📚 Review set 12B - CALCULATOR | 复习题集 12B - 计算器专项
Graphical display calculators (GDC) are a central tool in IB Mathematics, especially for Paper 2 and Paper 3 exams where technology is permitted. This revision set focuses on building fluency with calculator techniques across the AA and AI syllabuses, covering equation solving, calculus, statistics, matrices, sequences, complex numbers, and more. Each section provides worked examples and reminders of the keystroke logic needed to save time under exam pressure.
图形计算器 (GDC) 是 IB 数学考试中至关重要的工具，尤其是在允许使用技术的试卷二和试卷三中。本题集专注于提升计算器操作熟练度，涵盖解方程、微积分、统计、矩阵、数列、复数等多个模块。每个小节提供典型例题与操作逻辑提示，帮助考生在考试压力下高效作答。
1. Equation Solving with GDC | 使用图形计算器解方程
the left-hand and right-hand sides of an equation as two separate functions, then use the 'intersect' or 'numerical solver' he x-intercepts, or use the polynomial root finder. Always round your answer to three significant figures unless told otherwise.
先将方程左右两边分别定义为两个函数，然后使用“交点”或“数值求解器”功能。例如，解方程 2x³ - 5x² + 3 = 0 时，可绘制 y = 2x³ - 5x² + 3 并寻找 x 轴截距，或使用多项式求根器。除非另有说明，答案一律四舍五入至三位有效数字。
e system in matrix form or use the equation solver app if available. A 3×3 system such as:
2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3
can be solved instantly by inputting the augmented matrix and applying reduced row echelon form (rref).
对于联立方程组，可将其保存为矩阵形式，或使用（如果可用）方程求解器应用。一个 3×3 方程组，如：
2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3
可通过输入增广矩阵并应用简化行阶梯形 (rref) 快速求解。
Remember to check for extraneous solutions when equations involve square roots or rational expressions, as the solver may return values outside the domain.
当方程涉及平方根或有理式时，务必检查增根，因为求解器可能会返回域外的值。
2. Graphing and Analyzing Functions | 绘制与分析函数图像
ical asymptote mentally and check the table of values to confirm the behaviour near x = 1.
调整合适的观察窗口至关重要。使用 ZoomFit 或手动设置 xmin, xmax, ymin, ymax，以捕捉截距、转折点和渐近线等关键特征。对于有理函数，如 f(x) = (x² - 4)/(x - 1)，务必在脑海中画出垂直渐近线，并通过数值表确认 x = 1 附近的行为。
The 'zero', 'minimum', 'maximum' and 'intersect' trace tools allow you to extract coordinates directly. When a question asks for the coordinates of a local minimum, navigate to the point and copy the displayed values with full precision before rounding.
“零点”、“最小值”、“最大值”和“交点”等追踪工具可直接提取坐标。题目要求局部最小值坐标时，先导航至该点，复制显示的全精度值，再进行四舍五入。
Use the table feature to evaluate functions at specific integer values, which is especially helpful when sketching by hand or completing a table of signs.
利用表格功能计算特定整数点的函数值，这在手绘草图或填写符号表时尤其有用。
3. Calculus Tools: Derivatives and Integrals | 微积分工具：导数与积分
Typing d/dx (function) | at a point, which is faster than using the gradient formula. For symbolic derivatives, many GDCs can handle the derivative of expressions like eˣ·sin(x) and display the result in simplified form – but always check if the exam expects algebraic working alongside the calculator output.
输入 d/dx (函数) | x = a 可得到某点的导数值，这比使用斜率公式更快。对于符号导数，许多图形计算器可以处理 eˣ·sin(x) 这类表达式的导数，并显示简化结果——但务必检查考试是否要求同时提供代数过程。
) command. For area between two curves, set up the integral of (top function) - (bottom function) or simply evaluate the intersection points and use absolute values if necessary. Pay attention to units when the function models a physical quantity.
定积分同样直接：使用 ∫(f(x), x, a, b) 命令。求两曲线间的面积时，设置“上方函数”减“下方函数”的积分，或直接求出交点，必要时使用绝对值。当函数模拟物理量时，请留意单位。
For kinematics problems, remember that the calculator can find displacement and total distance travelled from a velocity-time graph, but the total distance requires integrating |v(t)|, which you can handle by splitting the integral at sign changes.
在运动学问题中，计算器可从速度-时间图求出位移和总路程，但总路程需要积分 |v(t)|，可通过在符号变化点拆分积分来处理。
4. Statistics and Probability Distributions | 统计与概率分布
Enter data into lists and use the 1-Var Stats or 2-Var Stats commands to obtain mean, standard deviation, quartiles and the linear regression line. Always check whether you are working with a sample or a population – the sx and σx symbols distinguish sample and population standard deviations.
将数据输入列表，使用单变量统计或双变量统计命令获取平均值、标准差、四分位数和线性回归线。务必辨别正在处理的是样本还是总体—— sx 和 σx 符号分别表示样本和总体标准差。
For normal distributions, use the built-in normalcdf(lower, upper, μ, σ) to find probabilities, and invNorm(area, μ, σ) to find z-scores or raw values. Binomial and Poisson probabilities can be obtained via binompdf/binomcdf and poissonpdf/poissoncdf. Avoid the common mistake of using pdf when the question asks for a cumulative probability.
对于正态分布，使用内置的 normalcdf(下限, 上限, μ, σ) 求概率，invNorm(面积, μ, σ) 求 z 分数或原始值。二项分布和泊松分布的概率可通过 binompdf/binomcdf 和 poissonpdf/poissoncdf 获得。避免常见错误：当题目要求累积概率时，却使用了概率密度函数 pdf。
When conducting chi-squared tests, carefully input the observed matrix and store the expected matrix. The calculator will display the test statistic χ², the p-value and degrees of freedom, making it unnecessary to look up critical value tables.
进行卡方检验时，仔细输入观测值矩阵并存储期望值矩阵。计算器会显示检验统计量 χ²、p 值和自由度，无需查临界值表。
5. Matrix Operations | 矩阵运算
te X = A⁻¹B, but verify that det(A) ≠ 0 first. The rref command on the augmented matrix [A| earer picture of the solution set, especially for systems with infinite solutions or none.
存储矩阵 [A]、[B] 等，直接执行 A + B、A × B、A⁻¹ 和 det(A) 等运算。解方程组 AX = B 时，计算 X = A⁻¹B，但先确认 det(A) ≠ 0。对增广矩阵 [A|B] 使用 rref 命令通常能更清晰地呈现解集，尤其是对于无穷多解或无解的系统。
Eigenvalues and eigenvectors are required in some HL topics; use the eigVl and eigVc commands if supported. Always double-check the dimension agreement before multiplication – the number of columns in the first matrix must equal the number of rows in the second.
某些 HL 主题需要特征值和特征向量；如果支持，可使用 eigVl 和 eigVc 命令。乘法前务必检查维度匹配——第一个矩阵的列数必须等于第二个矩阵的行数。
6. Sequences and Series | 数列与级数
Arithmetic and geometric sequences can be explored efficiently using sequence graphing mode or the seq( command. For the sum of the first n terms, use the sum(seq( formula )) structure. For infinite geometric series, check that |r| < 1 before applying the formula S∞ = a/(1 - r); the calculator can confirm the limiting value numerically.
使用数列绘图模式或 seq( 命令可以高效地探索等差数列和等比数列。求前 n 项和时，可使用 sum(seq(公式)) 结构。对于无穷等比级数，应用公式 S∞ = a/(1 - r) 前，先检查 |r| < 1；计算器可数值验证极限值。
When modelling compound interest or population growth, the TVM (Time Value of Money) solver handles N, I%, PV, PMT, FV, P/Y, C/Y. Pay careful attention to the sign conventions: money received is positive, money paid out is negative.
模拟复利或人口增长时，TVM（资金时间价值）求解器可处理 N、I%、PV、PMT、FV、P/Y、C/Y。特别注意符号约定：收到的钱为正，付出的钱为负。
7. Complex Numbers | 复数
ex mode and solve z³ = -8 to obtain three distinct roots, then display them in both forms.
图形计算器对复数的支持允许直接输入 a + bi，进行直角坐标与极坐标形式转换，以及共轭、模和辐角等运算。例如，求 -8 的立方根时，将计算器设为复数模式，解 z³ = -8 得到三个不同的根，然后以两种形式显示。
De Moivre's theorem calculations become quicker: to compute (√2 cis 45°)⁶, enter the complex number in polar form, raise to the power 6, and convert back. Always isolate the principal argument in (-π, π] unless a different range is specified.
棣莫弗定理的计算变得更快：要计算 (√2 cis 45°)⁶，以极坐标形式输入复数，求六次幂，再转换回来。除非另有说明，主辅角应限定在 (-π, π] 内。
8. Trigonometric Equations and Identities | 三角方程与恒等式
raightforward with the solver, but understanding the periodic nature is still essential. Use the graph to identify all solutions within the domain, then verify by substituting each candidate back into the original equation.
用求解器解 sin(2x) = 0.5 (0 ≤ x ≤ 2π) 很直接，但理解周期性仍然必要。利用图像找出域内所有解，然后将每个候选解代回原方程进行验证。
identity holds. However, IB requires algebraic proof, so use this only as a checking tool.
证明恒等式时，计算器可通过绘制两表达式之差来快速检验相等性；若图像为 y = 0 的水平直线，则表明恒等式成立。但 IB 要求代数证明，因此仅将此作为检查工具。
9. Optimization Problems | 最优化问题
Optimization often starts by writing a function of one variable A(x) for area or V(x) for volume. Graph the function on an appropriate domain derived from the constraints, then use the 'maximum' feature to locate the optimal x-value. For absolute extrema on a closed interval, evaluate the function at the endpoints and at interior critical points; the calculator can find derivatives numerically if the algebraic derivative is messy.
最优化问题通常从写出单变量函数入手，如面积 A(x) 或体积 V(x)。根据约束条件确定合适的定义域并绘制函数图像，然后使用“最大值”功能找到最优 x 值。对于闭区间上的绝对极值，同时计算端点和内部临界点的函数值；若代数导数复杂，计算器可数值求导。
Remember to justify that the critical point yields a maximum or minimum by either checking the sign change of the derivative or using the second derivative test (f ''(x) < 0 indicates a maximum).
记得通过检查导数符号变化或使用二阶导数检验（f ''(x) < 0 表示极大值）来证明极值点的性质。
10. Vector Geometry | 向量几何
s are quickly computed. To find the acute angle between two lines, use the formula cos θ = |a · b| / (|a||b|) and apply the calculator's built-in norm and dot product functions.
向量可存储为列表或使用专用向量模式。点积 a·b、叉积 a×b 和向量夹角可快速计算。求两直线间的锐角时，使用公式 cos θ = |a·b| / (|a||b|)，并应用计算器内置的范数和点积功能。
When finding the intersection of two lines in 3D, set up the parametric equations and solve the system of three equations for two parameters. The calculator can handle this via the simultaneous equation solver; if the system is inconsistent, the lines are skew.
求三维空间两直线交点时，建立参数方程并解含有两个参数的三方程系统。计算器可通过联立方程求解器处理；若系统无解，则两直线为异面直线。
11. Probability Simulations | 概率模拟
Random number generators rand() and randInt() allow simulation of experiments, such as tossing a coin or rolling a die. By storing many trials in a list and analysing the results, you can estimate probabilities and compare them with theoretical values, a skill especially useful for IA explorations.
随机数生成器 rand() 和 randInt() 允许模拟抛硬币、掷骰子等试验。将大量试验结果存储在列表中并分析，可以估算概率并与理论值比较，这对内部评估 (IA) 研究尤为有用。
For conditional probability simulations, use sequences of random selections without replacement. The calculator can mimic drawing balls from a bag by shuffling a list and picking the first few elements.
对于条件概率模拟，使用不放回的随机选择序列。计算器可通过打乱列表并选取前几个元素来模拟从袋中取球。
12. Exam Tips and Common Errors | 考试技巧与常见错误
1. Angle mode: Always check whether the calculator is in degrees or radians before starting a trigonometry or calculus problem. A radian symbol near the x-axis on the graph screen indicates radian mode. 2. Stored variables: Clear variables or use dedicated memory resets to avoid carrying forward wrong values from a previous question. 3. Rounding: Store intermediate results to full precision and round only the final answer. 4. Battery: Ensure the GDC is fully charged and backup batteries are available. 5. Syntax: Missing parentheses and confusion between the subtraction and negative signs are the most common input errors – double-check every line.
1. 角度模式： 开始解决三角或微积分问题前，务必检查计算器处于度还是弧度模式。图屏幕上 x 轴附近的弧度符号表示弧度模式。2. 已存变量： 清除变量或使用专用内存重置，避免前一题的数值延续到下一题。3. 舍入： 中间结果保留全精度，仅对最终答案进行四舍五入。4. 电池： 确保图形计算器充满电并备有备用电池。5. 语法： 漏掉括号以及混淆减号与负号是最常见的输入错误——请逐行仔细检查。
A well-practised calculator workflow transforms the GDC from a simple computation device into a powerful reasoning partner. By mastering these twelve topics, students can move through Paper 2 and Paper 3 questions with confidence, reserving mental energy for interpretation and proof.
经过充分练习的计算器工作流能将 GDC 从简单的计算工具转变为强大的推理伙伴。通过掌握这十二个专题，考生可以自信地应对试卷二和试卷三中的问题，将脑力节省下来用于诠释与证明。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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