📚 AS Further Mathematics: Experimental Operation Guide | AS 进阶数学：实验操作指南
In AS Further Mathematics, hands-on experiments using calculators, spreadsheets, or programming tools bridge the gap between abstract theory and tangible insight. This guide walks you through key numerical methods, matrix transformations, and simulation techniques that appear across syllabi, with step-by-step instructions designed to reinforce your understanding and prepare you for both coursework and examinations.
在 AS 进阶数学中，借助计算器、电子表格或编程工具进行动手实验，能将抽象理论与直观认识连接起来。本指南将带你逐一操作常见的数值方法、矩阵变换与模拟技术，这些内容贯穿各大考纲，分步说明旨在巩固理解，并为课程作业与考试做好准备。
1. Newton-Raphson Method for Root Finding | 牛顿-拉弗森法求根实验
Objective: Use the iterative formula xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) to approximate a root of f(x) = 0. Choose a starting value x₀ near the suspected root and perform iterations until successive values agree to a required accuracy. This experiment visually demonstrates quadratic convergence.
目标：利用迭代公式 xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) 逼近方程 f(x)=0 的根。选取在猜测根附近的初值 x₀，反复迭代直到相邻值达到所需精度。本实验能直观展示二次收敛的特性。
Procedure: Define f(x) = x³ − 2x − 5 and its derivative f'(x) = 3x² − 2. Start with x₀ = 2. On a graphical calculator or in a spreadsheet, enter the recursive formula. After three iterations you should observe the sequence: 2.1, 2.0946, 2.09455…, converging to the root near 2.09455.
步骤：设 f(x)=x³−2x−5，导数 f'(x)=3x²−2。取初值 x₀=2。在图形计算器或电子表格中输入递推公式。三次迭代后可观察到序列：2.1、2.0946、2.09455…，收敛至根约 2.09455。
Extension question: What happens if you start with x₀ = 0? The derivative becomes −2, leading to a much slower approach or divergence, illustrating sensitivity to the initial guess.
拓展问题：若从 x₀=0 开始会怎样？导数为 −2，导致收敛很慢或发散，体现了初值敏感性。
2. Trapezium Rule for Numerical Integration | 梯形法则数值积分实验
Objective: Approximate the definite integral ∫ₐᵇ f(x) dx by dividing the interval into n strips of equal width h = (b−a)/n and applying the trapezium rule: Area ≈ h/2 [y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ]. Increase n to observe improved accuracy.
目标：将区间 [a,b] 等分为 n 份，每份宽 h=(b−a)/n，用梯形法则近似定积分：面积 ≈ h/2 [y₀+2(y₁+y₂+…+yₙ₋₁)+yₙ]。增大 n 以观察精度提升。
Use f(x) = √(1+x³) from x=0 to x=2 with n=4. Compute h=0.5, tabulate y-values, and sum them. The approximation should be compared with the exact value obtained from a CAS tool to estimate the error.
取 f(x)=√(1+x³)，区间 0 到 2，n=4。h=0.5，列表计算 y 值并求和。将近似值与 CAS 工具得出的精确值比较，估计误差。
In a spreadsheet, you can set up columns for x, y, and the weighted sum. Conditional formatting can highlight how the error roughly halves when n doubles, illustrating the second-order accuracy of the method.
在电子表格中可建立 x、y 和加权和列。条件格式可突出显示当 n 翻倍时误差约减半，演示该方法的二阶精度。
3. Fixed-Point Iteration | 不动点迭代实验
Objective: Solve equations of the form x = g(x) by iterating xₙ₊₁ = g(xₙ). Success depends on |g'(x)| < 1 near the root. Experiment with different rearrangements to see when convergence occurs.
目标：通过迭代 xₙ₊₁ = g(xₙ) 求解 x=g(x) 形式的方程。收敛条件是在根附近 |g'(x)|<1。尝试不同的变形，观察何时收敛。
Take the equation x³ − 3x + 1 = 0. Rearranged as x = (x³+1)/3 gives g(x) = (x³+1)/3. Starting with x₀=1, compute several terms. You will see a slow spiral convergence to approximately 0.3473. Try another rearrangement x = (3x−1)^(1/3)—this may diverge for certain initial values.
以方程 x³−3x+1=0 为例。变形为 x=(x³+1)/3，g(x)=(x³+1)/3。从 x₀=1 开始计算若干项，会看到缓慢螺旋收敛至约 0.3473。尝试另一变形 x=(3x−1)^(1/3)，对某些初值可能发散。
Plot the line y = x and the curve y = g(x). Use a cobweb diagram tool to visualise the iteration steps. This geometric illustration helps you understand why certain slopes produce convergence.
绘制直线 y=x 与曲线 y=g(x)，使用蛛网图工具可视化迭代过程。这一几何图示有助于理解为什么某些斜率会产生收敛。
4. Bisection Method Experiment | 二分法实验
Objective: Find a root of f(x)=0 in an interval [a,b] where f(a) and f(b) have opposite signs. Repeatedly halve the interval and retain the subinterval where the sign changes. The method guarantees linear convergence.
目标：在 f(a) 与 f(b) 异号的区间 [a,b] 内寻找 f(x)=0 的根。不断将区间二分，保留符号变化的那一半。该方法保证线性收敛。
For f(x) = eˣ − 4x, with a=1, b=2 (f(1)≈ −1.28, f(2)≈3.39). At each step calculate the midpoint m = (a+b)/2 and evaluate f(m). Replace a or b with m accordingly. After 10 iterations the root is pinned to about 1.4296.
对于 f(x)=eˣ−4x，a=1, b=2（f(1)≈−1.28，f(2)≈3.39）。每一步计算中点 m=(a+b)/2 并求 f(m)，相应替换 a 或 b。10 次迭代后根被锁定在约 1.4296。
Implement this on a spreadsheet with conditional logic. Track the bracket width after each iteration: it halves each time. This steady reduction contrasts with the variable speed of Newton-Raphson, giving you a feel for different convergence rates.
在电子表格中利用条件逻辑实现该方法。跟踪每次迭代后的区间宽度：每次都减半。这种稳定的缩窄与牛顿法变化的速度形成对比，让你感受不同的收敛速率。
5. Euler’s Method for Differential Equations | 欧拉方法解微分方程实验
Objective: Approximate the solution to a first-order ODE dy/dx = f(x,y) with initial condition y(x₀)=y₀. Use step length h and update yₙ₊₁ = yₙ + h×f(xₙ, yₙ). Reduce h to see improved accuracy at the cost of more steps.
目标：近似求解一阶常微分方程 dy/dx=f(x,y)，初值 y(x₀)=y₀。用步长 h，按 yₙ₊₁ = yₙ + h×f(xₙ, yₙ) 逐步更新。减小 h 可看到精度提升但步骤增多。
Solve dy/dx = x + y, y(0)=1, from x=0 to x=1 using h=0.2. Construct a table: x₀=0, y₀=1; x₁=0.2, y₁=1+0.2×(0+1)=1.2; x₂=0.4, y₂=1.2+0.2×(0.2+1.2)=1.48; … Compare with the exact solution y=2eˣ−x−1.
解 dy/dx=x+y，y(0)=1，从 x=0 到 x=1，h=0.2。建表：x₀=0, y₀=1；x₁=0.2, y₁=1+0.2×(0+1)=1.2；x₂=0.4, y₂=1.2+0.2×(0.2+1.2)=1.48；… 与精确解 y=2eˣ−x−1 比较。
Try h=0.1 and observe the global error reduction. This experiment underscores the trade-off between computational effort and accuracy inherent in numerical analysis.
尝试 h=0.1，观察整体误差减小。本实验强调了数值分析中计算量与精度之间的权衡。
6. Matrix Transformation of Geometric Figures | 矩阵变换的几何图形实验
Objective: Apply 2×2 matrices to sets of coordinates representing a unit square or other shapes, then plot the image to understand linear transformations such as stretches, shears, rotations, and reflections.
目标：将 2×2 矩阵作用于表示单位正方形或其他图形的坐标集，绘制像点以理解线性变换，如拉伸、剪切、旋转和反射。
Define the unit square with vertices (0,0), (1,0), (1,1), (0,1). Apply the matrix M = [[2, 0], [0, 1]] to scale horizontally by factor 2. Plot original and image on dynamic geometry software. The area scale factor is |det(M)| = 2.
取单位正方形顶点 (0,0)、(1,0)、(1,1)、(0,1)。应用矩阵 M=[[2,0],[0,1]] 进行水平方向比例因子 2 的拉伸。在动态几何软件中绘制原图与像。面积放大因子为 |det(M)|=2。
Experiment with shear matrix [[1, 1], [0, 1]], rotation matrix [[cosθ, −sinθ], [sinθ, cosθ]] for θ=30°, and a combined transformation. Note how the order of multiplication affects the final result. This visual approach cements the geometric interpretation of matrix algebra.
尝试剪切矩阵 [[1,1],[0,1]]、θ=30° 的旋转矩阵 [[cosθ, −sinθ],[sinθ, cosθ]]，以及复合变换。注意乘法顺序如何影响最终结果。这种视觉化方式巩固了矩阵代数的几何解释。
7. Monte Carlo Estimation of π | 蒙特卡罗方法估计 π 实验
Objective: Use random points in a unit square to estimate π/4 by counting the proportion falling inside a quarter circle of radius 1. This introduces stochastic simulation and the law of large numbers.
目标：在单位正方形内产生随机点，统计落入半径 1 的四分之一圆内的比例来估计 π/4。这引入了随机模拟与大数定律。
Generate N pairs of random numbers (x,y) uniformly distributed in [0,1]. Count those where x²+y² ≤ 1. The ratio count/N approximates π/4. Multiply by 4 to get an estimate of π. Increase N to reduce sampling error.
生成 N 对在 [0,1] 上均匀分布的随机数 (x,y)，统计满足 x²+y²≤1 的点数。比值 count/N 近似 π/4，乘以 4 即得 π 估计值。增大 N 可减小抽样误差。
Implement this in a spreadsheet with RAND() functions. For N=1000 you might obtain 3.12; N=10000 could yield 3.144. Plot the running estimate to see convergence. Discuss the standard error proportional to 1/√N.
在电子表格中用 RAND() 函数实现。N=1000 时可能得到 3.12；N=10000 可能得到 3.144。绘制运行估计值观察收敛。讨论标准差与 1/√N 成正比。
8. Exploring Complex Numbers with Argand Diagrams | Argand 图与复数运算实验
Objective: Visualise addition, subtraction, multiplication, and division of complex numbers on an Argand diagram using dynamic geometry or a graphical calculator. Observe the geometric effects: translation, scaling, rotation.
目标：使用动态几何或图形计算器在 Argand 图上可视化复数的加减乘除。观察几何效果：平移、缩放、旋转。
Represent z₁ = 3+2i, z₂ = 1−i as vectors. Addition z₁+z₂ = 4+i is the diagonal of a parallelogram. Multiplication by i (rotation by 90°) and by a real number (scaling) can be demonstrated sequentially.
将 z₁=3+2i, z₂=1−i 表示为向量。加法 z₁+z₂=4+i 为平行四边形对角线。依次演示乘以 i（旋转 90°）和乘以实数（缩放）。
Use the transformation w = (z+1)/(z−1) to map a grid of points and discover Möbius transformations. This gives a preview of further pure topics and deepens understanding of the complex plane.
使用变换 w=(z+1)/(z−1) 映射网格点，探索莫比乌斯变换。这将预览进阶纯数主题，加深对复平面的理解。
9. Solving Linear Systems with Gaussian Elimination on a Spreadsheet | 用电子表格进行高斯消元法实验
Objective: Solve a system of three linear equations by systematically performing row operations. Build a spreadsheet template that exemplifies the elimination and back-substitution steps.
目标：通过系统执行行变换求解三元线性方程组。建立一个电子表格模板，展示消元与回代步骤。
System: 2x+ y− z=8, −3x− y+2z=−11, −2x+ y+2z=−3. Create an augmented matrix in cells. Use row operations to produce zeros below the pivot in column 1, then column 2, and finally back-substitute. Check the solution x=2, y=3, z=−1.
方程组：2x+y−z=8, −3x−y+2z=−11, −2x+y+2z=−3。在单元格中建立增广矩阵。使用行操作在列 1 的主元下方消零，再列 2，最后回代。验证解 x=2, y=3, z=−1。
Recording each row operation with notes mimics an examiner’s expectation for showing working. Automating this process through basic formulas reduces arithmetic errors and builds confidence in matrix methods.
记录每一步行操作并加注释，模仿考官对展现步骤的要求。通过基础公式自动执行该过程可减少算术错误，增强对矩阵方法的信心。
10. Difference Equation Modelling of Population Growth | 差分方程模型与人口增长实验
Objective: Model population change using a recurrence relation uₙ₊₁ = r uₙ (1 − uₙ) (logistic map). Explore the long-term behaviour for various values of r between 2 and 4, observing steady states, periodic cycles, and chaos.
目标：使用递推关系 uₙ₊₁ = r uₙ (1−uₙ)（逻辑斯蒂映射）模拟人口变化。探索 r 在 2 到 4 之间不同取值时的长期行为，观察稳态、周期循环与混沌。
Set u₀ = 0.5, run iterations for r=2.5 (converges to a stable point), r=3.2 (period-2 cycle), r=3.5 (period-4), and r=3.9 (chaos). Plot uₙ vs n to visualise the dynamics. This experiment bridges AS Further Maths with the concept of deterministic chaos.
取 u₀=0.5，对 r=2.5（收敛至稳定点）、r=3.2（2 周期）、r=3.5（4 周期）、r=3.9（混沌状态）进行迭代。绘制 uₙ 关于 n 的图像可视化动态过程。本实验将 AS 进阶数学与确定性混沌概念联系起来。
In a spreadsheet you can quickly copy down the formula and generate a bifurcation diagram by plotting the long-term values against r. This offers a glimpse of more advanced dynamical systems studies.
在电子表格中可快速向下复制公式，并绘制长期值关于 r 的分岔图，提供对更高级动力系统研究的一瞥。
11. Using the Modulus Function and Graph Sketching with Technology | 使用技术进行绝对值函数与图像绘制实验
Objective: Investigate the graphs of y = |f(x)|, y = f(|x|), and related piecewise-linear functions. Technology enables rapid comparison and deepens understanding of transformations.
目标：研究 y=|f(x)|, y=f(|x|) 及相关分段线性函数的图像。技术工具使得快速比较成为可能，加深对变换的理解。
Start with f(x)=x²−4. Sketch y = |x²−4|: the portion below the x-axis is reflected above. Then sketch y = (|x|)²−4 = x²−4, which is even. Compare with y = |x²−4| visually. Extend to modulus equations like |2x−1| = |x+3|; solve graphically by intersecting V-shaped graphs.
从 f(x)=x²−4 开始。绘制 y=|x²−4|：x 轴下方的部分被翻折到上方。再绘制 y=(|x|)²−4=x²−4，这是一个偶函数。与 y=|x²−4| 做视觉对比。拓展到绝对值方程如 |2x−1|=|x+3|，通过 V 形图的交点图解求解。
Use sliders to see the effect of translating or stretching modulus graphs. This hands-on manipulation builds intuition for inequalities like |x−a| < b and their geometric meaning.
使用滑块观察平移或拉伸绝对值图像的效果。这种实际操控为不等式如 |x−a|
12. Numerical Solution of Second-Order ODEs Using the Auxiliary Equation and Verification | 用辅助方程解二阶常微分方程及验证实验
Objective: Solve a homogeneous second-order ODE analytically, then verify the solution by plotting and by numerical approximation. Compare the exact and Euler-improved solutions.
目标：解析求解齐次二阶常微分方程，再通过绘图和数值近似进行验证。比较精确解与改进欧拉法解的差异。
For d²y/dx² − 3 dy/dx + 2y = 0, y(0)=1, y'(0)=0, the auxiliary equation m²−3m+2=0 gives roots m=1,2, so y = Aeˣ + Be²ˣ. Apply initial conditions to find A=2, B=−1. Plot this exact solution. Then apply the Euler or Runge-Kutta method to generate approximate points; overlay them to see the fit.
对于 d²y/dx² − 3 dy/dx + 2y = 0，y(0)=1, y'(0)=0，辅助方程 m²−3m+2=0 得根 m=1,2，故 y = Aeˣ + Be²ˣ。代入初始条件得 A=2, B=−1。绘制该精确解。然后用欧拉法或龙格-库塔法生成近似点并叠加，观察拟合程度。
This experiment links pure analytical skills with computational checks, a vital practice in modern applied mathematics. It also highlights the importance of step length in stability.
该实验将纯解析技能与计算验证相结合，是现代应用数学的关键实践。同时凸显了步长对稳定性的重要性。
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