📚 Review Set 6C: Mastering Calculus Challenges | 复习题集6C：掌握微积分挑战
Review Set 6C is designed to test and strengthen your calculus skills within the IB Mathematics: Analysis and Approaches syllabus. This set focuses on differentiation techniques, applied rates of change, optimisation, and integral calculus. Each problem has been carefully selected to mirror the style of Paper 1 and Paper 2 questions, emphasising analytical thinking and precise mathematical communication.
复习题集6C旨在检验并强化你在IB数学分析与方法课程中的微积分技能。本套题目聚焦于求导方法、变化率的应用、最优化以及积分计算。每道题都精心挑选，以反映试卷一和试卷二的出题风格，着重考查分析思维和严谨的数学表达。
1. Introduction to Review Set 6C | 复习题集6C简介
Review Set 6C is a compilation of 7 multi‑part problems that progress from straightforward derivative calculations to complex modelling and integration tasks. The set is intended for both Standard Level and Higher Level students, with HL candidates expected to handle more sophisticated algebraic manipulation. Work through each item systematically, showing all reasoning steps as required in IB mark schemes.
复习题集6C由7道多小问题目组成，从基础求导逐步过渡到复杂的建模与积分任务。本套题同时适用于标准水平和高级水平学生,高级水平考生需掌握更复杂的代数变形。请按照IB评分方案的要求，系统解答每一道题，并展示完整的推理过程。
2. Key Concepts Revision | 核心概念回顾
Before attempting the set, review these essential calculus concepts: the chain rule, product rule and quotient rule; implicit differentiation; connected rates of change; optimisation of functions on closed intervals; indefinite and definite integrals; integration by substitution; and the fundamental theorem of calculus linking differentiation and integration. Make sure you can switch confidently between function notation and Leibniz notation.
在动手解题之前，请回顾以下核心微积分概念：链式法则、乘法法则和除法法则；隐函数求导；相关变化率；闭区间上函数的最优化；不定积分和定积分；换元积分法；以及联系微分与积分的微积分基本定理。确保你能在函数记法与莱布尼茨记法之间自如转换。
3. Problem 1: Derivative of a Composite Function | 问题1：复合函数求导
Given f(x) = ln(sin(2x + 1)), find f' ddle function is sine, and the innermost is 2x + 1. Apply the chain rule layer by layer: f' ) · 2, which simplifies to 2cot(2x+1). Remember, derivatives are often asked in exact form — resist the urge to evaluate numerically unless specified.
已知 f(x) = ln(sin(2x + 1))，求 f'(x)。外层函数是自然对数，中间层是正弦，最内层是 2x + 1。逐层运用链式法则：f'(x) = (1/sin(2x+1)) · cos(2x+1) · 2，可化简为 2cot(2x+1)。请记住，导数常要求保留精确形式——除非题目明确要求，否则不要数值求值。
4. Problem 2: Implicit Differentiation | 问题2：隐函数求导
²). Always check that the denominator is not zero before claiming the expression is valid.
对于方程 x²y + y³ = 3x，求 dy/dx。两边对 x 求导，并对 x²y 使用乘法法则：2xy + x²(dy/dx) + 3y²(dy/dx) = 3。分离含 dy/dx 的项：x²(dy/dx) + 3y²(dy/dx) = 3 − 2xy。提取公因式得 dy/dx = (3 − 2xy) / (x² + 3y²)。在声称表达式有效之前，始终检查分母是否非零。
5. Problem 3: Related Rates | 问题3：相关变化率
s⁻¹. This is a classic example of how a constant volume rate leads to a variable radius rate.
一个球形气球正在充气，体积以 100 cm³ s⁻¹ 的速率增加。求当半径为 5 cm 时半径增加的速率。回顾 V = (4/3)πr³，因此 dV/dt = 4πr² (dr/dt)。代入已知值：100 = 4π(5)² (dr/dt) → dr/dt = 100 / (100π) = 1/π cm s⁻¹。这是一个典型例题，说明体积变化率恒定时，半径变化率却是变化的。
6. Problem 4: Optimization | 问题4：最优化问题
, then the side parallel to the wall is 40 − 2x. Area A(x) = x(40 − 2x) = 40x − 2x². Set A' e confirms a maximum. The optimal dimensions are 10 m by 20 m, yielding an area of 200 m².
一个矩形花园一边靠墙，另三边用 40 米围栏围成。求使面积最大化的尺寸。设垂直于墙的边长为 x，则平行于墙的边长为 40 − 2x。面积 A(x) = x(40 − 2x) = 40x − 2x²。令 A'(x) = 40 − 4x = 0 → x = 10。二阶导数检验确认为极大值。最优尺寸为 10 米乘 20 米，面积为 200 平方米。
7. Problem 5: Integration by Substitution | 问题5：换元积分法
Evaluate ∫ 2x √x²+1 dx. Choose u = x² + 1, so du = 2x dx. The integral becomes ∫ √u du = (2/3)u3/2 + C = (2/3)(x²+1)3/2 + C. Always change back to the original variable when presenting a final indefinite integral. For definite integrals, remember to adjust the limits of integration accordingly.
计算 ∫ 2x √x²+1 dx。令 u = x² + 1，则 du = 2x dx。积分变为 ∫ √u du = (2/3)u3/2 + C = (2/3)(x²+1)3/2 + C。在写出最终不定积分结果时，务必换回原变量。对于定积分，请记住相应地调整积分上下限。
8. Problem 6: Area Between Curves | 问题6：曲线间的面积
2x²) dx = [x² − (2/3)x³]₀¹ = 1 − 2/3 = 1/3. Sketching the region helps avoid sign errors.
求由 y = x² 与 y = 2x − x² 围成的面积。首先求交点：x² = 2x − x² → 2x² − 2x = 0 → 2x(x − 1) = 0，得 x = 0 和 x = 1。在 (0,1) 区间上，上方函数是 2x − x²。面积为 ∫₀¹ [(2x − x²) − x²] dx = ∫₀¹ (2x − 2x²) dx = [x² − (2/3)x³]₀¹ = 1 − 2/3 = 1/3。画出示意图有助于避免符号错误。
9. Problem 7: Differential Equations | 问题7：微分方程
a constant, given y(0) = y₀. Separate variables: (1/y) dy = k dx. Integrate both sides: ln|y| = kx + C. Exponentiate: |y| = ekx+C = eC · ekx. Let A = ±eC, so y = A ekx. Using the initial condition, A = y₀. The particular solution is y = y₀ ekx. This exponential model describes growth or decay crucial in IB applications.
求解 dy/dx = ky，其中 k 为常数，已知 y(0) = y₀。分离变量：(1/y) dy = k dx。两边积分：ln|y| = kx + C。取指数：|y| = ekx+C = eC · ekx。令 A = ±eC，得 y = A ekx。利用初始条件，A = y₀。特解为 y = y₀ ekx。这一指数模型描述了增长或衰减，在IB应用题中至关重要。
10. Common Mistakes and Tips | 常见错误与技巧
Frequent errors include forgetting to multiply by the derivative of the inner function when using the chain rule, mixing up dv/dt and dr/dt in related rates, and mishandling absolute values when integrating 1/x. In optimization, always verify that your answer is indeed a maximum or minimum using the second derivative test or a sign diagram, and state the answer in the context of the question.
常见错误包括：使用链式法则时忘记乘以内层函数的导数，在相关变化率问题中混淆 dv/dt 和 dr/dt，以及在积分 1/x 时处理绝对值不当。在最优化问题中，始终使用二阶导数检验或符号图验证答案确实为极大值或极小值，并结合题意陈述最终答案。
11. Practice Makes Perfect | 熟能生巧
After completing Review Set 6C, attempt similar problems from past IB papers. Pay attention to the command terms: "Find", "Hence", "Show that" — each requires a different level of justification. Time yourself: Paper 1 (non‑calculator) demands mental arithmetic agility; Paper 2 (calculator‑allowed) often tests how you set up integrals and then use technology wisely.
完成复习题集6C之后，尝试解答历年IB真题中的类似题目。注意指令词：“Find”、“Hence”、“Show that”——每个词要求的论证程度不同。为自己计时：试卷一（不可用计算器）要求心算敏捷；试卷二（可用计算器）常考查你如何建立积分式并合理使用技术工具。
12. Conclusion | 结语
Review Set 6C is more than a collection of problems — it is a diagnostic tool. Identify which questions cause you trouble and revisit the underlying theory before moving on. Consistent practice, combined with reflective self‑assessment, is the surest path to a high score in IB Mathematics calculus components.
复习题集6C不仅仅是一套练习题——它还是一个诊断工具。找出哪些题目令你困扰，并在继续前进之前重温相应的基础理论。持续练习加上反思性自我评估，是在IB数学微积分部分取得高分的最可靠途径。
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