Review set 16A - NON-CALCULATOR | 复习集 16A - 非计算器

📚 Review set 16A – NON-CALCULATOR | 复习集 16A – 非计算器

Working through a non-calculator review set is one of the best ways to build fluency with exact values, algebraic simplification and clear reasoning. This article breaks down the key ideas that underpin the typical questions in Review Set 16A, covering differentiation, integration, tangents, stationary points and the algebra that holds it all together – all without touching a calculator.

在非计算器的复习集里磨炼，是提升精确值处理、代数化简和清晰推理能力的最佳方式之一。本文拆解了复习集16A中常见题型背后的核心思想，涵盖了微分、积分、切线、驻点以及串联一切的代数运算——全程不碰计算器。

1. Differentiation Rules and Algebraic Manipulation | 微分法则与代数化简

Before attempting any derivative in a non-calculator paper, you must be able to rewrite functions in a form that makes the power rule easy to apply. For example, expressions like 1/x² should become x⁻², and √x should become x^1/2. This reduces the mental load and avoids mistakes with negative or fractional exponents.

在非计算器考试中求导之前，你必须能先把函数改写为便于使用幂法则的形式。比如，像 1/x² 这样的表达式应该改写为 x⁻²，√x 要写成 x^1/2。这能减轻心算负担，避免在负指数或分数指数上出错。

If f(x) = xⁿ then f'(x) = n xⁿ⁻¹

Combine this with the sum rule and constant multiple rule. Write every term clearly before you differentiate. Never try to differentiate a messy fraction directly – always split it into sum of powers of x. For instance, (x²+3)/√x becomes x^3/2 + 3x^⁻1/2.

结合和法则与常数倍法则使用。在求导前，清楚地写出每一项。绝不要直接对混乱的分式求导——总是先将其拆分为 x 的幂之和。例如 (x²+3)/√x 应化为 x^3/2 + 3x^⁻1/2。

Simplification makes derivative steps shorter and more reliable.

化简使求导步骤更短也更可靠。
Always convert roots and denominators into index form before differentiating.

求导前务必把根号和分母转换为指数形式。

2. Finding Derivatives from First Principles | 用第一原理求导

The first principles definition is foundational and often appears in the non-calculator section. You must recall the limit definition and apply it securely, expanding and simplifying without numeric substitution until the final step.

第一原理定义是根基，常出现在非计算器部分。你必须牢记极限定义并扎实地运用，在不代入数值的情况下展开、化简，直到最后一步。

f'(x) = limₕ→₀ [f(x+h) − f(x)] / h

For a quadratic like f(x)=x², you expand (x+h)² = x² + 2xh + h², cancel x², divide through by h and let h → 0 to obtain 2x. The same discipline is needed for rational functions – use a common denominator and simplify the numerator carefully.

对于像 f(x)=x² 这样的二次函数，你展开 (x+h)² = x² + 2xh + h²，消去 x²，除以 h 并令 h → 0 即得 2x。处理有理函数时也需要同样的严谨——通分后仔细化简分子。

Common pitfalls include forgetting to cancel h correctly or mis-handling negative terms. Writing every line of algebra on paper is essential even when you are confident, because tiny sign errors cost marks.

常见失误包括未能正确地约去 h 或处理负项时出错。即使你很自信，也必须在纸上写出每一行代数推导，因为微小的符号错误就会导致失分。

3. Tangents and Normals without a Calculator | 切线方程与法线方程（手算）

Once you have a derivative, finding the equation of a tangent or normal is a purely algebraic task. The non-calculator setting means you must handle fractions and surds accurately. Keep your working systematic: find the y-coordinate, then the gradient, and only then plug into y − y₁ = m(x − x₁).

一旦求得导数，求切线或法线方程就完全是一个代数任务了。在非计算器环境中，你必须准确处理分数与根式。保持解题步骤系统化：先求 y 坐标，再求斜率，然后再代入 y − y₁ = m(x − x₁)。

Normal gradients are the negative reciprocal of the tangent gradient. If the tangent gradient is 2, the normal gradient is −1/2. When dealing with surds, rationalise only at the end if the question explicitly requires it, but keep exact values throughout.

法线斜率是切线斜率的负倒数。如果切线斜率为 2，法线斜率就是 −1/2。处理根式时，除非题目明确要求，否则到最后才进行分母有理化，但全程要保持精确值。

Tangent: y − f(a) = f'(a)(x − a)
Normal: y − f(a) = −1/f'(a) (x − a)

4. Stationary Points and Their Nature | 驻点及其性质

To locate stationary points, set f'(x) = 0 and solve the resulting equation. Without a calculator, you may need to factorise quadratics, solve cubic by grouping, or handle simple trigonometric equations. Always find both x- and y-coordinates.

要找驻点，就令 f'(x) = 0 并解所得的方程。没有计算器的情况下，你可能需要分解二次式、通过分组解三次方程，或处理简单的三角方程。务必同时求出 x 坐标和 y 坐标。

Determining the nature uses either the second derivative f”(x) or a sign chart for f'(x). With f”(a) > 0, the point is a local minimum; with f”(a) < 0, a local maximum. If f''(x)=0 or is hard to evaluate, use the first derivative test – note the sign change on either side.

判断驻点性质可用二阶导数 f”(x)，或用 f'(x) 的符号表。若 f”(a) > 0，该点是局部极小值点；若 f”(a) < 0，则是局部极大值点。如果 f''(x)=0 或不易求值，就用一阶导数检验——观察符号在驻点两侧的变化。

Simplify f'(x) fully before setting it equal to zero.

令 f'(x)=0 之前先充分化简 f'(x)。
Keep coordinates in exact form; do not approximate.

坐标保持精确形式，不要使用近似值。

5. Integration as Reverse Differentiation | 积分作为微分的逆运算

When calculators are removed, indefinite integration relies entirely on recognising standard derivatives. The power rule for integration is simply the reverse process: add one to the exponent and divide by the new exponent. Always include the constant of integration +c.

在去掉计算器后，不定积分完全依赖于识别标准导数。积分的幂法则就是逆过程：指数加一，再除以新指数。永远要加上积分常数 +c。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, n ≠ −1

Rewrite the integrand just as carefully as you do for differentiation. Split sums, convert roots and bring constants outside the integral. For example, ∫(4x² − 3/√x) dx becomes 4∫ x² dx − 3∫ x⁻¹/² dx.

像做微分时一样仔细地改写被积函数。拆分和式，转换根号，把常数提到积分号外。例如，∫(4x² − 3/√x) dx 化为 4∫ x² dx − 3∫ x⁻¹/² dx。

Remember that the integral of 1/x is ln|x| + c. This special case is frequently tested in non-calculator contexts, often combined with properties of logarithms.

记住 1/x 的积分是 ln|x| + c。这一特殊情况频频在非计算器考题中出现，经常结合对数的性质一同考查。

6. Definite Integrals and Exact Areas | 定积分与精确面积

Evaluating a definite integral without a calculator means substituting exact limits into an exact antiderivative. Work in stages: find the indefinite integral first, write it in brackets with the limits, then substitute. Keep fractions and surds as they are; do not convert to decimals.

在非计算器情况下计算定积分，就是把精确的上下限代入精确的原函数。分阶段处理：先求出不定积分，把原函数连同积分上下限写在方括号里，再代入。保留分数与根式，不要化为小数。

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)

Area calculations often require you to determine where a curve crosses the x-axis before integrating. Solve f(x)=0 to find intercepts, then create separate integrals if the function goes below the axis, because area treats regions under the axis as positive.

面积计算常常需要先确定曲线与 x 轴的交点再积分。解 f(x)=0 求出截距，如果函数到了轴下方，就需要分段积分，因为面积把轴下区域视为正值。

When an integral yields a negative value, take the absolute value when interpreting it as an area. Clearly state when you are computing a signed integral versus an actual area.

当积分值为负时，在将其解释为面积时要取绝对值。要清楚地说明你是在计算带号积分还是实际面积。

7. Using Trigonometric Identities in Differentiation and Integration | 在微积分中使用三角恒等式

Non-calculator problems with trigonometric functions assume you know exact values for sin, cos and tan at 0, π/6, π/4, π/3, π/2 and their multiples. Additionally, you should use identities to simplify expressions before differentiating or integrating.

含三角函数的非计算器问题预设你熟知 sin、cos、tan 在 0, π/6, π/4, π/3, π/2 及其整数倍角的精确值。此外，你应在求导或积分前使用恒等式化简表达式。

d/dx (sin x) = cos x, d/dx (cos x) = − sin x
∫ sin x dx = − cos x + c, ∫ cos x dx = sin x + c

Essential identities include sin²x + cos²x = 1, tan x = sin x / cos x, and the double angle formulas: sin 2x = 2 sin x cos x, cos 2x = cos²x − sin²x. Use these to rewrite products like sin x cos x into forms that are easier to integrate.

必备的恒等式包括 sin²x + cos²x = 1，tan x = sin x / cos x，以及倍角公式：sin 2x = 2 sin x cos x，cos 2x = cos²x − sin²x。利用它们可以把 sin x cos x 这类乘积改写为更容易积分的形式。

Memorise the derivatives of tan x = sec² x, and cot x = − csc² x for higher level.

牢记 tan x 的导数是 sec² x，cot x 的导数是 − csc² x（适用于高水平）。
Rewrite integrals like ∫ sin²x dx using the power-reducing identity cos 2x = 1 − 2 sin²x.

利用降幂公式 cos 2x = 1 − 2 sin²x 改写如 ∫ sin²x dx 的积分。

8. Integrating Simple Rational Functions | 简单有理函数的积分

When the integrand is a rational expression, try splitting the numerator over the denominator first. For ∫ (ax + b)/(cx + d) dx, perform polynomial division or rewrite as a constant plus a fraction with constant numerator.

当被积函数是有理式时，先尝试将分子拆开分配到分母上。对于 ∫ (ax + b)/(cx + d) dx，可进行多项式除法或改写为一个常数加上一个分子为常数的分式。

If you see a numerator that is the derivative of the denominator, recall that ∫ f'(x)/f(x) dx = ln|f(x)| + c. This pattern appears frequently in non-calculator reviews, often disguised by a prefactor that needs adjusting.

如果遇到分子是分母的导数，要想起 ∫ f'(x)/f(x) dx = ln|f(x)| + c。这种模式在非计算器复习题中频繁出现，经常藏在一个需要调整的系数后面。

∫ (2x)/(x²+1) dx = ln|x²+1| + c

For more complex rational expressions, partial fractions may be tested but only with distinct linear factors in the denominator. Set up the decomposition and solve for constants by equating coefficients – no calculator needed.

对于更复杂的有理式，可能会考查部分分式，但仅涉及分母为不同线性因子的情形。设出分解式，通过比较系数解出常数——全程无需计算器。

9. Applications to Kinematics | 运动学中的应用

Displacement s(t), velocity v(t) and acceleration a(t) are connected by differentiation and integration. v = ds/dt, a = dv/dt. To find displacement from velocity, integrate v(t) and use initial conditions to determine the constant of integration.

位移 s(t)、速度 v(t) 和加速度 a(t) 通过微分与积分相联系。v = ds/dt，a = dv/dt。要从速度求位移，对 v(t) 积分并利用初始条件确定积分常数。

Non-calculator questions often ask for the distance travelled in a given time interval. You must check if the velocity changes sign; if it does, integrate the absolute value of velocity piecewise. Always interpret the physical meaning rather than blindly integrating.

非计算器题目常会问给定时间区间内的路程。你必须检查速度是否变号；如果变号，就要分段积分速度的绝对值。始终要解读物理意义，而不是盲目积分。

Distance = ∫ |v(t)| dt over the interval

When velocity is zero, the particle may change direction – find those times.

速度为零时，质点可能改变方向——先找出这些时刻。
Use exact fractions for time values and simplify position expressions fully.

时间值使用精确分数，充分化简位置表达式。

10. Non-Calculator Strategy for Accuracy | 非计算器题型的准确率策略

Success in a non-calculator review set depends on organised written steps and frequent checking. Write the derivative rule you are using beside the line. When integrating, mentally differentiate your answer to verify it returns to the integrand.

在非计算器复习集中取得成功，取决于有条理的书面步骤和频繁的检查。在行旁写出你正在使用的求导法则。积分时，在心算中对你得到的答案求导，验算它能否回到原被积函数。

Manage fractions by keeping them factored rather than expanding large numbers prematurely. For example, leave 2/6 as is until you can cancel with other terms. Factorise the numerator and denominator before cancelling common factors.

处理分数时，尽量保持因式分解的形式，不要过早展开大数字。例如，先把 2/6 留作原样，直到能与其他项约分。在约去公因子前，对分子分母进行因式分解。

Finally, use exact values for trigonometric functions: sin(π/6)=1/2, cos(π/4)=√2/2. Keeping answers in square root and π notation is not only correct but often reveals simplifications that decimal approximations would hide.

最后，使用三角函数的精确值：sin(π/6)=1/2，cos(π/4)=√2/2。将答案保留为根号和 π 的形式不仅是正确的，而且常常能揭示那些小数近似会掩盖的化简可能。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Review set 16A – NON-CALCULATOR | 复习集 16A – 非计算器