IB Math Review Set 12A: Non-Calculator Essentials | IB数学复习题集12A：非计算器核心技巧

📚 IB Math Review Set 12A: Non-Calculator Essentials | IB数学复习题集12A：非计算器核心技巧

Review Set 12A is designed to test your ability to solve mathematical problems without a calculator. This article breaks down the key topics, techniques, and common pitfalls, helping you build the algebraic fluency and conceptual understanding required for the IB Mathematics non-calculator papers.

复习题集12A旨在考查你在无计算器情况下解决数学问题的能力。本文梳理了关键主题、技巧与常见易错点，帮助你建立IB数学非计算器试卷所需的代数熟练度和概念理解。

1. Simplifying Algebraic Expressions | 代数表达式化简

Always begin by looking for a common factor, a difference of two squares, or a quadratic trinomial that can be factorised. Cancel only after factoring completely.

始终先寻找公因式、平方差或可分解的二次三项式。只有完全因式分解后才能约分。

For example, simplify (x² - 9)/(x - 3) by recognising the numerator as (x - 3)(x + 3), giving x + 3 for x ≠ 3.

例如，化简 (x² - 9)/(x - 3) 时，将分子视为 (x - 3)(x + 3)，得到 x + 3（x ≠ 3）。

When dealing with surds, rationalise denominators by multiplying the numerator and denominator by the conjugate. For instance, 1/(√2 - 1) becomes √2 + 1 after rationalisation.

当涉及根式时，通过分子分母同乘共轭来有理化分母。例如，1/(√2 - 1) 有理化后变为 √2 + 1。

Completing the square transforms a quadratic into a(x - h)² + k. This form reveals the vertex and is essential for integrating certain rational functions.

配方法将二次式转化为 a(x - h)² + k 的形式。这种形式能揭示顶点，对于积分某些有理函数至关重要。

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2. Solving Equations Without a Calculator | 无计算器解方程

s the fastest method. If factorisation is not obvious, use the quadratic formula x = [-b ± √b² - 4ac]/(2a) and simplify the square root surd.

对于二次方程，因式分解是最快的方法。如果难以直接分解，使用求根公式 x = [-b ± √b² - 4ac]/(2a) 并化简根号。

When solving exponential equations like 2^2x-1 = 8, rewrite both sides with the same base: 2^2x-1 = 2³, hence 2x - 1 = 3 and x = 2.

解指数方程如 2^2x-1 = 8 时，将两边改写为同底数：2^2x-1 = 2³，因此 2x - 1 = 3，得 x = 2。

) = 2 with the domain x > 1. Solve the resulting quadratic and check for extraneous roots.

对数方程常常需要合并对数项：ln(x) + ln(x - 1) = ln(2) 导出 x(x - 1) = 2，并注意定义域 x > 1。解出二次方程后必须排除增根。

Trigonometric equations require exact solutions within a given interval. For 2 sin θ - √3 = 0, solve sin θ = √3/2 and give θ = π/3, 2π/3 etc., without a decimal approximation.

三角方程要求在一个给定区间内给出精确解。对于 2 sin θ - √3 = 0，解得 sin θ = √3/2，并给出 θ = π/3, 2π/3 等，而不是小数近似值。

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3. Mastering Exponential and Logarithmic Functions | 掌握指数与对数函数

Recall the core properties: e^{ln x} = x and ln(e^a) = a. These are indispensable for reversing operations in equations.

牢记核心性质：e^{ln x} = x 以及 ln(e^a) = a。它们在方程中逆转运算时不可或缺。

The change-of-base rule, log_a / ln a, lets you evaluate logs with unusual bases or express answers in a simpler ln-form.

换底公式 log_a b = ln b / ln a 使你可以计算非惯用底数的对数，或者用更简洁的 ln 形式表达答案。

When sketching y = e^x and y = ln x, show the horizontal asymptote y = 0 for e^-x and the vertical asymptote x = 0 for ln x. Label intersections with axes exactly.

在绘制 y = e^x 和 y = ln x 的草图时，标明 e^-x 的水平渐近线 y = 0 以及 ln x 的垂直渐近线 x = 0。准确标出坐标轴交点。

Solving 3^x+1 = 5^2x = 2x ln 5, then rearranging to x(ln 3 - 2 ln 5) = -ln 3, giving an exact expression for x.

解 3^x+1 = 5^2x 需要两边取自然对数：(x+1)ln 3 = 2x ln 5，然后整理为 x(ln 3 - 2 ln 5) = -ln 3，得出 x 的精确表达式。

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4. Trigonometric Functions and Exact Values | 三角函数与精确值

Memorise the exact values from the unit circle: sin(π/6) = 1/2, cos(π/4) = √2/2, tan(π/3) = √3. They are the foundation of all non-calculator trigonometry.

熟记单位圆上的精确值：sin(π/6) = 1/2，cos(π/4) = √2/2，tan(π/3) = √3。它们是无计算器三角函数的基础。

Use Pythagorean identities like sin²x + cos²x = 1 and 1 + tan²x = sec²x to rewrite and simplify trigonometric expressions.

运用勾股恒等式如 sin²x + cos²x = 1 以及 1 + tan²x = sec²x 来重写和化简三角表达式。

to find x in the required interval. Always give radian answers unless degrees are specified.

解 cos 2x = 1/2 时，先解出 2x = π/3, 5π/3, …，然后除以 2 得到要求区间内的 x。除非题目指定角度，否则一律用弧度作答。

± cosA sinB. These allow expansion of expressions like sin(π/3 + x) without a calculator.

熟悉复合角公式，如 sin(A ± B) = sinA cosB ± cosA sinB。这些公式让你能在没有计算器的情况下展开像 sin(π/3 + x) 这样的式子。

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5. Differentiation from First Principles | 从第一原理求导

The limit definition f'(x) = lim_h→0 [f(x+h) - f(x)]/h is the foundation of differential calculus. You may be asked to derive the derivative of a simple function from first principles.

极限定义 f'(x) = lim_h→0 [f(x+h) - f(x)]/h 是微分学的基础。你可能会被要求用第一原理推导一个简单函数的导数。

For f(x) = x², write f(x+h) - f(x) = (x+h)² - x² = 2xh + h², divide by h to get 2x + h, and take the limit as h → 0 to obtain f'(x) = 2x.

对于 f(x) = x²，写出 f(x+h) - f(x) = (x+h)² - x² = 2xh + h²，除以 h 得到 2x + h，并令 h → 0 取极限，得出 f'(x) = 2x。

(x+h) - 1/x = -h/[x(x+h)], divide by h, and simplify to -1/[x(x+h)]. The limit yields -1/x².

对于有理函数如 f(x) = 1/x，通分得到 1/(x+h) - 1/x = -h/[x(x+h)]，除以 h 并化简为 -1/[x(x+h)]。取极限得到 -1/x²。

Work carefully with algebraic simplifications; the common mistake is forgetting to cancel h before taking the limit.

小心处理代数化简；常见的错误是在取极限之前忘记约去 h。

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6. Applying Derivative Rules | 应用求导法则

Rule / 法则	Formula / 公式
Power Rule / 幂法则	d/dx (xn) = n xn-1
Product Rule / 乘积法则	d/dx (uv) = u'v + uv'
Quotient Rule / 商法则	d/dx (u/v) = (u'v - uv')/v2
Chain Rule / 链式法则	dy/dx = dy/du · du/dx

ys simplify the function before differentiating if possible. For instance, write f(x) = (x²+1)/x as x + x^-1 before applying the power rule.

尽可能在求导前先化简函数。例如，将 f(x) = (x²+1)/x 写成 x + x^-1 后再用幂法则。

sin x; then dy/dx = (1/u) · cos x = cot x. Keep the derivative in a simplified exact form.

链式法则对复合函数至关重要。对于 y = ln(sin x)，设 u = sin x；则 dy/dx = (1/u) · cos x = cot x。将导数保留为简化的精确形式。

When using the quotient rule, avoid expanding unnecessarily. It is often useful to factor the numerator afterward to locate stationary points.

使用商法则时，不要不必要地展开。通常之后应将分子因式分解，以便寻找驻点。

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7. Tangents, Normals, and Stationary Points | 切线、法线与驻点

The gradient of the tangent at x = a is f'(a). The tangent line equation is y - f(a) = f'(a)(x - a). The normal is perpendicular, so its gradient is -1/f'(a), provided f'(a) ≠ 0.

在 x = a 处的切线斜率为 f'(a)。切线方程为 y - f(a) = f'(a)(x - a)。法线与切线垂直，因此其斜率为 -1/f'(a)（假设 f'(a) ≠ 0）。

Stationary points occur where f'(x) = 0. To classify them, use the second derivative: f''(x) > 0 indicates a local minimum, f''(x) < 0 a local maximum, and f''(x) = 0 could be an inflection point.

驻点出现在 f'(x) = 0 处。要判断其性质，使用二阶导数：f''(x) > 0 表示局部极小值，f''(x) < 0 表示局部极大值，f''(x) = 0 可能是一个拐点。

For a function like f(x) = x³ - 6x² + 9x, find f'(x) = 3x² - 12x + 9 = 3(x-1)(x-3). Stationary points at x=1 and x=3. Evaluate f''(x) = 6x - 12: negative at x=1 (max), positive at x=3 (min).

对于函数 f(x) = x³ - 6x² + 9x，求出 f'(x) = 3x² - 12x + 9 = 3(x-1)(x-3)。在 x=1 和 x=3 处有驻点。计算 f''(x) = 6x - 12：在 x=1 处为负（极大），在 x=3 处为正（极小）。

Always show coordinates of the points of tangency or stationary points as exact (x, y) pairs, not decimal approximations.

始终将切点或驻点的坐标表示为精确的 (x, y) 形式，而不是小数近似值。

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8. Integration as Reverse Differentiation | 积分作为反求导

The indefinite integral reverses differentiation. For ∫ xⁿ dx, the formula is xⁿ⁺¹/(n+1) + C, with n ≠ -1. When n = -1, ∫ x^-1 dx = ln|x| + C.

不定积分是求导的逆运算。对于 ∫ xⁿ dx，公式为 xⁿ⁺¹/(n+1) + C，其中 n ≠ -1。当 n = -1 时，∫ x^-1 dx = ln|x| + C。

Before integrating, rewrite roots and fractions as powers: √x becomes x^1/2, 1/x³ becomes x^-3. This avoids careless errors.

积分前，将根式和分式改写为幂形式：√x 变为 x^1/2，1/x³ 变为 x^-3。这可以避免粗心错误。

More complex patterns like ∫ (ax + b)ⁿ dx use the reverse of the chain rule: (1/a)*(ax+b)ⁿ⁺¹/(n+1) + C. Always check by differentiating.

像 ∫ (ax + b)ⁿ dx 这样更复杂的模式使用链式法则的逆运算：(1/a)*(ax+b)ⁿ⁺¹/(n+1) + C。始终通过求导来检验。

(kx) + C should be memorised. Avoid sign mistakes with ∫ sin(kx) dx = - (1/k) cos(kx) + C.

三角函数积分如 ∫ cos(kx) dx = (1/k) sin(kx) + C 应当熟记。要避免 ∫ sin(kx) dx = - (1/k) cos(kx) + C 中的符号错误。

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9. Definite Integrals and Area Under Curves | 定积分与曲线下面积

A definite integral ∫_a^b = a to x = b. Use the Fundamental Theorem: F(b) - F(a), where F is an antiderivative of f.

定积分 ∫_a^b f(x) dx 计算的是曲线与 x 轴在 x = a 到 x = b 之间的带符号面积。使用微积分基本定理：F(b) - F(a)，其中 F 是 f 的一个原函数。

For example, to evaluate ∫₁⁴ (2x + ½ x²) dx, find F(x) = x² + (1/6) x³ tute: F(4) = 16 + 64/6, F(1) = 1 + 1/6, difference = 15 + 63/6 = 15 + 10.5 = 25.5 exactly.

例如，计算 ∫₁⁴ (2x + ½ x²) dx，先求 F(x) = x² + (1/6) x³。代入得：F(4) = 16 + 64/6，F(1) = 1 + 1/6，差值为 15 + 63/6 = 15 + 10.5 = 25.5（精确值）。

When the curve passes below the x-axis, split the integral into intervals where the function is positive and negative, and take absolute values to find total area.

当曲线穿过 x 轴下方时，将积分区间拆分为函数为正和为负的部分，并取绝对值来计算总面积。

Use symmetry to simplify calculations. For an odd function f, ∫_-a^a f(x) dx = 0; for an even function, it equals 2∫₀^a f(x) dx.

利用对称性简化计算。对于奇函数 f，有 ∫_-a^a f(x) dx = 0；对于偶函数，则等于 2∫₀^a f(x) dx。

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10. Function Transformations and Graph Analysis | 函数变换与图像分析

Key transformations include f(x) + k (vertical shift), f(x + k) (horizontal shift left for +k), -f(x) (reflection in x-axis), and f(-x) (reflection in y-axis). No calculator means you need to reason analytically.

关键变换包括 f(x) + k（上下平移），f(x + k)（+k 时向左平移），-f(x)（关于 x 轴反射）以及 f(-x)（关于 y 轴反射）。不能使用计算器意味着你必须进行分析推理。

y = (2x-1)/(x+3), the vertical asymptote is x = -3, and the horizontal asymptote is y = 2.

水平和垂直渐近线可以通过考察 x → ±∞ 或分母趋近于零时的行为来找到。对于 y = (2x-1)/(x+3)，垂直渐近线为 x = -3，水平渐近线为 y = 2。

essential for sketching. For f(x) = ln(2x + 1), set y=0 to find x-intercept: 0 = ln(2x+1) ⇒ 2x+1 = 1 ⇒ x

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