Area enclosed by a polar curve r = f(θ) is A = ½ ∫αβ [f(θ)]² dθ. Derivatives in polar form can be found via parametric conversion.
极坐标(r,θ)：x=r cosθ, y=r sinθ。极曲线r=f(θ)所围面积A = ½ ∫αβ [f(θ)]² dθ。极坐标下的导数可通过参数转换求得。
Infinite series: recognize geometric series Σ arⁿ converges to a/(1–r) if |r|<1. Taylor and Maclaurin series give polynomial approximations. The ratio test checks for absolute convergence: limn→∞ |aₙ₊₁/aₙ| = L < 1 converges, >1 diverges. Lagrange error bound estimates truncation error.
无穷级数：记住几何级数Σ arⁿ当|r|<1时收敛于a/(1–r)。泰勒和麦克劳林级数给出多项式逼近。比值检验判别绝对收敛：limn→∞ |aₙ₊₁/aₙ| = L < 1收敛，>1发散。拉格朗日误差界估计截断误差。
Published by TutorHao | AP Calculus Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导