📚 Review Set 21C: Calculus Revision | 复习题集21C：微积分复习
The IB Mathematics Review Set 21C is your final checkpoint before moving on from core calculus topics. It pulls together differentiation rules, integration techniques, and their real-world applications. Working through this set with a strategic mindset will help you identify weak spots and refine your problem-solving speed.
IB数学复习题集21C是你在结束微积分核心主题学习前的最后一个检测点。它汇集了求导法则、积分技巧以及它们的实际应用。带着策略性的思路完成这套题集，将帮助你发现薄弱环节，并提升解题速度。
1. Understanding the Review Set | 理解复习题集
Review Set 21C covers the full calculus syllabus typically assessed in IB Mathematics courses. You will encounter questions on basic derivatives, the chain rule, implicit differentiation, tangents and normals, second derivatives, indefinite and definite integrals, integration by substitution, and kinematics. The set is designed to mimic exam-style problems, so treat it as a timed practice session.
复习题集21C涵盖了IB数学课程中通常考查的全部微积分大纲内容。你将遇到关于基本导数、链式法则、隐函数求导、切线与法线、二阶导数、不定积分与定积分、换元积分法以及运动学的题目。这套题集旨在模拟考试风格的题目，因此把它当作一次限时练习来对待。
Before you dive in, quickly review your formula booklet and make sure you can recall derivative and integral formulas without hesitation. Many mistakes occur simply because the chain rule or the constant of integration was left out under pressure.
在你开始之前，快速回顾一下公式手册，确保你能毫不犹豫地回忆出求导和积分公式。许多错误的发生，仅仅是因为在压力下遗漏了链式法则或积分常数。
2. Mastering Basic Differentiation | 掌握基本求导
The first few problems in Review Set 21C test your fluency with the power rule, constant multiple rule, and sums of functions. For any term axn, the derivative is anxn-1. Remember that the derivative of a constant is zero, and the derivative of ex is simply ex.
复习题集21C的前几道题测试你对幂法则、常数乘数法则以及函数之和的熟练程度。对于任意一项 axn，其导数为 anxn-1。记住常数的导数为零，而 ex 的导数就是 ex 本身。
d/dx (x4 + 3x2 – 5x + 7) = 4x3 + 6x – 5
The quotient and product rules often appear next. Stay organised: write ‘u’ and ‘v’ clearly and calculate u’ and v’ before substituting into the formula.
商法则和积法则通常会接着出现。要保持条理：清晰地写下 ‘u’ 和 ‘v’，并在代入公式之前先计算出 u’ 和 v’。
3. The Chain Rule in Action | 链式法则应用
Many students lose marks by forgetting to multiply by the derivative of the inner function. When you see a composite function like (3x2+1)5, identify the ‘outside’ derivative first, then multiply by the ‘inside’ derivative.
许多学生因为忘记乘以内层函数的导数而失分。当你看到像 (3x2+1)5 这样的复合函数时，先找出“外层”导数，然后再乘以“内层”导数。
If f(x) = (3x2+1)5, then f ‘(x) = 5(3x2+1)4 × 6x = 30x(3x2+1)4
A useful check: write down the intermediate layer clearly on your working paper so you don’t skip a step. Over time, this process becomes second nature.
一个有用的检查方法：在草稿纸上清晰地写出中间层，这样你就不会跳过某个步骤。久而久之，这个过程会成为你的第二天性。
4. Implicit Differentiation | 隐函数求导
For equations that cannot be easily solved for y, such as x2 + y2 = 25, differentiate every term with respect to x. Remember that d/dx of y2 becomes 2y(dy/dx), because y is a function of x.
对于那些无法方便地解出 y 的方程，例如 x2 + y2 = 25，需要对每一项关于 x 求导。记住，y2 对 x 求导会变成 2y(dy/dx)，因为 y 是 x 的函数。
After differentiating, collect all terms involving dy/dx on one side and solve. Always present your answer in the simplest form. This technique is essential for IB exam questions involving related rates or finding tangents to circles and ellipses.
求导之后，把所有包含 dy/dx 的项移到同一边并求解。始终以最简形式呈现你的答案。此技巧对于涉及相关变化率或求圆与椭圆切线的IB考题至关重要。
5. Tangents and Normals | 切线与法线
Once you know the derivative f ‘(x), you can find the slope of the tangent at any point x = a. The equation of the tangent is y – f(a) = f ‘(a)(x – a). The normal line is perpendicular, so its slope is -1 / f ‘(a), provided f ‘(a) ≠ 0.
一旦你知道了导数 f ‘(x)，就能求出在任意点 x = a 处切线的斜率。切线方程为 y – f(a) = f ‘(a)(x – a)。法线是垂直的，因此它的斜率为 -1 / f ‘(a)，前提是 f ‘(a) ≠ 0。
Typical Review Set 21C problems ask for the tangent at a given x-value, or require you to find points where the tangent is horizontal. Always double-check your calculations for the y-coordinate—substituting into the original function avoids silly slip-ups.
复习题集21C中的典型问题会要求求出给定 x 值处的切线，或者要求你找出切线为水平的点。反复检查你对 y 坐标的计算——代回原函数可以避免低级失误。
6. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative f ”(x) tells you whether the graph is concave up (f ”(x) > 0) or concave down (f ”(x) < 0). Points of inflection occur where f ''(x) changes sign.
二阶导数 f ”(x) 告诉你图像是凹向上 (f ”(x) > 0) 还是凹向下 (f ”(x) < 0)。拐点出现在 f ''(x) 改变符号之处。
For optimisation, you often use f ‘(x) = 0 to find critical points and then check f ”(x) to confirm whether it is a maximum or minimum. If f ”(a) > 0, the point is a local minimum; if f ”(a) < 0, it is a local maximum.
对于优化问题，你通常先令 f ‘(x) = 0 来找到临界点，然后用 f ”(x) 来确认它是最大值还是最小值。如果 f ”(a) > 0，则该点为局部最小值；如果 f ”(a) < 0，则为局部最大值。
When the second derivative test is inconclusive (f ”(a) = 0), use the first derivative test or analyse the sign pattern. Review Set 21C often includes at least one such tricky case.
当二阶导数检验不能得出结论时 (f ”(a) = 0)，可使用一阶导数检验或分析符号变化模式。复习题集21C通常至少包含一个这样的复杂情况。
7. Indefinite Integrals and Constants | 不定积分与常数
Never forget ‘+ C’ in an indefinite integral. If the problem gives an initial condition, use it to solve for C. This tiny constant is the most common deduction point in IB calculus.
在不定积分中永远不要忘记 ‘+ C’。如果题目给出了一个初始条件，用它来解出 C。这个小小的常数是IB微积分中最常见的扣分点。
∫ (6x2 – 4x + 3) dx = 2x3 – 2x2 + 3x + C
Also, remember the basic integration rules: the integral of 1/x is ln|x|, not ln(x) without absolute value in some contexts; the integral of ekx is (1/k) ekx.
此外，记住基本的积分法则：1/x 的积分是 ln|x|，在某些上下文中不能遗漏绝对值；ekx 的积分是 (1/k) ekx。
8. Definite Integrals and Area | 定积分与面积
A definite integral computes the net signed area between a curve and the x-axis. If the graph goes below the x-axis, the integral yields a negative value for that portion. To find the total enclosed area, you must split the integral at x-intercepts and take absolute values of negative sections.
定积分计算的是曲线与 x 轴之间的净符号面积。如果图像在 x 轴下方，积分会为那部分给出一个负值。要计算所围成的总面积，你必须在 x 轴交点处拆分积分，并对负的部分取绝对值。
Area = ∫ab |f(x)| dx
Use your GDC or analytical methods to evaluate definite integrals accurately. In Review Set 21C, area problems are frequently combined with kinematics or shaded region tasks requiring you to subtract one integral from another.
使用你的图形计算器或分析方法来精确计算定积分。在复习题集21C中，面积问题常常与运动学或者需要用一个积分减去另一个积分的阴影区域任务相结合。
9. Integration by Substitution | 换元积分法
When an integral contains a function and its derivative in a disguised form, substitution is the key. Choose u = g(x) such that du/dx appears elsewhere in the integrand. Don’t forget to change the limits when dealing with a definite integral.
当一个积分包含了一个函数及其导数的隐式形式时，换元法就是关键。选择 u = g(x) 使得 du/dx 出现在被积函数的其他地方。在计算定积分时，千万别忘了更换积分上限和下限。
For example, to integrate ∫ 2x(x2+3)4 dx, set u = x2+3 so that du = 2x dx, turning the whole thing into a simple ∫ u4 du. This reduces complex expressions to basic power integrals.
例如，为了积分 ∫ 2x(x2+3)4 dx，设 u = x2+3，于是 du = 2x dx，从而把整个积分变成简单的 ∫ u4 du。这能将复杂表达式化为基本的幂积分。
10. Kinematics Applications | 运动学应用
IB Mathematics frequently connects calculus to motion. If s(t) is displacement, then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s”(t). Integrating acceleration gives velocity, and integrating velocity gives displacement, usually with initial conditions provided.
IB数学经常将微积分与运动联系起来。如果 s(t) 表示位移，那么速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s”(t)。对加速度积分得到速度，对速度积分得到位移，通常会给出初始条件。
When the question asks for total distance travelled, you need to identify when velocity changes sign and integrate the absolute value of velocity over each interval. This is a classic pitfall found in Review Set 21C kinematics exercises.
当题目要求计算总路程时，你需要判断速度何时改变符号，并对每个区间内的速度绝对值进行积分。这是复习题集21C运动学习题中的一个经典陷阱。
11. Common Pitfalls to Avoid | 常见错误要避免
After analysing hundreds of student responses, the most frequent errors in Review Set 21C are: forgetting the chain rule, dropping the ‘+ C’, mishandling negative areas, misapplying the quotient rule, and algebra mistakes when simplifying derivative expressions.
在分析了数百份学生答卷后，复习题集21C中最常见的错误是：忘记链式法则、遗漏 ‘+ C’、处理负面积不当、误用商法则，以及在化简导数表达式时的代数错误。
Another hidden trap is failing to draw a quick sketch. A rough graph can instantly reveal whether an area is positive or negative, or where a tangent should lie. Spend ten seconds sketching before solving—it saves minutes of frustration.
另一个隐藏的陷阱是未能快速画出一个草图。一个粗略的图形可以立刻揭示一个面积是正还是负，或者一条切线应该位于何处。花十秒钟画个草图再解题，可以避免数分钟的懊恼。
- Missing inner derivative in chain rule / 链式法则中遗漏内层导数
- Forgetting absolute value for area / 计算面积时忘记取绝对值
- Mixing up the signs in quotient rule / 商法则中混淆符号
- Omitting parentheses when substituting negative numbers / 代入负数时遗漏括号
12. Final Tips and Summary | 最后提示与总结
Review Set 21C is not just another homework; it is a self-diagnostic tool. Time yourself, mark your work honestly, and revisit any topic where you scored below 80%. Focus your revision on the exact sub-topics that caused trouble.
复习题集21C不仅仅是另一项家庭作业；它是一个自我诊断工具。给自己计时，诚实地批改作业，并重新复习任何得分低于80%的主题。将复习重点精确地放在那些造成困难的子主题上。
Keep a ‘common mistakes’ log and refer to it before every test. With consistent practice, the calculus processes become automatic, leaving you more time to think logically about application problems. Remember, even the most complex IB calculus problem can be broken down into simple rules you already know.
准备一本“常见错误”日志，并在每次考试前翻阅它。通过持续练习，微积分运算过程会变得自动化，为你留下更多时间对应用问题进行逻辑思考。请记住，即使是最复杂的IB微积分问题，也可以分解为你已经掌握的简单规则。
Believe in your preparation, and let Review Set 21C be your final proof of readiness.
相信你的准备，让复习题集21C成为你准备就绪的最后证明。
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