📚 Mixed Exercise 7: Differentiation & Applications Review | 混合练习7：微分及应用复习
Mixed Exercise 7 in the Edexcel AS Pure Mathematics course draws together all the key differentiation techniques and their applications. It is designed to test your ability to differentiate a variety of functions, interpret gradients, locate stationary points, and apply calculus to real-world scenarios such as optimisation and motion. This article provides a structured walkthrough of the core skills needed to tackle similar mixed exercises with confidence, offering bilingual explanations to ensure you fully grasp both the procedures and the underlying concepts.
在Edexcel AS纯数课程中，混合练习7汇集了所有关键的微分技巧及其应用。它的目的是检验你对各类函数求导、解读梯度、寻找驻点以及将微积分应用于优化和运动等实际情境的能力。本文将通过双语讲解，系统地梳理攻克此类混合练习所需的核心技能，帮助你既掌握解题步骤，也理解深层概念。
1. Overview of the Exercise | 练习总体概览
Mixed Exercise 7 typically combines straightforward differentiation of polynomials, fractional powers, and negative powers with contextual problems. You will be asked to find equations of tangents and normals, determine intervals of increase and decrease, locate stationary points, and use second derivatives to classify them. Word problems involving maximum and minimum values, such as optimisation of area or volume, also appear regularly. The exercise reinforces the idea that differentiation is not just a set of rules but a powerful tool for analysing functions.
混合练习7通常将多项式、分数幂和负指数幂的直接求导与各种情境问题结合在一起。题目要求你求出切线和法线方程，确定函数的递增递减区间，找出驻点，并利用二阶导数进行分类。涉及极值（如面积或体积的最优化）的应用题也经常出现。这些练习旨在强调：微分不仅是一套规则，更是一种分析函数的强大工具。
2. Key Differentiation Rules | 核心求导法则
Before attempting Mixed Exercise 7, make sure you can apply the power rule confidently. For a term of the form axⁿ, the derivative is naxⁿ⁻¹. This rule works for any real n, including fractions and negative numbers. You must also be comfortable differentiating constant multiples and sums term by term. The second derivative, d²y/dx², is found by differentiating the first derivative again.
在尝试混合练习7之前，请确保你能熟练运用幂规则。对于形如 axⁿ 的项，其导数为 naxⁿ⁻¹。这一规则适用于任何实数 n，包括分数和负数。你还必须能轻松地逐项求导常数倍和和式。二阶导数 d²y/dx² 是对一阶导数再次求导得到的结果。
| Function f(x) | Derivative f'(x) |
|---|---|
| x⁴ | 4x³ |
| 3x⁻² | -6x⁻³ |
| 5√x (5x^½) | (5/2)x⁻½ |
| 2/x³ (2x⁻³) | -6x⁻⁴ |
Always rewrite terms like √x tiating. This eliminates common mistakes with algebraic manipulation. Once the power form is established, the rule f' ¹ can be applied mechanically, and you can then rewrite the result in the required format.
求导前务必将 √x 或 1/x² 这类项写成 x 的幂形式。这样可以消除代数操作中常见的失误。变成幂形式后，就可以机械地运用 f'(x) = naxⁿ⁻¹ 这一规则，然后把结果改写为题目要求的格式。
3. Tangents and Normals | 切线与法线
To find the equation of a tangent at a point (x₁, y₁), you first compute the gradient mt by substituting x₁ into f'(x). The tangent line then has the equation y - y₁ = mt(x - x₁). For the normal, which is perpendicular to the tangent, the gradient mn satisfies mt × mn = -1, so mn = -1/mt. Its equation follows the same point–gradient structure.
要求出点 (x₁, y₁) 处的切线方程，首先将 x₁ 代入 f'(x) 求出梯度 mt。然后切线方程就可写为 y - y₁ = mt(x - x₁)。法线与切线垂直，其梯度 mn 满足 mt × mn = -1，因此 mn = -1/mt。法线方程同样遵循点斜式结构。
xed Exercise 7 often includes rational functions where you must check the domain; a tangent may not exist at points where the derivative is undefined. Always simplify your final equation into the form ax + by + c = 0 or y = mx + c as requested.
要特别注意梯度为零（水平切线）或法线梯度无定义（垂直法线）的情况。混合练习7中经常出现有理函数，你需要检查定义域；在导数不存在的点上可能不存在切线。始终按照题目要求，将最终方程化简为 ax + by + c = 0 或 y = mx + c 的形式。
4. Increasing and Decreasing Functions | 函数的增减性
A function f(x) increases where f'(x) > 0 and decreases where f'(x) < 0. Solving these inequalities determines the intervals on which the function is strictly increasing or strictly decreasing. A common task is to give these intervals in set notation or as a union of intervals. Make sure to solve the inequality algebraically and interpret the sign of the derivative correctly.
当 f'(x) > 0 时函数递增，当 f'(x) < 0 时函数递减。求解这些不等式就能确定函数严格递增或严格递减的区间。常见的要求是用集合表示法或区间的并集给出这些区间。务必采用代数方法求解不等式，并正确解读导数的正负号。
In Mixed Exercise 7, you may encounter cubic or quadratic derivatives where factorising reveals the critical x-values. A sign diagram is particularly helpful: plot the roots on a number line, test a value in each interval, and read off where the derivative is positive or negative. This visual approach reduces errors when multiple intervals are involved.
在混合练习7中，你可能会遇到三次或二次导数，通过因式分解可以找到临界 x 值。符号图在此非常有用：在数轴上标出根，在每个区间内代入一个测试值，就能看出导数在何处为正、何处为负。这一可视化方法能减少处理多个区间时的错误。
5. Stationary Points | 驻点
Stationary points occur where f' he x-coordinates of all points where the tangent is horizontal. You must then substitute these x-values back into the original function f(x) to find the corresponding y-coordinates. Keep your answers as exact fractions or surds unless decimal approximations are requested.
驻点出现在 f'(x) = 0 之处。解此方程就能得到所有切线为水平位置的点的 x 坐标。然后你必须把这些 x 值代回原函数 f(x)，以求出对应的 y 坐标。除非题目明确要求，否则答案应保留为精确分数或根式，不要取小数近似值。
Watch out for hidden stationary points when the derivative is a rational function – the equation f' l algebraic handling. In Mixed Exercise 7, rational expressions sometimes hide stationary points that students miss by failing to consider the numerator separately. Always set the numerator equal to zero while ensuring the denominator is not zero at those points.
当导数是有理函数时，要留意隐藏的驻点——方程 f'(x) = 0 可能会有需要仔细代数处理的根。在混合练习7中，有理式有时会隐藏驻点，学生因未能单独处理分子而漏掉这些点。应始终令分子等于零，同时确保在这些点上分母不为零。
6. Classifying Turning Points | 对转折点进行分类
ion. The second derivative test is the quickest method: substitute the x-coordinate into f''(x). If f''(x) > 0, the point is a minimum; if f''(x) < 0, it is a maximum. If f''(x) = 0, the test is inconclusive and you should examine the sign of f'(x) on either side of the stationary point.
确定驻点后，你需要将其分类为局部极大值、局部极小值或拐点。二阶导数测试是最快捷的方法：将 x 坐标代入 f''(x)。若 f''(x) > 0，该点为极小值点；若 f''(x) < 0，则为极大值点。若 f''(x) = 0，则此测试无法得出结论，此时应检查驻点左右两侧 f'(x) 的符号。
arks by forgetting to state the nature of the stationary point using full sentences, e.g., 'f''(2) = 6 > 0, therefore the function has a local minimum at x = 2' the function at the turning point, so present your final answer clearly with both x and y.
许多考生因忘记用完整语句陈述驻点性质而失分，例如“f''(2) = 6 > 0，因此该函数在 x = 2 处有局部极小值”。在混合练习7中，清晰的理由陈述是必需的。你可能还需要给出精确坐标并指明转折点处的函数值，因此最终答案应清楚地同时写出 x 和 y 坐标。
7. Curve Sketching with Derivatives | 利用导数绘制曲线
Although complex sketches are not the main focus, Mixed Exercise 7 sometimes asks you to label stationary points and intercepts on a given set of axes. Use the derivative to confirm the shape: where f'(x) > 0 the curve rises, and where f'(x) < 0 it falls. The turning points help you position the peaks and troughs accurately.
尽管复杂的图形绘制并非重点，但混合练习7有时会要求你在给定的坐标系上标出驻点和截距。利用导数来确认曲线形状：f'(x) > 0 处曲线上升，f'(x) < 0 处曲线下降。转折点则有助于你准确定位波峰和波谷。
Annotate your sketch with the coordinates of intercepts (both x- and y-intercepts where possible) and label local maxima and minima. When sketching cubics, note the end behaviour determined by the leading term. Integrating information from f'(x) and f''(x) will make your sketch more accurate even without point-by-point plotting.
在草图中标注出截距坐标（尽可能同时标出 x 轴和 y 轴截距），并标识局部极大值与极小值。绘制三次曲线时，应注意由首项决定的末端走势。综合 f'(x) 与 f''(x) 的信息，即使不进行逐点描点，也能使你的草图更加准确。
8. Optimisation Problems | 优化问题
7 is devoted to optimisation: finding the maximum or minimum value of a quantity that depends on one variable. The typical strategy involves expressing the quantity to be optimised as a function of a single variable, often by using a constraint to eliminate another variable. Differentiate, set f' y that the solution gives a maximum or minimum using the second derivative or a sign test.
混合练习7中有相当一部分内容是优化问题：找出依赖于一个变量的某个量的最大值或最小值。典型的策略是先用约束条件消去另一个变量，将待优化量表示成单一变量的函数。然后求导，令 f'(x) = 0，再通过二阶导数或符号测试验证该解给出的是最大值还是最小值。
Geometry-based optimisation, such as maximising the area of a rectangle inscribed in a triangle or minimising the surface area of a cylinder given a fixed volume, is common. Translate the word description into algebraic expressions with a clear diagram. Always define your variable at the start and check that the domain is physically reasonable (e.g., positive lengths). The final answer should state the optimum value and the corresponding dimensions.
基于几何的优化题很常见，例如最大化内接于三角形中的矩形面积，或在体积固定时最小化圆柱体的表面积。借助清晰的图示，把文字描述转化为代数表达式。务必在起始处定义变量，并验证其定义域在物理上是合理的（如长度必须为正数）。最终答案应给出最优值及其对应的尺寸。
9. Kinematics Applications | 运动学应用
t². Problems may ask for the times when a particle is at rest (v = 0), the distance from the origin at a given instant, or the maximum speed. Apply differentiation exactly as with any other function, but interpret the results in context.
如果混合练习7包含运动学问题，你会遇到位移函数 s(t)，速度 v(t) = ds/dt，以及加速度 a(t) = dv/dt = d²s/dt²。题目可能要求计算质点静止的时刻（v = 0）、某一瞬间距原点的距离，或最大速率。求导方法与任何其他函数完全相同，但要结合情境解读结果。
equires checking when the particle changes direction (v = 0) and summing absolute displacements. Mixed Exercise 7 tests whether you can move flexibly between displacement, velocity, and acceleration using differentiation.
单位处理必须严谨：如果 s 以米为单位，t 以秒为单位，那么 v 的单位是 m/s，a 的单位是 m/s²。题目还可能会要求总路程，这就需要检查质点何时改变运动方向（v = 0）并累加位移的绝对值。混合练习7考查你能否熟练地通过微分在位移、速度和加速度之间灵活转换。
10. Common Pitfalls and Exam Tips | 常见陷阱与应试技巧
negative or fractional powers after differentiation, especially when simplifying results. Always double-check your algebra when rewriting expressions. Another error is failing to distinguish between finding the gradient of a curve at a point and finding the equation of a tangent – the gradient is only the first step. Also, remember that f'' oes not automatically imply a point of inflection; you must confirm a change in concavity.
一个常见错误是在求导后没有正确处理负指数或分数指数，尤其是在化简结果时。务必反复检查代数运算。另一个错误是混淆求曲线上一点处的梯度与求切线方程——梯度只是第一步。同时请记住，f''(x) = 0 并不意味着一定是拐点；你必须验证凹性的改变。
Under exam pressure, students often forget to find the y-coordinate of a stationary point or omit correct notation (e.g., writing 'max' instead of 'local maximum at (2, -3)'). In Mixed Exercise 7–style tasks, present your reasoning step by step. If a question involves a worded context, briefly restate the conclusion in the context (e.g., 'the maximum area is 36 cm² when the width is 6 cm').
在考试压力下，学生常常忘记求驻点的 y 坐标，或遗漏正确的表达方式（例如只写“max”而没写“在 (2, -3) 处为局部极大值”）。在处理混合练习7这一类题目时，要逐步展示你的推理过程。如果题目涉及文字情境，应简要结合情境重申结论（例如“当宽为6 cm时，最大面积为36 cm²”）。
Time management is crucial: allocate a few minutes to check critical solutions, especially those that lead to domain violations. Practising the full Mixed Exercise 7 under timed conditions will sharpen your algebraic fluency and help you become comfortable with the variety of contexts in which differentiation appears.
时间管理至关重要：安排几分钟来检查关键解，特别是那些可能导致定义域矛盾的解。在限时条件下完整操练混合练习7，将提高你的代数熟练度，并让你轻松应对微分所出现的各类情境。
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