IB Maths AA HL 极限 导数 链式法则 分部积分
AA HL 的微积分是整门课里最吃基本功的部分。按 IBO 公布的 syllabus，Topic 5 在 HL 占 55 个教学小时，在 SL 只占 28 小时，多出来的时间几乎都花在 SL 不考的内容上：第一原理、洛必达法则、隐函数求导、相关变化率、分部积分、旋转体体积、一阶微分方程与麦克劳林级数。很多学生以为 HL 只是把 SL 的题算得更难，实际上这些新增块各自有独立的套路，缺一块就会在 Paper 3 上直接掉分。
Calculus is where the HL paper rewards structure rather than speed. The guide allocates 55 teaching hours to Topic 5 at HL against 28 at SL, and the extra time goes to techniques that SL never meets: differentiation from first principles, L'Hôpital's rule, implicit differentiation, related rates, integration by parts, volumes of revolution, first-order differential equations and Maclaurin series. Treating HL as harder SL questions is the usual mistake, because each new block has its own entry point.
极限与连续：先判方向再动笔
极限是 Topic 5 的起点，判断方法比计算技巧更先用到。看一个点附近的极限，要先看左右两侧是否走向同一个值，再看函数在该点有没有定义。很多学生以为把数字代进去得到一个结果就算求完，碰到零比零就停了手。这类不定式先试因式分解约掉公因式，再试等价无穷小替换，最后才考虑洛必达法则。连续性的三条要分开核对：该点有定义、极限存在、极限值等于函数值，缺一条就断。分段函数在接缝处最容易失分，只算单侧极限是最常见的写法。洛必达法则要求分子分母都可导且求导后的极限存在，若求导后回到原式，说明这条路走不通。
A limit question starts with comparison, not computation: check the approach from the left and from the right before touching algebra. Substituting a value looks like progress until a zero over zero appears, at which point factorising, cancelling or an equivalent infinitesimal is usually the shorter route. Continuity is three separate conditions rather than one sentence to memorise, and piecewise functions fail at the join whenever only one side is checked; L'Hôpital's rule applies only when both parts are differentiable and the new limit exists.
第一原理求导：差商的极限
第一原理在 AA HL 是明确考点，写的是导数定义式 f'(x) 等于 h 趋于零时 [f(x+h) - f(x)] 除以 h 的极限。这类题的分不在最后那个结果，而在展开差商、约分、取极限这三步是否完整。常见的错误是直接写答案，跳掉化简约分的中间过程，方法分就没了。多项式、根式与简单分式都能用同一套流程做出来，展开时先把 f(x+h) 完整代进去，不要提前化简。取极限这一步要说明 h 趋于零而不是等于零，否则除以 h 没有意义。考试里这类题通常 4 到 6 分，写清每一步比追求速度划算。用第一原理验证过的结果，可以回头和公式册里的标准导数对照。
Differentiation from first principles is an HL requirement, and the marks sit in the algebra rather than the answer. Expand f(x+h) fully, simplify the quotient, then let h tend to zero while making clear that h never equals zero. Candidates who jump to the known derivative lose the method marks, which is why the working has to show each stage even when the result is obvious.
链式法则：把复合函数层层剥开
复合函数的导数靠链式法则，写成 dy/dx 等于 dy/du 乘以 du/dx，层数越多越要先标出每一层。做题时先认外层函数，再认内层函数，然后逐层求导相乘。很多学生以为括号里的东西不用管，直接把外层的导数写下来就结束了。三角、指数、对数套在一起的题目最容易漏掉内层的常数因子，比如 2x 求导得到 2，这一步丢掉就让整题错位。含参数或含隐式结构时，链式法则要跟着记号走，别把 d 和 dx 混在一起。练习时把复合结构写成 u 等于内层的替换形式，写多了就能直接看出层数。这类题在 Paper 1 里出现频率高，因为没有计算器可以帮忙检查。
The chain rule is easiest to keep straight when each layer is named before any differentiation happens. Identify the outer function, identify the inner function, differentiate each and multiply; skipping the inner factor is the single most common slip in trigonometry and in expressions with exponentials. Writing the substitution u explicitly costs one line and prevents most of these errors, especially in Paper 1 where no calculator will flag a wrong factor.
乘积法则与商法则：先看结构再套公式
两个函数相乘用乘积法则，写成 u 乘 v 的导数等于 u 的导数乘 v 加 u 乘 v 的导数。两个函数相除用商法则，分子是 u 的导数乘 v 减 u 乘 v 的导数，分母是 v 的平方。这两条法则本身不难，难在判断式子是乘积还是商，很多学生以为分数形式就一定是商法则，其实分母是常数的可以直接提出来。带负指数或根号的式子先改写成幂函数，再用乘积法则往往更省时间。运算中要保持顺序，商法则里减号的位置写反是扣分高发点。整理结果时把公因式提出来，答案的形式也会更接近评分标准。商的平方分母不要漏写，这是最容易被阅卷抓到的细节。
Product and quotient rules are mechanical once the structure is identified, so the real work is deciding which one applies. A constant denominator is not a quotient rule problem, and rewriting roots or negative powers as powers often turns a messy quotient into a short product. Keep the subtraction order in the quotient rule intact and factorise the finished expression, since a tidy form is easier to check and closer to the marking scheme.
隐函数求导与相关变化率
隐函数求导处理的是 y 没有单独解出来的方程，做法是对等式两边同时对 x 求导，凡是对 y 求导的项都要乘上 dy/dx。整理时把含 dy/dx 的项归到一边，再整体除出来。相关变化率是同一套技术在应用题里的样子，体积、半径、高度随时间变化，靠链式法则把几个变化率连起来。常见的错误是先代数值再求导，顺序反过来，结果一定不对。设变量、写关系式、两边对时间求导、代入已知量，这四步的顺序不能颠倒。单位要跟着走，立方厘米每秒和厘米每分钟混用会让答案看起来对但数值全错。这类题在 Paper 2 出现得多，因为图形计算器的数值解可以反过来检验。
Implicit differentiation treats y as a function of x, so every derivative taken with respect to y carries a dy/dx factor; collecting those terms and dividing gives the gradient. Related rates apply the same idea to quantities that change with time, and the order matters: differentiate first, substitute the numbers afterwards. A common failure is mixing units between a rate in cubic centimetres per second and a length in centimetres per minute, which leaves the arithmetic correct but the answer meaningless.
切线与法线：斜率与垂线关系
切线斜率就是该点的导数值，法线斜率是切线斜率的负倒数，两者相乘等于负一。写切线方程用点斜式，先把切点的坐标算出来，再代斜率。很多学生以为切点一定在曲线与坐标轴的交点上，题目给的往往是别的点。曲线与直线相切的条件是斜率相等且经过同一点，这类题要联立两个条件而不是只用一个。求切线与坐标轴围成的三角形面积时，先把截距算准，再套面积公式。法线容易在负倒数上算错，特别当切线斜率是分数时，倒过来再取负是最稳的写法。答案里的常数项要在最后整理，别留成点斜式就交卷。
The derivative at a point gives the tangent gradient, and the normal gradient is its negative reciprocal, a step that gets inverted under pressure whenever the tangent slope is a fraction. Tangency between a curve and a line needs two conditions, equal gradients and a shared point, so solving with only the gradient loses the question. Finish with the equation in a clean form and check the intercepts before computing any area.
驻点、拐点与二阶导数检验
驻点是导数为零的点，二阶导数在驻点的符号决定它是极大还是极小：二阶导数为负是极大，为正则是极小。二阶导数为零且符号在两侧改变的点才是拐点，单凭二阶导数为零不能下结论。很多学生以为驻点就等于极值点，实际上一阶导数为零只是候选。做表格把区间、导数的符号、函数的增减写在一起，判断会快很多。二阶导数还能说明凹凸性，凹凸性改变的位置就是拐点。含参数的题目要分别在参数的不同取值下讨论，别只算出一个答案。这类题的分通常按结论加理由给，写清依据比只写答案更安全。
A stationary point is only a candidate; the sign of the second derivative decides which kind, with negative meaning a maximum and positive a minimum. An inflection point requires the second derivative to change sign across the point, not merely to equal zero there. A sign table across the intervals settles increasing, decreasing and concavity in one pass, and it is also the clearest evidence to put on the page.
最值问题与优化建模
优化题的第一步是把要最大或最小的量写成单个变量的函数，约束条件用来消去多余的变量。定义域要跟着写出，长度和体积不能取负数，这一步漏掉常常丢掉定义域分。写完函数再求导、找驻点、比较端点值，闭区间上的最值一定在驻点或端点取到。常见的错误是只找驻点就宣布答案，忽略了端点可能更大。建模题的用词要跟着题意，面积、成本、表面积这些量单位不同，不要混在一个式子里。检验答案时可以代回原约束，看看是否满足物理意义。答案要写清单位和条件，只给一个数字通常拿不到满分。
Optimisation starts by writing the quantity to be maximised as a function of a single variable, with the constraint used to eliminate the rest. State the domain alongside the function, because a negative length is not a solution and the endpoint comparison is often where the actual extreme value sits. Checking the answer against the original constraint catches most modelling errors before they reach the page.
运动学：位移、速度与加速度
直线运动中速度是位移对时间的导数，加速度是速度对时间的导数，反过来积分可以从加速度回到速度与位移。方向问题最容易出错，速度为零的时刻要单独找出来，因为那往往是折返点。很多学生以为总路程等于末位移，实际要分段积分再把各段路程相加。定积分的下限要按时间顺序写，写反了会得到一个负数。速度的正负表示方向，速率只取大小，两个概念在答题时要分清。题目给的初值条件用来定积分常数，漏掉常数项是最常见的失分点。这类题在 Paper 2 与 Paper 3 都可能出现，计算量不大但步骤要完整。
In kinematics, velocity is the derivative of displacement and acceleration the derivative of velocity, so integration runs the chain backwards. Finding the moment when velocity is zero matters because that is where the motion reverses, and total distance is the sum of separate integrals rather than the final displacement. Keep the initial conditions visible, since dropping the constant of integration is the usual reason a correct method still loses marks.
不定积分与换元法
不定积分要先认结构：能写成某个函数的导数形式的，直接套公式册里的标准积分；不能的，用换元法把内层结构替换掉。换元时 dx 要跟着换成 du，这一步是换元法唯一真正的技术难点。很多学生以为把 u 写出来就完成换元，忘了处理微分，最后答案里混着两种变量。定积分换元时上下限要一起换，或者换回原变量再代值，两条路都对但不要混用。常数因子可以先提到积分号外面，这样公式更容易认出来。积分结果记得加常数 C，不定积分漏常数是固定扣分项。含三角函数的积分先看能否用恒等式化简，化简之后常常一步就能出结果。
Integration begins with pattern recognition rather than technique: if the integrand already looks like the derivative of something in the formula booklet, use it, and only then reach for substitution. Substitution fails when dx is left untouched, so replace the differential as carefully as the variable; with definite integrals, either change the limits too or convert back before substituting values. Do not drop the constant of integration, and simplify trigonometric integrands with identities before trying anything heavier.
分部积分：u 与 dv 怎么选
分部积分来自乘积法则的逆用，写成 u 乘 v 的积分等于 u 乘 v 减去 v 乘 du 的积分。选 u 的原则是让它求导后变简单，选 dv 的原则是让它容易积分。多项式乘指数、多项式乘三角、对数乘多项式这几类是最常见的组合。很多学生以为随便选也行，方向选反会让积分越来越复杂。循环出现的积分可以移项解出，两边同时加上同一个积分再除以系数。遇到对数或反三角函数，通常把它们当作 u，因为它们的导数会变成有理式。写完整过程时把 u、dv、du、v 四行列清楚，代公式时不容易错位。
Integration by parts is the product rule read backwards, so the choice of u and dv decides whether the problem gets shorter or longer. Pick u to differentiate into something simpler and dv to integrate easily, and write the four ingredients u, dv, du and v in a column before substituting. When the original integral reappears, treat it as an equation and solve for it rather than starting over.
定积分、面积与旋转体体积
定积分给的是带符号的面积，曲线在 x 轴下方时积分为负，求几何面积要分段取绝对值。两曲线之间的面积用上函数减下函数，交点要先解出来，不先找交点直接积分是常见错误。对 y 轴积分时要把函数改写成 x 关于 y 的形式，上下限也跟着换。旋转体体积绕 x 轴用 π 乘 y 平方的积分，绕 y 轴则换元到 x 的平方。很多学生以为体积公式里的平方可以放到积分外面，实际上平方写在被积函数里。计算前先画草图，确认哪条曲线在上，能省掉一半的返工。结果保留三位有效数字，这是 IBO 在试卷说明里写明的通用要求。
A definite integral returns a signed area, so any region below the axis needs splitting or absolute values before a geometric answer is quoted. Areas between curves require the intersection points first, and volumes of revolution put the square inside the integrand rather than outside the integral sign. Sketching the region before integrating is the cheapest way to avoid choosing the wrong upper and lower functions.
Paper 1 无计算器与 Paper 3 长题的分数配置
AA HL 的三份卷子分工很清楚：Paper 1 两小时 110 分、不允许使用计算器，Paper 2 两小时 110 分、需要用图形计算器，Paper 3 一小时 55 分、只考两道延伸题，内部评估占 20 分。按 IBO 的权重，Paper 1 与 Paper 2 各占 30%，Paper 3 与内部评估各占 20%。Paper 1 的微积分题偏重精确值与代数整理，根号、分数、对数都要保留精确形式，提前化成小数会丢分。Paper 3 的两道题通常围绕一个新情境逐步展开，前一问的结论往往是后一问的入口。常见的错误是在 Paper 3 里回头重算前面的结果，浪费时间还容易带入错误。答题顺序按分值走，先拿稳前两问，再攻最后一问的证明或推广。
The three papers split the work: Paper 1 runs two hours for 110 marks without a calculator, Paper 2 matches that with a graphic calculator allowed, and Paper 3 gives one hour for 55 marks of extended response, all weighted alongside a 20-mark exploration. Exact values belong on Paper 1, where an early decimal conversion usually costs accuracy marks. On Paper 3, each part feeds the next, so carrying forward a result is faster and safer than starting the algebra again.
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