📚 Mastering IB HL Analysis and Approaches: Core Topics from Functions to Calculus | 掌握 IB HL 分析与方法：从函数到微积分的核心专题
The Oxford IB Mathematics: Analysis and Approaches HL course rewards students who combine conceptual understanding with precise algebraic technique. This revision guide covers the most frequently examined topics across Paper 1, Paper 2 and Paper 3, with emphasis on functions, calculus, proof, vectors and probability.
牛津版 IB 数学：分析与方法 HL 课程奖励能够将概念理解与严谨代数技巧相结合的学生。本复习指南涵盖 Paper 1、Paper 2 和 Paper 3 中最高频考查的专题，重点关注函数、微积分、证明、向量与概率。
1. Functions, Domain, Range and Inverses | 函数、定义域、值域与反函数
A function maps each input to a unique output. For a square-root function such as f(x) = √x − 2 nge is y ≥ 0. For rational functions, exclude any x-value that makes the denominator zero.
函数将每个输入映射到唯一输出。对于形如 f(x)=√x−2 的平方根函数，定义域为 x≥2，因为被开方数必须非负；其值域为 y≥0。对于有理函数，应排除使分母为零的一切 x 值。
fter solving x in terms of y, swap x and y and check both f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
反函数会反转原函数的映射关系。只有原函数是一一对应时反函数才存在；否则必须限制定义域，使其满足水平线检验。用 y 表示 x 后交换 x 与 y，并验证 f(f⁻¹(x))=x 和 f⁻¹(f(x))=x。
f(x) = x² − 4x + 3, x ≥ 2 → f(x) = (x − 2)² − 1 → f⁻¹(x) = 2 + √x + 1
When composing functions, (g∘ hen g. The domain of the composite function consists only of the inputs that are valid for f and whose outputs are valid for g.
对于复合函数，(g∘f)(x)=g(f(x)) 表示先作用 f，再作用 g。复合函数的定义域只包含那些对 f 有意义、且其输出对 g 有意义的输入。
2. Graph Transformations and Asymptotes | 图像变换与渐近线
l translation by c, a horizontal stretch by factor 1/b, a vertical stretch by factor a, and a vertical translation by d.
图像变换遵循可预测的顺序。一般形式 y=a f(b(x−c))+d 组合了水平平移 c、水平伸缩因子 1/b、竖直伸缩因子 a 以及竖直平移 d。
y = a f(b(x − c)) + d
A negative a reflects the graph in the x-axis, while a negative b reflects it in the y-axis. Always apply horizontal transformations first when the argument contains a coefficient, because they affect the input values before vertical changes.
当 a 为负时图像关于 x 轴反射，当 b 为负时图像关于 y 轴反射。当自变量含有系数时，应始终先应用水平变换，因为水平变换在竖直变化之前作用于输入值。
Rational functions often have vertical, horizontal or slant asymptotes. If the degree of the numerator is one greater than the degree of the denominator, divide the polynomial to find the slant asymptote. If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients.
有理函数常具有竖直、水平或斜渐近线。若分子次数比分母次数大 1，可通过多项式除法求得斜渐近线。若次数相等，水平渐近线为最高次项系数之比。
3. Polynomials, Factor Theorem and Rational Functions | 多项式、因式定理与有理函数
m says that when p(x) is divided by (x − a), the remainder is p(a). Use these results to factorise cubics and quartics quickly.
因式定理指出，(x−a) 是多项式 p(x) 的因式当且仅当 p(a)=0。余数定理表明，当 p(x) 除以 (x−a) 时，余数为 p(a)。可利用这些结论快速分解三次或四次多项式。
s −b/a, pairwise product is c/a, and product of roots is −d/a. These relations can help solve systems of equations without finding the roots explicitly.
对于二次方程 ax²+bx+c=0，若两根为 α 与 β，则 α+β=−b/a，αβ=c/a。对于三次方程，根之和为 −b/a，两两乘积之和为 c/a，根之积为 −d/a。这些关系有助于在不显式求根的情况下解方程组。
Partial fractions decompose a rational function with a factorised denominator into simpler terms. Use linear denominators for distinct factors and repeated linear factors, and use a linear numerator for an irreducible quadratic factor. This is essential preparation for integrating rational functions.
部分分式将分母已分解的有理函数拆分为更简单的项。对于不同的因式使用线性分母，对重复线性因式对应地使用多个项，对不可约二次因式使用线性分子。这是积分有理函数的重要准备。
4. Exponentials and Logarithms | 指数函数与对数函数
The natural logarithm ln x is the inverse of eˣ These laws are powerful for solving equations where the variable appears in the exponent.
自然对数 ln x 是 eˣ 的反函数。对数运算律为：ln(ab)=ln a+ln b，ln(a/b)=ln a−ln b，ln(aⁿ)=n ln a。当变量出现在指数中时，这些运算律是解方程的有力工具。
To solve an exponential equation such as 2e²ˣ lution lies in the domain of the original expression, especially when logarithms are involved.
求解指数方程如 2e²ˣ=5 时，先分离出指数项，再两边取对数：2x=ln(5/2)，因此 x=½ln(5/2)。始终要检验解是否在原表达式的定义域内，尤其是涉及对数时。
ln b / ln a. This converts any logarithmic base into natural logarithms, which can then be evaluated with a calculator or manipulated algebraically.
对于非自然对数，常需要换底公式：logₐ b=ln b / ln a。这可将任意对数底转换为自然对数，从而用计算器求值或进行代数运算。
5. Trigonometry: Identities and Equations | 三角学：恒等式与方程
the periodic properties sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ to find all solutions.
在 HL 分析与方法中，角度通常以弧度为单位。单位圆提供了关键角如 π/6、π/4、π/3 和 π/2 处 sin θ 与 cos θ 的精确值。利用周期性 sin(θ+2π)=sin θ 和 cos(θ+2π)=cos θ 可求出所有解。
sin²θ. These identities reduce a trigonometric equation to a polynomial in sin θ or cos θ.
核心恒等式包括 sin²θ+cos²θ=1，tan θ=sin θ/cos θ，sin 2θ=2 sin θ cos θ，以及 cos 2θ=cos²θ−sin²θ=2cos²θ−1=1−2sin²θ。这些恒等式可将三角方程化为关于 sin θ 或 cos θ 的多项式方程。
s in that interval. After using the inverse trigonometric function, add or subtract multiples of 2π and consider symmetries around the y-axis for sine and around the x-axis for cosine.
在区间 0≤θ<2π 内求解 θ 时，务必列出该区间内的所有解。使用反三角函数后，要加上或减去 2π 的整数倍，并利用正弦关于 y 轴的对称性以及余弦关于 x 轴的对称性。
6. Differentiation: Rules and Applications | 微分法：法则与应用
The chain rule is central in HL differentiation: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). Product and quotient rules are also essential: d/dx [uv] = u'v + uv' and d/dx [u/v] = (u'v − uv')/v².
链式法则在 HL 微分中处于核心地位：若 y=f(g(x))，则 dy/dx=f'(g(x))·g'(x)。积法则与商法则同样关键：d/dx[uv]=u'v+uv'，d/dx[u/v]=(u'v−uv')/v²。
Implicit differentiation is used when y is not explicitly given as a function of x. Differentiate both sides with respect to x, treating y as an implicit function, then solve for dy/dx. This is especially useful for curves such as circles and related rates problems.
当 y 没有明确表示为 x 的函数时，用隐函数求导。两边对 x 求导，将 y 视为隐函数，然后解出 dy/dx。这对于圆等曲线以及相关变化率问题尤为有用。
Applications of differentiation include finding tangents and normals, determining increasing and decreasing intervals, locating stationary points with the first derivative, and classifying maxima, minima and points of inflection with the second derivative. Optimisation problems require expressing the quantity to be optimised as a function of one variable and then finding its stationary points.
微分的应用包括求切线与法线、判断递增与递减区间、用一阶导数找驻点，以及用二阶导数对极大值、极小值和拐点进行分类。最优化问题需要将待优化量表示为单变量函数，然后求其驻点。
7. Integration: Techniques and Definite Integrals | 积分法：技巧与定积分
Integration by substitution reverses the chain rule. If u = g(x), then ∫ f(g(x))g' either change the limits to u-values or return to the original variable before evaluating.
换元积分法反转链式法则。若 u=g(x)，则 ∫ f(g(x))g'(x) dx=∫ f(u) du。对于定积分，要么将积分限改为 u 值，要么在求值前换回原变量。
d dv are chosen so that the remaining integral is simpler. It is commonly used for products of polynomials with exponentials, logarithms or trigonometric functions.
分部积分法基于微分的积法则。公式为 ∫ u dv=uv−∫ v du，其中 u 与 dv 的选取应使剩余积分更简单。它常用于多项式与指数、对数或三角函数的乘积。
∫ u dv = uv − ∫ v du
Definite integrals give the signed area between a curve and the
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