📚 Review set 24B - CALCULATOR | 复习套题24B - 计算器部分
Calculator-based problems in IB Mathematics require a blend of conceptual understanding and technical fluency. This review set covers the essential calculator techniques you need to master, from solving equations and analysing graphs to performing statistical tests and evaluating definite integrals. Each section pairs an English explanation with a Chinese translation to reinforce learning in both languages.
IB数学中基于计算器的题目要求学生既要理解概念，又要熟练操作。这份复习套题涵盖了你需要掌握的核心计算器技巧，从解方程、分析图像到执行统计检验和计算定积分。每个要点都配有中英文对照解释，帮助你彻底掌握。
1. Solving Equations Numerically | 数值解方程
ke 2x³ − 5x² + 3x − 7 = 0 on your graphing calculator, open the equation solver or use the "solve" by substituting it back into the original equation. For systems of equations, use the simultaneous equation solver, which handles up to three variables in most IB-approved models.
要在图形计算器上求解像 2x³ − 5x² + 3x − 7 = 0 这样的方程，可以打开方程求解器或使用 "solve" 功能。输入表达式并指定一个初始猜测值，计算器将返回小数近似解。务必通过代入原方程来验证解。对于方程组，使用联立方程求解器，大多数IB认可的计算器最多可处理三个变量。
When a question asks for answers correct to three significant figures, make sure to round only at the final step. The calculator can store intermediate values in memory (e.g., using the Ans or variable keys). This prevents cumulative rounding errors. In exam settings, it is good practice to write down the unrounded value first, then give the rounded answer.
当题目要求答案保留三位有效数字时，务必只在最后一步四舍五入。计算器可以将中间值存储在记忆中（例如使用 Ans 或变量键），这样可以避免累积舍入误差。在考试中，好的做法是先写下未四舍五入的值，再给出舍入后的答案。
2. Graphing Functions and Analysis | 函数作图与分析
n appropriate viewing window, and use the trace or G-Solv functions to find x-intercepts (zeros), y-intercepts, local maxima, and minima. For rational functions, a zoom-out often reveals asymptotic behaviour.
图形计算器能让你可视化函数并快速识别截距、转折点和渐近线等关键特征。在 Y= 编辑器中输入函数，设置合适的显示窗口，然后使用 trace 或 G-Solv 功能找到 x 截距（零点）、y 截距、局部极大值和极小值。对于有理函数，缩小视图通常能揭示渐近行为。
algebraic manipulation is cumbersome. You can also analyse intervals where one function is greater than another by observing which graph lies above.
要找到两个图像的交点，同时画出两个函数并使用交点命令。计算器会要求输入交点附近的猜测值，然后计算坐标。对于像 f(x) = g(x) 这样难以代数化简的方程，这尤其有用。你还可以通过观察哪个图像在上方来分析一个函数大于另一个函数的区间。
3. Statistical Calculations | 统计计算
Your calculator's statistics mode can compute summary statistics from a list of data. Enter the raw data into a list (e.g., L1), then select 1-Variable Statistics to obtain the mean, standard deviation, median, quartiles, and range. Always check whether the question requires the sample standard deviation (sₓ) or the population standard deviation (σₓ). For grouped data, enter midpoints in one list and frequencies in another, then use the frequency option.
计算器的统计模式可以从一组数据中计算摘要统计量。将原始数据输入列表（如 L1），然后选择单变量统计即可获得均值、标准差、中位数、四分位数和极差。务必检查题目要求的是样本标准差（sₓ）还是总体标准差（σₓ）。对于分组数据，在一个列表中输入组中值，另一个列表中输入频数，然后使用频数选项。
Linear regression is straightforward: enter x-values in one list, y-values in another, and perform linear regression (ax + b). The calculator provides the correlation coefficient r and the coefficient of determination r². Use these to comment on the strength and direction of a linear relationship. For non-linear models, such as exponential or power regression, apply a logarithmic transformation or use the built-in regression types. Always plot the data first to choose an appropriate model.
线性回归非常简单：在一个列表中输入 x 值，另一个列表输入 y 值，然后执行线性回归（ax + b）。计算器会给出相关系数 r 和判定系数 r²。利用它们来评述线性关系的强度和方向。对于非线性模型，如指数或幂回归，可以进行对数变换，或直接使用内置的回归类型。务必先画出数据散点图以选择合适的模型。
4. Probability Distributions | 概率分布
se binomial cdf. Input the number of trials n, the success probability p, and the value k. When the question involves 'at least' or 'more than', remember to convert to the appropriate cumulative form (e.g., 1 − P(X ≤ k−1)).
IB认可的计算器中的分布菜单涵盖二项分布、正态分布、泊松分布等。对于二项概率 P(X = k)，使用二项分布概率密度函数；对于累积概率 P(X ≤ k)，使用二项分布累积分布函数。输入试验次数 n、成功概率 p 和数值 k。当题目涉及“至少”或“多于”时，要转化为合适的累积形式（例如 1 − P(X ≤ k−1)）。
For normal distribution problems, use the normal cdf to find the probability between two bounds, entering the lower bound, upper bound, mean μ, and standard deviation σ. To find an unknown mean or standard deviation, use the inverse normal function together with a standardised score z. Always sketch a bell curve and shade the relevant area to avoid direction errors. Tables are rarely needed with a calculator, but the standard normal (z) cumulative is available for checking.
对于正态分布问题，使用正态累积分布函数求两个边界之间的概率，需要输入下界、上界、均值 μ 和标准差 σ。若要求未知的均值或标准差，可以使用逆正态函数结合标准化得分 z。务必画出钟形曲线并涂黑相应区域，以免方向错误。有了计算器很少需要查表，但仍然可以调用标准正态（z）累积函数进行核对。
5. Numerical Differentiation and Integration | 数值微分与积分
al derivative command (often nDeriv or d/dx). Enter the function, the variable, and the point. The calculator uses a symmetric difference quotient and provides a high-accuracy estimate. This is useful for verifying the gradient of a tangent or the instantaneous rate of change when algebraic differentiation is complex.
要近似求函数在 x = a 处的导数，使用数值微分命令（通常是 nDeriv 或 d/dx）。输入函数、变量和所求点。计算器使用对称差分商，能给出高精度的估计值。这在验证切线斜率或代数微分复杂时的瞬时变化率时非常有用。
variable, and the lower and upper limits. The calculator applies an adaptive quadrature method and returns the area under the curve. Remember that the integral gives the signed area; for total area, you must split the interval where the function crosses the x-axis. Use the integral function together with the graph to find areas between curves.
定积分通过数值积分函数（∫ f(x) dx）计算。输入被积函数、变量以及上下限。计算器应用自适应求积法，返回曲线下的面积。要记住积分给出的是有符号面积；求总面积时，必须在函数穿过 x 轴的地方拆分区间。结合图像使用积分函数可以求出曲线之间的面积。
6. Matrix Operations | 矩阵运算
he same as B×A. The calculator will display an error if the dimensions are not compatible.
在矩阵编辑器中，通过输入维度和各个元素来定义矩阵。你可以存储多个矩阵（如 [A]、[B]），并进行加法、减法、标量乘法和矩阵乘法。注意顺序：A×B 与 B×A 不同。如果维度不兼容，计算器会显示错误。
as either no unique solution or infinitely many solutions. For 3×3 matrices, you can also compute eigenvalues and eigenvectors if your syllabus requires them.
要解矩阵形式为 AX = B 的线性方程组，可以求逆矩阵 A⁻¹ 再与 B 相乘，或者直接使用联立方程求解器。方阵的行列式（det）能告诉你逆矩阵是否存在；如果 det = 0，矩阵是奇异的，方程组要么无唯一解，要么有无穷多解。对于 3×3 矩阵，如果教学大纲要求，还可以计算特征值和特征向量。
7. Complex Numbers | 复数
handles addition, subtraction, multiplication, division, and exponentiation. For division, it automatically rationalises the denominator and expresses the result in rectangular form.
将计算器设为复数模式（a+bi）后，即可进行复数运算。你可以输入虚数单位 i，并计算像 (3+2i)×(1−4i) 这样的表达式。计算器支持加、减、乘、除和乘方。进行除法时，它会自动分母有理化并以直角坐标形式显示结果。
Use the Pol and Rec commands to convert between Cartesian (rectangular) and polar forms. Arg and Abs functions give the argument (angle) and modulus of a complex number. When plotting complex roots of unity or solving polynomial equations, the calculator can list all real and complex solutions. De Moivre's theorem patterns can be verified by evaluating (r∠θ)n numerically.
使用 Pol 和 Rec 命令可以在笛卡尔坐标（直角坐标）和极坐标形式之间转换。Arg 和 Abs 函数能给出复数的辐角（角度）和模。在绘制单位根的复数根或解多项式方程时，计算器可以列出所有实根和复根。棣莫弗定理的规律可以通过数值计算 (r∠θ)n 来验证。
8. Financial Mathematics | 金融数学
The TVM (Time Value of Money) solver is a powerful tool for loans, annuities, and investments. You input the number of periods N, annual interest rate I%, present value PV, payment PMT, future value FV, and the number of compounding periods per year (P/Y). The calculator solves for the unknown variable. Always draw a timeline and clearly identify the direction of cash flows: payments out are usually negative, payments received are positive.
TVM（资金时间价值）求解器是处理贷款、年金和投资的有力工具。你输入期数 N、年利率 I%、现值 PV、每期付款额 PMT、终值 FV 以及每年复利次数（P/Y），计算器即可求解未知变量。务必画出时间轴并明确现金流方向：支出通常为负，收入为正。
tion, you can generate a schedule to see how each payment is split between interest and principal. When interest is compounded monthly but payments are annual, adjust P/Y and N accordingly. IB questions often require you to interpret the calculator output in the context of a financial scenario, so write a concluding sentence that explains the result in real-world terms.
对于没有定期付款的复利，将 PMT 设为 0。对于分期偿还，可以生成还款表以查看每笔付款如何分为利息和本金。如果按月复利但按年付款，需相应调整 P/Y 和 N。IB题目经常要求你结合金融情景解释计算器输出结果，因此要写一句总结，用现实语言解释该结果。
9. Graphing Sequences and Series | 数列与级数绘图
d view the scatter plot of points (n, aₙ). This helps you visualise convergence, divergence, or oscillatory behaviour. Use the summation (Σ) function to find partial sums Sₙ.
许多计算器可以绘制数列的项并计算级数的和。在数列模式下，显式定义第 n 项（aₙ = ...）或递归定义（aₙ = f(aₙ₋₁)）。设置 n 的范围并查看点 (n, aₙ) 的散点图，这有助于你观察收敛、发散或振荡行为。使用求和（Σ）函数可求部分和 Sₙ。
For arithmetic and geometric series, you can check your formula-based answers by directly summing a few terms. The calculator can also evaluate infinite series to a large number of terms to approximate the sum to a required precision. When testing for convergence of an infinite geometric series, verify that |r| < 1 and observe that the partial sums approach a/(1−r).
对于等差和等比级数，你可以通过直接对几项求和来验算按公式计算出的答案。计算器还能计算无穷级数的大量项，从而按要求的精度逼近级数和。在检验无穷等比级数的收敛性时，确认 |r| < 1 并观察部分和趋近于 a/(1−r)。
10. Testing Hypotheses and Confidence Intervals | 假设检验与置信区间
IB analysis and approaches (AA) students use the calculator to perform z-tests, t-tests, and construct confidence intervals. Enter the summary statistics (sample mean, standard deviation, sample size) or raw data. The calculator provides the test statistic (z or t), the p-value, and the conclusion bounds. Always compare the p-value to the significance level α to decide whether to reject H₀.
IB分析与方法（AA）的学生使用计算器执行 z 检验、t 检验并构建置信区间。输入摘要统计量（样本均值、标准差、样本量）或原始数据，计算器会给出检验统计量（z 或 t）、p 值以及结论界限。始终将 p 值与显著性水平 α 进行比较，从而决定是否拒绝原假设 H₀。
For a two-sample t-test, check whether the variances are assumed equal or unequal, as this affects the degrees of freedom. The calculator offers both options. Confidence intervals are equally easy: input the confidence level (e.g., 95%) and the calculator outputs the lower and upper bounds. When asked to interpret an interval, say 'We are 95% confident that the true population mean lies between …' – avoid saying there is a 95% probability the mean is in the interval.
对于双样本 t 检验，需检查是否假设方差相等，因为这会影响自由度。计算器两种选项都提供。置信区间同样简便：输入置信水平（如 95%），计算器会输出下界和上界。当要求解释区间时，应说“我们有95%的信心认为真实总体均值介于……之间”——避免说均值落入区间的概率是95%。
11. Avoiding Common Calculator Pitfalls | 避免常见计算器使用误区
One frequent mistake is having the calculator in radian mode when degrees are required, or vice versa. Always check the angle setting before starting trigonometric calculations. Another trap is forgetting to close parentheses when entering compound fractions or nested functions; use the fraction template and bracket keys deliberately. Clear any previous entries from memory to avoid interference with new calculations.
一个常见的错误是：当需要度数时计算器却处于弧度模式，反之亦然。在开始三角计算之前，一定要检查角度设置。另一个陷阱是在输入复合分数或嵌套函数时忘记关闭括号；刻意使用分数模板和括号键。清除记忆中的所有先前输入，以免干扰新的计算。
When graphing, an inappropriate window can hide important features. If you do not see the expected intercepts or asymptotes, zoom out or adjust Xmin, Xmax, Ymin, Ymax. In statistical work, double-check that the data lists are correctly paired and that no extra entries exist at the end of a list. Finally, never trust a calculator result blindly—use estimation and algebraic sense to verify that the answer is plausible.
在绘图时，不合适的窗口设置会隐藏重要特征。如果看不到预期的截距或渐近线，请缩小视图或调整 Xmin、Xmax、Ymin、Ymax。在统计工作中，请仔细检查数据列表是否正确配对，以及列表末尾是否有多余条目。最后，永远不要盲目相信计算器的结果——使用估算和代数直觉验证答案是否合理。
12. Exam Strategy for Calculator Papers | 计算器试卷的考试策略
In IB Mathematics Paper 2 and Paper 3 (for HL), the calculator is permitted, but its use must be efficient. Begin a question by identifying the relevant calculator tool—solver, graph, stat, or matrix—and structure your working on paper accordingly. Even when the calculator does the heavy lifting, you must show the setup: the equation entered, the lists used, or the function defined. Screen captures are not required; clear written communication of your process is essential.
在IB数学Paper 2和Paper 3（HL）中允许使用计算器，但必须高效使用。开始答题时，先确定相关的计算器工具——求解器、图像、统计或矩阵——并据此在答题纸上组织解题过程。即使计算器完成了繁重的运算，你也必须展示设置过程：输入的方程、使用的列表或定义的函数。不需要屏幕截图；清晰地将过程书面表达出来才是关键。
Manage your time by avoiding over-reliance on graphical trial-and-error. If you can quickly set up an equation and solve it directly, do so. Reserve graphing for when visual insights make the solution faster. Save your work frequently, as calculators can reset during exams. Finally, practice under timed conditions with past papers to become fluent in switching between different calculator apps. Proficiency with your device can save several minutes, which you can use to check answers.
合理安排时间，避免过度依赖图形试错。如果能快速建立方程并直接求解，那就直接解。将绘图留给那些能通过视觉洞察让解题更快的情形。经常保存你的工作，因为考试中计算器可能会重置。最后，在计时条件下使用历年真题进行练习，以熟练地在不同的计算器应用程序之间切换。熟练操作设备能为你节省宝贵的几分钟，用来检查答案。
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