📚 AP Calculus Derivatives & Integrals: Master 100 Selected Problems | AP微积分求导与积分精选100题
AP Calculus AB and BC demand fluency in both differentiation and integration. To help you reach that level, we have curated 100 carefully selected problems that target every essential rule, technique, and application. This article walks you through the core concepts covered in those problems and shows how to use them for effective exam preparation.
AP微积分AB与BC都要求学生熟练掌握求导与积分运算。为了帮助你达到这一水平，我们精选了100道题目，涵盖所有关键法则、技巧与应用。本文带你梳理这些题目背后的核心概念，并展示如何高效利用它们备考。
1. Mastering Basic Differentiation Rules | 掌握基本求导法则
The power rule, product rule, and quotient rule form the backbone of AP Calculus. In the 100 selected problems, you will apply d/dx (xⁿ) = n xⁿ⁻¹ to polynomials and combine rules for products and quotients.
幂法则、乘法法则及商法则是AP微积分的基础。在精选100题中，你将运用 d/dx (xⁿ) = n xⁿ⁻¹ 处理多项式，并组合使用乘法与商法则。
For example, when differentiating f(x) = (3x²)(sin x), the product rule gives f'(x) = 6x sin x + 3x² cos x. The problem set reinforces these patterns through repetition.
例如，对 f(x) = (3x²)(sin x) 求导，乘法法则给出 f'(x) = 6x sin x + 3x² cos x。该题集通过重复练习强化这些模式。
Many students forget to simplify after applying the quotient rule. The selected problems often ask for the derivative at a specific point, reminding you to evaluate carefully.
很多学生在使用商法则后会忘记化简。精选题目经常要求在某一点处的导数值，提醒你仔细求值。
2. Chain Rule & Implicit Differentiation | 链式法则与隐函数求导
The chain rule d/dx f(g(x)) = f'(g(x))·g'(x) is tested in nearly every free-response question. Our 100 problems include composite functions such as y = √(2x³+1) and y = e^(tan x), forcing you to identify the inner function.
链式法则 d/dx f(g(x)) = f'(g(x))·g'(x) 几乎出现在每一道自由响应题中。我们的100题涵盖了 y = √(2x³+1) 和 y = e^(tan x) 等复合函数，迫使你识别内层函数。
Implicit differentiation is essential when you cannot solve for y explicitly. Problems like x²+xy+y² = 7 require differentiating both sides with respect to x and then solving for dy/dx.
当无法显式解出y时，隐函数求导至关重要。像 x²+xy+y² = 7 这样的问题需要对x求导左右两边，然后解出 dy/dx。
The selected exercises also include finding second derivatives implicitly, a common AP free-response twist that combines chain rule with algebraic manipulation.
精选练习还包含求隐函数的二阶导数，这是AP自由响应题中常见的转折，将链式法则与代数运算相结合。
3. Higher Order Derivatives & Motion | 高阶导数与运动
Position, velocity, and acceleration are connected through derivatives: v(t)=s'(t) and a(t)=v'(t)=s”(t). The 100 problems dedicate a section to interpreting these relationships from graphs and equations.
位置、速度和加速度通过导数相关联：v(t)=s'(t) 且 a(t)=v'(t)=s”(t)。精选100题专门有一部分要求通过图形和方程解读这些关系。
You will encounter statements like “find the instantaneous velocity at t=2” or “determine when the particle changes direction.” These train you to set v(t)=0 and test intervals.
你会遇到像“求 t=2 时的瞬时速度”或“确定质点何时改变方向”的表述。这些训练你设 v(t)=0 并检验区间。
Higher order derivatives also appear in purely symbolic forms; for instance, finding the fourth derivative of a polynomial helps you spot patterns in successive differentiation.
高阶导数也以纯符号形式出现；例如，求多项式的四阶导数有助于你发现逐次求导的模式。
4. Antiderivatives & Indefinite Integrals | 原函数与不定积分
Antidifferentiation is the inverse of differentiation. The power rule for integrals is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, provided n ≠ -1. The 100 selected problems reinforce this with polynomials, roots, and negative exponents.
求原函数是微分的逆运算。积分的幂法则为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ -1。精选100题通过多项式、根式和负指数强化这一法则。
Special attention is given to the integral ∫ (1/x) dx = ln|x| + C. Mixed problems ask you to differentiate and integrate the same family of functions, highlighting the inverse nature.
特别注意积分 ∫ (1/x) dx = ln|x| + C。混合题要求你对同一类函数同时进行求导和积分，凸显可逆性。
Constant multiples and sums are straightforward, but the selected set ensures you practice splitting integrals: ∫ (3x² – 4 cos x) dx = ∫ 3x² dx – ∫ 4 cos x dx.
常数倍与和的性质很直接，但精选题目确保你练习拆分积分：∫ (3x² – 4 cos x) dx = ∫ 3x² dx – ∫ 4 cos x dx。
5. Definite Integrals & Riemann Sums | 定积分与黎曼和
A definite integral ∫ₐᵇ f(x) dx represents the signed area under a curve. The selected problems begin with left, right, and midpoint Riemann sums to build conceptual understanding before evaluating exact integrals.
定积分 ∫ₐᵇ f(x) dx 表示曲线下的带符号面积。精选题目从左边、右边和中间黎曼和入手，建立概念理解，然后再计算精确积分。
You will compute Riemann sums from tables of values, a classic AP question. The 100 problems emphasize that increasing the number of subintervals improves the approximation.
你将根据函数值表计算黎曼和，这是一道经典AP题。100题强调增加子区间数量能提高近似程度。
Properties of definite integrals, such as ∫ₐᵇ f(x) dx = – ∫ᵇₐ f(x) dx and the additivity property, are applied in multiple-choice and free-response settings.
定积分的性质，比如 ∫ₐᵇ f(x) dx = – ∫ᵇₐ f(x) dx 和区间可加性，在选择题和自由响应题中都被应用。
6. The Fundamental Theorem of Calculus | 微积分基本定理
The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. Part 1 states that if F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x). The 100 selected problems test your ability to apply this to functions defined as integrals with variable upper limits.
微积分基本定理将微分与积分联系起来。第一部分指出若 F(x) = ∫ₐˣ f(t) dt，则 F'(x) = f(x)。精选100题考察你将此应用于以积分定义的上限为变量的函数的能力。
When the upper limit is a function of x, such as ∫ₐᵍ⁽ˣ⁾ f(t) dt, the derivative becomes f(g(x))·g'(x). Our problem set includes variations like ∫ₓ¹ cos t dt that require reversing limits and applying the chain rule.
当上限是x的函数时，如 ∫ₐᵍ⁽ˣ⁾ f(t) dt，导数变为 f(g(x))·g'(x)。我们的题集包含像 ∫ₓ¹ cos t dt 这样的变体，需要反转上下限并应用链式法则。
FTC Part 2 is used to evaluate definite integrals: ∫ₐᵇ f(x) dx = F(b)-F(a), where F is any antiderivative. The selected exercises mix FTC applications to secure true mastery.
基本定理第二部分用于计算定积分：∫ₐᵇ f(x) dx = F(b)-F(a)，其中 F 是任意原函数。精选练习将各种FTC应用混合，确保真正掌握。
7. Integration Techniques: Substitution | 积分技巧：换元法
U-substitution is the most important integration technique in AP Calculus. The 100 problems systematically present integrals where you set u equal to an inner function and adjust the differential du.
u-换元法是AP微积分中最重要的积分技巧。100题系统地呈现了需要设u等于某个内层函数并调整微分du的积分。
For example, with ∫ x·√(x²+1) dx, let u = x²+1, du = 2x dx, so the integral becomes (1/2)∫ √u du. Practice builds the intuition to spot suitable substitutions.
例如，对于 ∫ x·√(x²+1) dx，令 u = x²+1，du = 2x dx，积分变为 (1/2)∫ √u du。练习能培养发现合适代换的直觉。
Definite integrals with substitution require changing the limits: if u = g(x), then ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du. Selected problems highlight this critical step to avoid back-substitution errors.
使用换元法的定积分需要改变积分限：若 u = g(x)，则 ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du。精选题目强调这一关键步骤以避免回代错误。
8. Applications: Area, Volume & Accumulation | 应用：面积、体积与累积量
Calculus shines in real-world applications. Our 100 problems cover the area bounded by two curves: A = ∫ₐᵇ [f(x)-g(x)] dx, where f(x) ≥ g(x). Both horizontal and vertical slicing are practiced.
微积分在现实应用中大放光彩。我们的100题涵盖两曲线围成的面积：A = ∫ₐᵇ [f(x)-g(x)] dx，其中 f(x) ≥ g(x)。横向和纵向切片均有练习。
Volumes of solids of revolution are found using the disk method V = π∫ₐᵇ [R(x)]² dx or the washer method when there is an inner radius. The problem set includes rotation around both axes and lines like y = -1.
旋转体的体积使用圆盘法 V = π∫ₐᵇ [R(x)]² dx 或有内半径时的垫圈法。题集包含绕坐标轴及绕 y = -1 等直线的旋转。
Accumulation functions using integrals model net change. Problems involving water flow or population growth connect derivatives and integrals in a concrete way.
使用积分的累积函数模拟净变化。涉及水流或人口增长的问题以具体方式将导数与积分联系起来。
9. Differential Equations & Slope Fields | 微分方程与斜率场
Many AP problems ask you to solve separable differential equations. The 100 selected problems guide you from dy/dx = ky to exponential growth/decay models, stressing the constant of integration.
许多AP题目要求解可分离变量的微分方程。精选100题引导你从 dy/dx = ky 过渡到指数增长/衰减模型，强调积分常数的作用。
Slope fields provide a qualitative view. You are asked to match a differential equation to its field or sketch a solution curve through a given point, reinforcing the meaning of dy/dx.
斜率场提供了定性的视角。你需要将微分方程与它的斜率场匹配，或画出经过某点的解曲线，强化 dy/dx 的含义。
Specific initial value problems force you to use given conditions to find the particular solution. Clear algebraic steps are practiced across multiple examples.
具体的初值问题要求你利用给定条件求出特解。大量例子练习了清晰的代数步骤。
10. How to Use the 100 Selected Problems Effectively | 如何高效利用精选100题
Start by working through the problems in groups of ten, each focused on a single topic. After completing a set, review your errors and write down the rule you missed before moving on.
开始时以十题为一组进行练习，每组专注一个主题。完成一组后，复习错题并写下你遗漏的法则，然后再继续下一组。
Simulate test conditions once you are two-thirds through the collection. Time yourself and practice pacing, as the AP exam rewards both accuracy and speed.
当你完成题集的三分之二时，模拟考试情境。计时并练习节奏，因为AP考试对准确度和速度都有要求。
Finally, re-solve every tenth problem a week later to strengthen long-term retention. The spaced repetition built into the 100- problem layout helps knowledge move from short-term to long-term memory.
最后，一周后重新解答每隔十题的题目以增强长期记忆。这100道题的布局融入了间隔重复，有助于知识从短期记忆转入长期记忆。
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