Review Set 1B – Calculator | 复习题集 1B – 计算器应用

📚 Review Set 1B – Calculator | 复习题集 1B – 计算器应用

Review Set 1B in the IB Mathematics course is specifically designed to be tackled with a graphing calculator. This set tests your ability to apply numerical, graphical and algebraic techniques using technology, covering equations, functions, calculus, statistics and more. The following guide walks you through the key calculator skills needed to solve typical problems efficiently and accurately.

复习题集 1B 在 IB 数学课程中是专门为使用图形计算器而设计的。这部分内容考察你利用技术手段进行数值、图像和代数处理的能力，涵盖方程、函数、微积分、统计等多个主题。以下指南将带你掌握解答典型题目所需的关键计算器技巧，帮助你高效、准确地完成作业。

1. Solving Equations and Inequalities | 解方程与不等式

Many items in Review Set 1B ask you to solve polynomial or trigonometric equations. Your calculator’s equation solver or poly‑solver can find real and complex roots instantly. For example, to solve 2x³ – 5x + 2 = 0, input the equation and let the calculator return x ≈ 1.366, x ≈ 0.366, and x ≈ –1. Two real roots and one negative root might surprise you, so always check the domain specified in the question.

复习题集 1B 中许多题目要求解多项式或三角方程。你计算器中的方程求解器或多项式求解器可以立即给出实根和复根。例如，求解 2x³ – 5x + 2 = 0，输入方程后计算器返回 x ≈ 1.366、x ≈ 0.366 和 x ≈ –1。两个正根和一个负根可能会让你意想不到，因此务必核对题目所给的区间。

When dealing with inequalities like x² – 4x > 5, rewrite as x² – 4x – 5 > 0, graph y = x² – 4x – 5, and use the zero or intersect function to find where the graph lies above the x‑axis. The solution is x < –1 or x > 5. Being able to read intervals directly from the graph saves time.

在处理 x² – 4x > 5 这类不等式时，先改写为 x² – 4x – 5 > 0，绘制 y = x² – 4x – 5 的图像，然后利用零点或交点功能找到图像在 x 轴上方的区域。解为 x < –1 或 x > 5。直接从图像读取区间能节省大量时间。

2. Graphing Functions and Finding Intersections | 绘制函数图像与求交点

Graphing is often the fastest way to understand function behaviour. In Review Set 1B you may be asked to compare f(x) = eˣ and g(x) = x³ on the same axes. Adjust the viewing window to show the region where they meet. Use the intersect tool to locate the points of intersection: approximately (–0.724, 0.485) and (1.857, 6.406). Always round answers to three significant figures unless told otherwise.

绘制图像通常是理解函数行为最快的方法。复习题集 1B 可能会让你在同一坐标系中比较 f(x) = eˣ 和 g(x) = x³。调整窗口以显示它们相交的区域。使用交点工具定位交点：坐标约为 (–0.724, 0.485) 和 (1.857, 6.406)。除非另有说明，答案通常保留三位有效数字。

If a question involves the composition of functions, you can define h(x) = f(g(x)) directly in the Y= editor and graph it. For instance, with f(u) = ln(u) and g(x) = 2x + 1, enter Y1 = ln(2X+1) to visualise the domain and asymptote. The graph makes it clear that the function is undefined for x ≤ –0.5.

若题目涉及复合函数，你可以在Y=编辑器中直接定义 h(x) = f(g(x)) 并作图。例如，设 f(u) = ln(u) 且 g(x) = 2x + 1，输入 Y1 = ln(2X+1) 即可观察定义域和渐近线。从图像可以清楚看出，x ≤ –0.5 时函数无定义。

3. Numerical Derivatives | 数值求导

Calculators can evaluate derivatives at a point without finding the algebraic expression. The nDeriv or d/dx feature is invaluable for checking tangent slopes. Suppose f(x) = √(x² + 3). To find f'(2), use nDeriv(√(X²+3), X, 2); the result is approximately 0.554. You can also graph the derivative function to study monotonicity and critical points.

计算器可以直接在某点处求导而无需获得代数表达式。nDeriv 或 d/dx 功能对于检验切线斜率极为有用。设 f(x) = √(x² + 3)，要求 f'(2)，使用 nDeriv(√(X²+3), X, 2)，结果约为 0.554。你还可以绘制导函数图像以研究单调性和临界点。

Be careful with the step size: most IB questions accept the default setting, but if the result seems erratic, check whether the function is differentiable at that point. Always confirm with a trace or table of values. This is especially useful when dealing with piecewise functions that appear in Review Set 1B.

注意步长设置：大多数 IB 题目接受默认值，但如果结果出现异常，请检查函数在该点是否可导。始终通过图像追踪或数值表加以确认。对于复习题集 1B 中可能出现的分段函数，这一点尤为重要。

4. Numerical Integration | 数值积分

Definite integrals that lack elementary antiderivatives can be tackled with the ∫f(x)dx function. For example, to approximate ∫ₐᵇ sin(x²) dx from 0 to 2, enter the integral command; the calculator returns about 1.202. This is often required when finding areas between curves or accumulated change.

对于缺乏初等原函数的定积分，可以使用∫f(x)dx 功能求解。例如，要计算从 0 到 2 的 ∫₀² sin(x²) dx，输入积分命令后计算器给出约 1.202。在求曲线间面积或累积变化量时常需要这样做。

When finding the area between f(x) and g(x), do not simply integrate f(x) – g(x); first determine the intersection points using the graph, then set up the integral over each subinterval. The calculator can handle the absolute value by integrating |f(x)–g(x)| directly if your model supports it, but it is safer to split the domain manually.

在求 f(x) 与 g(x) 之间的面积时，不要简单地积分 f(x) – g(x)；应先用图像确定交点，然后在各子区间上建立积分。如果你的计算器支持直接积分 |f(x)–g(x)|，也可以使用，但更稳妥的做法是手动划分区间。

5. Sequences and Series | 数列与级数

Review Set 1B often includes arithmetic and geometric sequences. Use the seq( command to generate terms: seq(3n+2, n, 1, 10) produces the first ten terms of the arithmetic sequence 5, 8, 11, … . For geometric sequences, try seq(2×(1.5)^(n–1), n, 1, 8). You can also sum these lists with cumSum( or the sigma summation function.

复习题集 1B 常包含等差数列和等比数列。利用 seq( 命令生成项：seq(3n+2, n, 1, 10) 可生成等差数列 5, 8, 11, … 的前十项。对于等比数列，可尝试 seq(2×(1.5)^(n–1), n, 1, 8)。你还可以用 cumSum( 或求和函数 Σ 对这些列表求和。

To check the sum of an infinite geometric series, simply sum a large number of terms and observe convergence. For Σ 4×(0.6)^(k–1) from k=1 to a large n, the sum approaches 10. This visual approach reinforces the concept of limits and can be verified by the formula S∞ = a/(1–r).

要检验无穷等比级数的和，只需对足够多项求和并观察收敛情况。例如对 Σ 4×(0.6)^(k–1) 从 k=1 到较大 n 求和，和趋近于 10。这种直观方法能加深对极限概念的理解，并可用公式 S∞ = a/(1–r) 来检验。

6. Statistics and Probability | 统计与概率

A major portion of the calculator paper involves statistics. Enter data into lists and use 1-Var Stats or 2-Var Stats to compute mean, standard deviation, quartiles and correlation coefficient. For a small data set: 12, 15, 18, 17, 14, the sample standard deviation sx is about 2.30. Remember that IB expects the sample standard deviation unless the population symbol is used.

计算器试卷中有很大一部分涉及统计。将数据输入列表，然后使用单变量统计或双变量统计计算均值、标准差、四分位数和相关系数。对于小数据集 12, 15, 18, 17, 14，样本标准差 sx 约为 2.30。记住，IB 通常要求样本标准差，除非题目使用了总体符号。

Probability distributions can be explored with built-in functions. The binomial probability P(X = 3) for n = 10, p = 0.35 is found with binompdf(10,0.35,3) ≈ 0.252. For cumulative probabilities, use binomcdf. The normal distribution functions invNorm( and normalcdf( are essential for inverse normal and tail probability questions.

概率分布可通过内置函数探索。二项分布概率 P(X = 3) 在 n = 10, p = 0.35 时，用 binompdf(10,0.35,3) 得到约 0.252。累积概率则使用 binomcdf。正态分布函数 invNorm( 和 normalcdf( 对于反查正态分布和求尾部概率至关重要。

7. Matrix Operations | 矩阵运算

If you are taking Applications and Interpretation (AI) or have matrix topics in your syllabus, the calculator handles matrices up to the built-in limit. Enter a 2×2 matrix A = [[3, –1], [2, 5]] and find its determinant (17) and inverse. Matrix multiplication can solve systems of equations or transform coordinates.

如果你选修的是应用与解释（AI）或大纲中包含矩阵内容，计算器可处理上限以内的矩阵。输入 2×2 矩阵 A = [[3, –1], [2, 5]]，求出其行列式（17）和逆矩阵。矩阵乘法可用于解方程组或变换坐标。

To solve the system 3x – y = 7 and 2x + 5y = –1, store the coefficient matrix and the constant vector, then compute A⁻¹×B to obtain x = 2, y = –1. Always double‑check by substituting back into the original equations. Calculator errors often come from mismatched dimensions.

要解方程组 3x – y = 7 和 2x + 5y = –1，将系数矩阵和常数向量储存起来，然后计算 A⁻¹×B 得到 x = 2, y = –1。务必代回原方程进行验证。计算器报错常常是由于维度不匹配造成的。

8. Complex Numbers | 复数计算

Even on a calculator paper, complex numbers appear: the quadratic formula may yield non‑real roots. Use the complex mode to compute √(–9) = 3i and to perform arithmetic. For z₁ = 2 + 3i, z₂ = 1 – i, find z₁/z₂ = –0.5 + 2.5i. The polar form can be obtained with angle and magnitude functions.

即使在计算器试卷中，复数也时有出现：二次公式可能会产生非实根。利用复数模式计算 √(–9) = 3i 并进行四则运算。对于 z₁ = 2 + 3i, z₂ = 1 – i，求 z₁/z₂ 得到 –0.5 + 2.5i。利用角度和模长函数可以得到极坐标形式。

Review Set 1B might ask you to plot complex numbers on an Argand diagram; you can quickly check the real and imaginary parts with the calculator. Some models also convert between rectangular and polar form, which is helpful when finding arguments and modulus.

复习题集 1B 可能会要求你在阿尔冈图上标出复数；你可以用计算器快速查看实部和虚部。某些型号还能在直角坐标和极坐标形式之间转换，这对于求辐角和模长很有帮助。

9. Solving Systems of Equations Graphically | 用图像法解方程组

While algebraic methods are fundamental, the calculator allows you to verify solutions to systems of non‑linear equations. Graph both equations side by side and use the intersect function. For the system x² + y² = 25 and y = 2x – 3, plot them implicitly or solve for y. The intersections are approximately (–2.00, –7.00) and (4.00, 5.00).

虽然代数方法是基础，但计算器可以帮你验证非线性方程组的解。将两个方程并排绘图，然后使用交点功能。对于方程组 x² + y² = 25 和 y = 2x – 3，可隐式绘图或解出 y。交点坐标约为 (–2.00, –7.00) 和 (4.00, 5.00)。

Some questions may require you to solve a system of three equations. The simultaneous equation solver (often under the algebra menu) handles linear and even some polynomial systems. Enter the equations exactly as written and let the calculator do the heavy lifting. Always label your solutions clearly.

有些题目可能要求解三元方程组。联立方程求解器（通常位于代数菜单下）可处理线性乃至部分多项式方程组。照原样输入方程，让计算器完成繁琐的计算。答案始终要标注清楚。

10. Regression Analysis | 回归分析

Data modelling is a favourite in the IB calculator exam. Enter bivariate data into two lists, then run LinReg(ax+b), QuadReg, or ExpReg to find the best‑fit model. The correlation coefficient r tells you how well the model fits; values close to 1 or –1 indicate a strong linear relationship. Don’t forget to plot the residuals to check for patterns.

数据建模是 IB 计算器考试的常见题型。将双变量数据输入两个列表，然后运行线性回归、二次回归或指数回归找到最佳拟合模型。相关系数 r 告诉你模型拟合的程度；接近 1 或 –1 的值表明线性关系强。别忘了绘制残差图以查看是否有模式可循。

If a question asks you to predict a value, use the regression equation stored in Y1. For example, after finding Y1 = 3.2X + 7.8, evaluate Y1(12) to get a predicted output. Extrapolation beyond the data range should be treated with caution and always commented upon.

如果题目要求预测某个值，可以直接使用储存于 Y1 中的回归方程。例如，在得到 Y1 = 3.2X + 7.8 后，计算 Y1(12) 即可得到预测值。超出数据范围的外推应当谨慎使用并加以说明。

11. Calculus Applications – Optimisation and Area | 微积分应用——优化与面积

Optimisation problems in Review Set 1B often reduce to finding maximum or minimum points on a graph. Use the maximum or minimum feature on a graphed function to locate local extrema. For instance, optimise P = x (20 – 2x)(10 – x) by graphing and finding the peak: x ≈ 3.94 gives a maximum volume.

复习题集 1B 中的优化问题通常归结为寻找图像上的最大值或最小值点。在已绘制的函数上使用最大值或最小值功能定位局部极值。例如，优化 P = x(20 – 2x)(10 – x)，通过绘图并寻找峰值，得到 x ≈ 3.94 时体积最大。

Area under a curve can be found precisely with the integral function, but also verify using geometric methods when possible. When a question asks for the area bounded by two curves and a vertical line, the calculator can compute the definite integral after storing the bounding x‑values from intersection points.

曲线下的面积可用积分功能精确求出，但也可以尽可能用几何方法验证。当题目要求计算由两条曲线和一条垂直线围成的面积时，计算器可在存储交点 x 值后直接计算定积分。

12. Common Mistakes and Pro Tips | 常见错误与实用技巧

Even experienced students overlook calculator settings. Always set the angle mode (degrees or radians) according to the question. Use the fraction or decimal toggle to present answers in the required form. Ensure the stat plots are turned off when returning to regular graphing, otherwise you might see stray points.

即使是经验丰富的学生也会忽略计算器设置。始终根据题目设置角度模式（度或弧度）。利用分数/小数切换功能按要求的形式呈现答案。返回常规绘图时务必关闭统计图，否则可能会看到散点干扰。

Develop a habit of estimating the answer before relying on the calculator. Rough mental arithmetic or a sketch graph helps catch input errors. Also, use the table function to examine function values at specific points—this is particularly useful for limits and continuity questions that appear in Review Set 1B.

养成在依赖计算器之前先估算答案的习惯。粗略的心算或草图有助于发现输入错误。同时，利用表格功能检查特定点的函数值——这对于复习题集 1B 中的极限和连续性问题尤其有用。

Finally, store intermediate results in variables rather than retyping long decimals. The memory keys (A, B, C, …) prevent rounding errors from accumulating and keep your work organised. With these strategies, Review Set 1B becomes a confidence‑building exercise rather than a struggle.

最后，将中间结果储存在变量中，而不是重新输入一长串小数。记忆键（A, B, C, …）能防止舍入误差累积，并使解题过程井井有条。掌握这些策略后，复习题集 1B 将成为建立信心的练习，而非一场鏖战。

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Review Set 1B – Calculator | 复习题集 1B – 计算器应用