Review Set 18B - Calculator | IB 数学复习题集18B（计算器部分）

📚 Review Set 18B - Calculator | IB 数学复习题集18B（计算器部分）

Welcome to this focused revision guide for IB Mathematics, designed around the calculator‑active Review Set 18B. Whether you are taking Analysis & Approaches or Applications & Interpretation, mastering your graphing calculator (GDC) is essential for both Papers that allow it. In this article we revisit the key calculator skills tested in Set 18B — from solving equations and graphing functions to statistics, calculus, and financial maths. Use the paired explanations to work through each skill step by step, and you’ll be ready to tackle any calculator question with confidence.

欢迎来到这篇为 IB 数学量身定制的复习指南，围绕计算器允许使用的 Review Set 18B 编写。无论你修读的是分析与方法（AA）还是应用与解释（AI），掌握图形计算器（GDC）对允许使用计算器的试卷都至关重要。本文将重温 18B 复习题集中考查的核心计算器技能——从方程求解和函数作图，到统计、微积分和金融数学。借助中英对照的讲解逐步掌握每一项技能，你就能自信地应对任何计算器题目。

1. Solving Equations Numerically | 方程数值求解

One of the most basic yet vital GDC skills is solving equations that cannot easily be rearranged algebraically. In Review Set 18B you will encounter exponential, logarithmic and trigonometric equations where you must use the equation solver or the graphical intersection method.

最基础但也至关重要的 GDC 技能之一是求解无法简单代数变形的方程。在复习题集 18B 中，你会碰到指数、对数和三角方程，此时必须使用方程求解器或图像交点法。

To solve an equation like 2^x − 3x = 0, access the solver (often labeled “Solver” or “Numeric Solver”), enter the equation as 2^x et the calculator iterate to a solution. Always check that the displayed root makes sense in the original context.

要求解例如 2^x − 3x = 0 这样的方程，可进入求解器（常标为“Solver”或“数值求解器”），输入方程 2^x - 3x = 0，给出一个初始猜测值，然后让计算器迭代求出解。务必核实所得根在原题背景中是否合理。

When multiple roots exist, you can graph both sides as Y₁ = 2^x and Y₂ = 3x, then use the intersect function to find each crossing point. This visual approach often reveals how many solutions you should expect, reducing the chance of missing a root.

当存在多个根时，可将两边分别绘图设为 Y₁ = 2^x 和 Y₂ = 3x，然后使用交点功能找出每个交点。这种直观的方法常常能显示出你应该得到多少个解，从而降低遗漏根的风险。

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2. Graphing and Analyzing Functions | 函数图形与分析

Set 18B frequently tests your ability to sketch and interpret functions such as quadratic, cubic, rational, and piecewise models. Your GDC can plot these instantly, allowing you to focus on domain, range, intercepts, asymptotes and turning points.

18B 复习题常考查你绘制并解读二次函数、三次函数、有理函数和分段函数等模型的能力。GDC 可以即刻画出图像，让你能专注于定义域、值域、截距、渐近线和拐点等特征。

Enter the function into [Y=], set a suitable viewing window, and press [GRAPH]. Use the Zero (or Root) feature to find x‑intercepts, Value to evaluate f(x) at specific x, and Maximum/Minimum to locate local extrema. For a piecewise function, write separate formulas with domain restrictions using “|” or “when” commands.

在 [Y=] 中输入函数，设置合适的观察窗口，然后按 [GRAPH]。使用零点（或求根）功能找出 x 轴截距，求值功能计算指定 x 处的 f(x)，以及最大值/最小值功能定位局部极值。对于分段函数，可用 “|” 或 “when” 指令书写带域限制的不同公式。

When an asymptote is suspected, inspect the table of values for rapid growth or undefined outputs. Many GDCs will display “ERROR” for x‑values where the function is undefined, helping you identify vertical asymptotes.

当怀疑有渐近线时，可查看数值表有无急剧增长或无定义输出。许多 GDC 会在函数无定义的 x 值处显示 “ERROR”，助你识别垂直渐近线。

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3. Finding Intersections and Roots | 求交点与根

core topics. You might be asked to solve f(x) = g(x) or find where a function meets the x‑axis. The GDC provides two robust methods: graph intersect and table comparison.

交点问题遍布核心专题。你可能需要解 f(x) = g(x) 或找出函数与 x 轴的交点。GDC 提供两种可靠方法：图像交点和表格比较。

After plotting Y₁ = f(x) and Y₂ = g(x), use the Intersect command. The calculator will ask for a guess near each intersection; provide a cursor location close to the desired point. The resulting coordinates give your solution to a high accuracy.

绘制 Y₁ = f(x) 和 Y₂ = g(x) 后，使用交点指令。计算器会在每个交点附近要求猜测值；将光标放在所需点旁边即可。所得坐标能以高精度给出你的解。

For root finding, activate Zero from the CALC menu. Move the cursor to the left bound, press ENTER, set the right bound, ENTER, and then provide a guess. The displayed root can then be stored or used in further calculations.

选择 CALC 菜单中的零点。将光标移到左界，按 ENTER，设定右界，ENTER，再给出猜测值。显示的根可以存储或用于后续计算。

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4. Numerical Differentiation | 数值微分

When a derivative is needed at a point and the algebraic derivative is messy or unknown, numerical differentiation on the GDC is a powerful check. Set 18B includes questions where you estimate gradients or verify differentiation rules.

当需要某点导数但代数导数繁琐或未知时，GDC 上的数值微分是强大的检验工具。18B 练习中包含了估计梯度或验证求导法则的题目。

Access the numerical derivative function, often nDeriv( or d/dx. The syntax is typically nDeriv(expression, variable, value). For f(x) = √x³+1, computing nDeriv(√X³+1, X, 2) will give f´(2) directly. Remember the calculator uses a symmetric difference quotient, so the result is very close to the true derivative.

进入数值导数功能，常见命令如 nDeriv( 或 d/dx。语法通常为 nDeriv(表达式, 变量, 值)。对 f(x) = √x³+1，计算 nDeriv(√X³+1, X, 2) 直接得出 f´(2)。请记住，计算器使用对称差商，因此结果非常接近真实导数。

Use this feature to check the slope at a turning point is approximately zero, or to verify that the derivative you found by hand matches the calculator’s output. It is also handy when answering contextual questions about instantaneous rate of change.

可以利用这一功能检验拐点处斜率约等于零，或校对你手算的导数是否与计算器输出一致。回答关于瞬时变化率的应用题时它也格外好用。

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5. Definite Integrals and Area | 定积分与面积

Calculating areas under curves is a standard requirement. The GDC’s numerical integrator, fnInt( or ∫f(x)dx, allows you to compute definite integrals without antidifferentiation.

计算曲线下方的面积是常规要求。GDC 的数值积分器 fnInt( 或 ∫f(x)dx 令你不必寻找原函数就能计算定积分。

To evaluate ∫₀^π sin(x) dx, enter fnInt(sin(X), X, 0, π). The calculator returns 2 immediately. For area between curves, integrate the absolute difference or use two separate integrals when the upper/lower curve swaps.

要计算 ∫₀^π sin(x) dx，输入 fnInt(sin(X), X, 0, π)，计算器立即返回 2。对于曲线间的面积，可将差值取绝对值积分，或当上下曲线交换时拆分为两个积分。

If the area is bounded by more than two functions or the x‑axis, graph all equations first. Identify the intersection points, then set up integrals from one intersection to the next. The GDC lets you sum several integrals quickly, or use the Shade function to visualise the region.

如果面积由多于两条函数或 x 轴围成，先绘制所有方程图像。确定交点后，从一个交点到另一个交点上设立积分。GDC 让你能快速累加多个积分，或用着色功能将区域可视化。

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6. Statistics: Data Entry and Regression | 统计：数据输入与回归

Set 18B features bivariate data sets where you must find the least‑squares regression line, correlation coefficient, and make predictions. Efficient data entry and regression skills save time in the exam.

18B 练习涉及双变量数据集，要求你找出最小二乘回归线、相关系数并进行预测。高效的数据输入和回归技能能在考试中节约时间。

Enter the explanatory variable into List 1 (L₁) and the response variable into List 2 (L₂). After clearing old data, type values carefully. Then access the Stat CALC menu, choose LinReg ecify L₁, L₂. The output gives y = a + bx, r, and r², which you must interpret in context.

将解释变量录入列表 1 (L₁)，响应变量录入列表 2 (L₂)。清除旧数据后，仔细键入数值。接着进入统计计算菜单，选择LinReg（或合适的模型），指定 L₁ 和 L₂。输出结果给出 y = a + bx、r 和 r²，你必须结合情境进行解读。

For non‑linear models, you can linearise data by taking logs, or use built‑in regressions such as QuartReg, ExpReg, or Logistic. Always verify the model fit by inspecting the residual plot if time permits.

对于非线性模型，可通过取对数将数据线性化，或使用内置回归如四次回归、指数回归或逻辑回归。如果时间允许，应查看残差图以验证模型拟合度。

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7. Probability Distributions (Normal, Binomial) | 概率分布（正态、二项）

Many questions in Set 18B involve the normal and binomial distributions. GDC commands like normalcdf, invNorm, binompdf, and binomcdf allow you to compute probabilities and critical values instantly.

18B 练习中许多题目涉及正态分布和二项分布。normalcdf、invNorm、binompdf 和 binomcdf 等 GDC 命令令你能即时计算概率和临界值。

For a normally distributed variable X~N(50, 7²), to find P(45 < X < 55), use normalcdf(45, 55, 50, 7). To find the 90th percentile, use invNorm(0.9, 50, 7). Always distinguish between left‑tailed, right‑tailed and interval probabilities.

对于服从正态分布的变量 X~N(50, 7²)，求 P(45 < X < 55) 可使用 normalcdf(45, 55, 50, 7)。求第 90 百分位数，使用 invNorm(0.9, 50, 7)。务必区分左尾、右尾和区间概率。

(X = k) exactly, while binomcdf(n, p, k) gives P(X ≤ k). Use the complement rule to handle “greater than” ilities, and always check that your syntax matches the calculator’s requirements (e.g., TI‑84 expects trials, probability, x‑value).

对于二项分布 B(n, p)，binompdf(n, p, k) 给出 P(X = k) 的精确概率，而 binomcdf(n, p, k) 给出 P(X ≤ k)。用补集规则处理“大于”概率，并始终检查语法是否符合计算器要求（例如 TI‑84 要求顺序为试验次数、概率、x 值）。

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8. Financial Mathematics: TVM Solver | 金融数学：TVM 求解器

IB Applications & Interpretation heavily relies on the TVM (Time Value of Money) Solver for annuities, loans, and investments. Even if you are in AA, Set 18B may include financial contexts to test your calculator fluency.

IB 应用与解释课程高度依赖 TVM（资金时间价值）求解器处理年金、贷款和投资。即便你修读 AA，18B 练习也可能包含金融背景以考查计算器使用的熟练度。

Open the TVM Solver. Key parameters: N = total number of payments, I% = annual interest rate (as a percentage, not decimal), PV = present value, PMT = payment amount, FV = future value, P/Y = payments per year, C/Y = compounding periods per year. Set PMT: END or BEGIN to indicate ordinary annuity or annuity due.

打开 TVM 求解器。关键参数：N = 付款总期数，I% = 年利率（以百分比而非小数形式），PV = 现值，PMT = 每期付款额，FV = 终值，P/Y = 每年付款次数，C/Y = 每年复利次数。设置 PMT: END 或 BEGIN 以表示普通年金或预付年金。

To find the monthly payment on a $200 000 loan at 5.2% compounded monthly over 30 years, enter N=360, I% PMT, : END. Solve for PMT, which gives a negative value (money out). Always interpret the sign according to cash‑flow convention.

要计算一笔 $200 000 贷款、年利率 5.2%、月复利、30 年期下的月付款额，输入 N=360, I%=5.2, PV=200000, FV=0, P/Y=12, C/Y=12, PMT: END。求解 PMT 会得到负值（代表支出）。始终根据现金流惯例理解符号。

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9. Matrices and Vector Operations | 矩阵与向量操作

For AA and some AI topics, matrix algebra and vector calculations are calculator‑friendly tasks. Set 18B may ask you to find the determinant, inverse, or solve a system of linear equations using matrices.

对于 AA 和部分 AI 专题，矩阵代数和向量计算是适合用计算器的任务。18B 复习题可能要求你求行列式、逆矩阵或用矩阵求解线性方程组。

t gives x = 1.8, y = 1.4. Alternatively, use the simultaneous equation solver if built in.

通过矩阵编辑器将矩阵存入 [A]、[B] 等。求解方程组 2x + y = 5, 3x − y = 4，构造系数矩阵 [A] = [[2,1],[3,-1]] 和常数矩阵 [B] = [[5],[4]]。解为 [A]⁻¹ * [B]，[A]⁻¹ 为逆矩阵。结果给出 x = 1.8, y = 1.4。也可使用内置的联立方程求解器。

GDC’s dotP( function automates this. Check your formula booklet for the correct notation.

对于向量运算如点积、叉积或两条向量的夹角，使用向量菜单。向量常以列表形式输入。对 a = (2, −3) 和 b = (5, 1)，点积 a·b = 2×5 + (−3)×1 = 7。GDC 的 dotP( 函数可自动完成。请参照公式手册确认正确表示法。

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10. Sequences and Series: Summation | 数列与级数：求和

Arithmetic and geometric sequences feature regularly in IB, and the GDC can sum series and generate terms efficiently. Set 18B includes both explicit formulas and recurrence relations.

等差数列和等比数列在 IB 中经常出现，GDC 可高效地求级数和以及生成项。18B 练习包含显式公式和递推关系。

of the first 20 terms of the geometric progression uₙ = 3·2ⁿ⁻¹, use the summation function Σ( or sum(seq(. The syntax Σ(3·2^X-1, X, 1, 20) will compute the sum. Many calculators also have a sequence mode where you can enter the nth term explicitly and then evaluate sums.

要计算等比数列 uₙ = 3·2ⁿ⁻¹ 前 20 项的和，使用求和函数 Σ( 或 sum(seq(。语法 Σ(3·2^X-1, X, 1, 20) 将算出总和。许多计算器还有数列模式，可显式输入第 n 项然后求值。

g the recursive formula in Y= mode with u(nMin)=5, u(n)=0.8u(n-1)+2 to view and sum terms.

对于递推关系 u₁ = 5, uₙ₊₁ = 0.8uₙ + 2，可将 5 存入变量然后反复使用规律：0.8×ANS + 2 来生成项。通过在 Y= 模式中输入 u(nMin)=5, u(n)=0.8u(n-1)+2 的递推公式，使用 TABLE 功能可查看并求项和。

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11. Complex Numbers: Quick Checks | 复数：快速验算

In Analysis & Approaches, complex numbers appear from roots of polynomials to polar form. Although you must often show algebraic steps, your GDC can verify answers or handle cumbersome arithmetic like multiplication and division in polar form.

在分析与方法中，复数从多项式求根到极式均有涉及。虽然常常需要展示代数步骤，但 GDC 可以核对答案或处理极式中的乘除等繁琐运算。

Set your calculator to a+bi or r∠θ mode as needed. To divide (3 + 4i) by (1 − 2i), simply type (3+4i)/(1−2i) and the result appears in the chosen format. For modulus and argument, use the abs( and angle( commands, or ▶Polar conversion.

根据需要将计算器设为 a+bi 或 r∠θ 模式。要计算 (3 + 4i) ÷ (1 − 2i)，直接键入 (3+4i)/(1−2i) 就会以选定格式显示结果。对于模长和辐角，使用 abs( 和 angle( 命令，或 ▶Polar 转换。

en solving polynomial equations like z³ = 8, you can use the polynomial root finder (e.g., PolySmlt or the solve function set to complex mode) to obtain all three cube roots: 2, −1+√3 i, −1−√3 i. This instant feedback helps you check manual methods and spot sign errors.

当求解如 z³ = 8 的多项式方程时，可用多项式求根器（如 PolySmlt 或设为复数模式的 solve 函数）得到全部三个立方根：2, −1+√3 i, −1−√3 i。这种即时反馈帮助你校验手动方法并发现符号错误。

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