Miscellaneous Exercises 7 | 综合练习七

📚 Miscellaneous Exercises 7 | 综合练习七

Miscellaneous exercises bring together several topics in one set of questions. They test your ability to choose the right method, to apply standard techniques accurately, and to interpret your results. In this article we will solve a selection of typical problems, with step-by-step reasoning.

综合练习将多个知识点汇集在同一组题目中，考查你选择正确方法、准确运用标准技巧以及解读结果的能力。本文将选取一组典型题目，进行分步讲解与推理。

1. Quadratic Inequalities | 二次不等式

Solve x² – 5x + 6 < 0. Start by factorising the quadratic. We look for two numbers that multiply to 6 and add to -5; these are -2 and -3, so x² - 5x + 6 = (x - 2)(x - 3). The critical values are x = 2 and x = 3. Test each interval on a number line: for x < 2 both factors are negative, so the product is positive; for 2 < x < 3 one factor is negative and the other positive, so the product is negative; for x > 3 both factors are positive, so the product is positive. The inequality asks for the interval where the product is less than zero, so the solution is 2 < x < 3.

解不等式 x² – 5x + 6 < 0。首先对二次式因式分解：找两个数相乘为 6 且相加为 -5，分别是 -2 和 -3，所以 x² - 5x + 6 = (x - 2)(x - 3)。临界值为 x = 2 与 x = 3。在数轴上测试各区间：当 x < 2 时，两个因子都为负，乘积为正；当 2 < x < 3 时，一负一正，乘积为负；当 x > 3 时，两个因子都为正，乘积为正。题干要求乘积小于零的区间，因此解为 2 < x < 3。

Tip: When solving quadratic inequalities, always factorise first and then use a sign diagram. Do not simply write the answer by looking at the critical values, because the direction of the sign changes only at single roots.

提示：解二次不等式时，先因式分解，再用符号图判断区间符号。不要只凭临界值直接写答案，因为在单根处符号才会改变。

2. Equation of a Circle | 圆的方程

A diameter has endpoints A(1,2) and B(5,6). The centre is the midpoint of AB: ((1+5)/2, (2+6)/2) = (3,4). The radius is half the length of the diameter. Distance AB = √[(5-1)² + (6-2)²] = √(16+16) = √32 = 4√2, so the radius is r = 2√2. Therefore the equation of the circle is:

一条直径的两个端点为 A(1,2) 与 B(5,6)。圆心是 AB 的中点：((1+5)/2, (2+6)/2) = (3,4)。半径等于直径长度的一半。AB 的长度 = √[(5-1)² + (6-2)²] = √(16+16) = √32 = 4√2，所以半径 r = 2√2。因此圆的方程为：

(x – 3)² + (y – 4)² = 8

Tip: If you are given the endpoints of a diameter, the centre is simply their midpoint. Remember to square the radius when writing the equation in standard form.

提示：若已知直径为端点坐标，圆心就是两点中点。写出标准方程时，半径需要平方。

3. Differentiation: Product Rule | 乘积法则求导

Differentiate y = x² sin x. Use the product rule: if y = u v, then dy/dx = u dv/dx + v du/dx. Set u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x. Hence:

求导 y = x² sin x。使用乘积法则：若 y = u v，则 dy/dx = u dv/dx + v du/dx。令 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x。因此：

dy/dx = x² cos x + 2x sin x

Tip: Keep the order of terms clear. The product rule is useful whenever two different functions are multiplied together, and it can be combined with the chain rule in later problems.

提示：注意保持各项顺序清晰。当两个不同函数相乘时，乘积法则非常有效；后续题目中还可与链式法则结合使用。

4. Integration by Parts | 分部积分法

Evaluate ∫ x eˣ dx. Use the integration by parts formula:

计算 ∫ x eˣ dx。使用分部积分公式：

∫ u dv = u v – ∫ v du

Let u =

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