📚 Review set 3B - CALCULATOR | 复习题集 3B - 计算器
Welcome to this focused review set designed to sharpen your calculator skills for IB Mathematics. Whether you are using a TI-84 Plus, TI-Nspire, Casio fx-CG50, or another approved graphing calculator, mastering GDC functions can save valuable time in exams and reduce algebraic errors. This set covers equation solving, graph analysis, calculus, statistics, finance, and more — all typical of Paper 2 or Paper 3 questions. Work through each section carefully, replicate the keystrokes on your own device, and build the confidence to tackle any calculator-active problem.
欢迎来到这个针对性复习题集，专门为提升你在 IB 数学考试中使用计算器的技能而设计。无论你用的是 TI-84 Plus、TI-Nspire、Casio fx-CG50 还是其他经批准的图形计算器，掌握 GDC 功能都能在考试中节省宝贵时间并减少代数错误。本题集涵盖方程求解、图像分析、微积分、统计、金融等内容——全都是卷二或卷三的典型题型。仔细完成每一部分，在自己的设备上操作对应的按键，逐步建立应对任何计算器题型的信心。
1. Solving Equations Numerically | 数值解方程
Many IB questions ask you to solve equations like 2x³ – 5x + 1 = 0 or e²ˣ an locate real roots quickly using the built-in equation solver or polynomial root finder.
许多 IB 题目要求解出如 2x³ – 5x + 1 = 0 或 e²ˣ + ln x = 4 这样的方程。你的图形计算器能借助内置的方程求解器或多项式求根工具迅速找到实数根。
On a TI-84 Plus, press MATH and scroll down to Solver. Enter the equation in the form 0 = E1 – E2 if needed. Provide an initial guess and press ALPHA [SOLVE]. For polynomial equations, use APPS > PlySmlt2 to find all roots at once.
在 TI-84 Plus 上，按 MATH 并向下滚动到 Solver。必要时将方程转化为 0 = E1 – E2 的形式。输入初始猜测值，然后按 ALPHA [SOLVE]。对于多项式方程，可使用 APPS > PlySmlt2 一次性求出所有根。
Casio fx-CG50 users can press MENU > Equation, select polynomial degree or solver, input coefficients, and press SOLVE. Always check that your solution makes sense in the context of the problem — extraneous solutions can appear when logarithms or radicals are involved.
Casio fx-CG50 用户可按 MENU > Equation，选择多项式次数或求解器，输入系数，然后按 SOLVE。务必检查你的解在题目情境中是否合理——涉及对数或根式时可能会出现增根。
2. Graphical Analysis: Roots, Intersections, and Turning Points | 图像分析：根、交点与转折点
the function(s) into the Y= editor. Choose a suitable window that shows all key features.
当题目要求求交点坐标或曲线上的最大值点时，GDC 的图像工具不可或缺。首先将函数输入 Y= 编辑器。选择一个能显示所有关键特征的合适窗口。
To find a root (x-intercept), use CALC > zero. Set a left bound, a right bound, and a guess. For intersections, use CALC > intersect, select the two curves, and move the cursor near the expected point.
要寻找根（x 轴截距），使用 CALC > zero。设定左边界、右边界和猜测值。对于交点，使用 CALC > intersect，选择两条曲线，将光标移到预期的交点附近。
Turning points can be analysed with CALC > maximum or CALC > minimum. On TI-Nspire, you can simply tap the graph, go to Analyze Graph, and select the appropriate option. Remember that the calculator only gives numerical approximations, so round your final answers as instructed by the IB.
转折点可通过 CALC > maximum 或 CALC > minimum 进行分析。在 TI-Nspire 上，只需点击图像，进入 Analyze Graph 并选择相应选项即可。请记住，计算器只给出数值近似，因此要按照 IB 的要求对最终答案进行舍入。
3. Numerical Derivatives and Definite Integrals | 数值导数与定积分
g results and for tackling functions that are messy to integrate analytically, such as ∫₀² √1 + e³ˣ dx.
即便你能手动求导，计算器仍然是检查结果以及处理那些难以解析积分的函数（如 ∫₀² √1 + e³ˣ dx）的绝佳工具。
On TI-84, use MATH > nDeriv to find the derivative at a point. The syntax is nDeriv(expression, variable, value). For a definite integral, use MATH > fnInt: fnInt(expression, variable, lower, upper). Always check that your GDC is in radian mode if the expression involves trigonometric functions.
在 TI-84 上，使用 MATH > nDeriv 求某一点处的导数值。语法为 nDeriv(表达式, 变量, 值)。对于定积分，使用 MATH > fnInt：fnInt(表达式, 变量, 下限, 上限)。如果表达式中含有三角函数，务必检查你的 GDC 是否处于弧度模式。
Casio users can press OPTN > CALC > d/dx or ∫dx. An accuracy tolerance can often be left at default. The IB exam may require you to write down the calculator display to a certain number of decimal places, so be precise.
Casio 用户可按 OPTN > CALC > d/dx 或 ∫dx。精度容差通常可保留默认值。IB 考试可能要求你抄写计算器显示值到若干位小数，因此要仔细。
4. Matrix Operations | 矩阵运算
Matrix arithmetic — addition, subtraction, multiplication, determinant, and inverse — is heavily tested in the applications and interpretation course. The GDC handles these calculations in seconds, but you must enter the dimensions correctly.
矩阵算术——加法、减法、乘法、行列式和逆——在应用与解释课程中被大量考查。GDC 可在几秒钟内完成这些计算，但你必须正确输入维度。
On TI-84, access 2ND > MATRIX. Choose EDIT to define a matrix, say [A] as a 3×3. After storing, go to NAMES, select [A], and perform operations like [A]⁻¹ or det([A]). Remember that multiplication is not commutative, so enter [A][B] or [B][A] carefully.
在 TI-84 上，进入 2ND > MATRIX。选择 EDIT 定义一个矩阵，例如将 [A] 设为 3×3。存储后，转到 NAMES，选择 [A]，并执行诸如 [A]⁻¹ 或 det([A]) 的运算。请记住乘法不满足交换律，因此要小心输入 [A][B] 或 [B][A]。
compute [A]⁻¹[B] or use the simultaneous equation solver built into the calculator (e.g., APPS > PlySmlt2 on TI-84, or Equation mode on Casio). Always verify your solution by substituting back or by checking the determinant is non-zero.
要求解线性方程组 AX = B，你可以计算 [A]⁻¹[B]，或使用计算器内置的联立方程求解器（如 TI-84 上的 APPS > PlySmlt2，或 Casio 上的 Equation 模式）。一定要通过回代或检查行列式非零来验证你的解。
5. Univariate Statistics | 单变量统计
Data analysis questions often begin with a list of numbers. Your calculator can produce summary statistics — mean, standard deviation, quartiles — with ease, but you need to know the difference between a sample and a population.
数据分析题常常从一列数字开始。你的计算器能轻松生成汇总统计量——平均值、标准差、四分位数——但你需要知道样本和总体的区别。
Enter the data into a list (e.g., L1 on TI-84, or List 1 on Casio). Then press STAT > CALC > 1-Var Stats and specify the list. The output shows x̄ (sample mean), Sx (sample standard deviation), σx (population standard deviation), and the five-number summary. Use Sx when your data is a sample; the IB usually expects the sample standard deviation unless stated otherwise.
将数据输入列表（例如 TI-84 的 L1，或 Casio 的 List 1）。然后按 STAT > CALC > 1-Var Stats 并指定列表。输出会显示 x̄（样本均值）、Sx（样本标准差）、σx（总体标准差）以及五数概括。当数据为样本时使用 Sx；除非另有说明，IB 通常期望使用样本标准差。
faster and less error-prone. Pay attention to outliers using the IQR rule: any value < Q1 – 1.5×IQR or > Q3 + 1.5×IQR is an outlier.
你也可以通过对列表排序来直接找到中位数和四分位数，但 1-Var Stats 更快且不易出错。请利用 IQR 法则关注异常值：任何 < Q1 – 1.5×IQR 或 > Q3 + 1.5×IQR 的值都是异常值。
6. Linear Regression and Correlation | 线性回归与相关性
x and the Pearson correlation coefficient r. The calculator outputs these values directly.
对于双变量数据，你经常需要求出回归直线方程 y = a + bx 和皮尔逊相关系数 r。计算器可直接输出这些值。
After storing x-values in one list and y-values in another, select STAT > CALC > LinReg(ax+b) or LinReg(a+bx). Make sure to specify the correct lists. The output includes a, b, r², and r. The IB requires you to report r to three significant figures and interpret its meaning in context.
在将一个列表存储 x 值、另一个列表存储 y 值之后，选择 STAT > CALC > LinReg(ax+b) 或 LinReg(a+bx)。确保指定正确的列表。输出包含 a、b、r² 和 r。IB 要求你将 r 报告到三位有效数字，并结合情境解释其意义。
Be cautious: a high r value does not imply causation. Also check the residual plot if you suspect a non-linear pattern. Your GDC can store residuals in a list after running the regression — use RESID from the LIST NAMES menu.
注意：高的 r 值并不意味着因果关系。如果你怀疑存在非线性模式，还应检查残差图。你的 GDC 在运行回归后可将残差存储到一个列表中——使用 LIST NAMES 菜单中的 RESID。
7. Probability Distributions: Binomial and Normal | 概率分布：二项分布与正态分布
Both binomial and normal distribution functions appear frequently on the calculator paper. You need to know when to use pdf (exact value) and cdf (cumulative value).
二项分布和正态分布函数在计算器试卷上频繁出现。你需要知道何时使用 pdf（精确值）和 cdf（累积值）。
For a binomial distribution X ~ B(n, p), to find P(X = k) use binompdf(n, p, k). For P(X ≤ k) use binomcdf(n, p, k). To find P(X > k), compute 1 – binomcdf(n, p, k). On TI-84 these are under DISTR (2ND + VARS). Casio users navigate via STAT > DIST > BINM.
对于二项分布 X ~ B(n, p)，若要计算 P(X = k)，使用 binompdf(n, p, k)。对于 P(X ≤ k)，使用 binomcdf(n, p, k)。若要计算 P(X > k)，则执行 1 – binomcdf(n, p, k)。在 TI-84 上，这些功能位于 DISTR (2ND + VARS)。Casio 用户则通过 STAT > DIST > BINM 进入。
For the normal distribution, use normalcdf(lower, upper, μ, σ) to find the probability. For inverse normal (finding the value given a left-tail probability), use invNorm(area, μ, σ). When standardising, ensure you enter an extremely negative number (e.g., –1E99) for a lower tail that goes to –∞.
对于正态分布，使用 normalcdf(下限, 上限, μ, σ) 求概率。对于逆正态（给定左尾概率求对应值），使用 invNorm(面积, μ, σ)。在进行标准化时，若左尾延伸到 –∞，务必输入一个极小的负数（例如 –1E99）。
8. Financial Mathematics: TVM Solver | 金融数学：货币时间价值求解器
Loan repayments, investments, and annuity problems are solved efficiently with the TVM (Time Value of Money) solver. The key is to understand the sign convention and the meaning of each variable.
贷款还款、投资和年金问题可通过 TVM（货币时间价值）求解器高效解决。关键在于理解符号约定以及每个变量的含义。
On TI-84, access APPS > Finance > TVM Solver. N = total number of compounding periods, I% = compounding per year. Set PMT to END unless the payment is at the beginning of a period.
在 TI-84 上，进入 APPS > Finance > TVM Solver。N = 总复利周期数，I% = 名义年利率，PV = 现值（若为支出则为负），PMT = 每期付款额，FV = 未来值，P/Y = 每年付款次数，C/Y = 每年复利次数。除非付款发生在一个周期的期初，否则将 PMT 设为 END。
Casio fx-CG50 users will find the TVM solver under FINANCIAL (from the main menu). The same parameters apply. Always draw a timeline and decide whether the unknown variable should be positive or negative according to the IB’s cash flow convention.
Casio fx-CG50 用户可在主菜单的 FINANCIAL 下找到 TVM 求解器，参数相同。始终画出时间线，并根据 IB 的现金流惯例判断未知变量应为正值还是负值。
9. Complex Number Calculations | 复数计算
Complex numbers appear in both AA and AI courses at HL. The GDC can perform arithmetic in rectangular form a + b i and polar form r∠θ, as well as convert between the two.
复数在 AA 和 AI 的高级课程中都会出现。GDC 可以执行直角坐标形式 a + b i 和极坐标形式 r∠θ 的运算，并在两者之间进行转换。
On TI-84, set the mode to a+bi to enable complex output. Then you can enter expressions like (3+4i) × (2–i) directly. Use MATH > CPX to find the conjugate, real part, imaginary part, argument, or absolute value. To convert to polar, use angle( for argument and abs( for modulus.
在 TI-84 上，将模式设为 a+bi 以启用复数输出。之后便可直接输入如 (3+4i) × (2–i) 这样的表达式。使用 MATH > CPX 求共轭、实部、虚部、辐角或绝对值。要转换为极坐标形式，使用 angle( 求辐角、abs( 求模。
Casio users can work in complex mode (RUN·MAT with SETUP > Complex Mode: a+bi or r∠θ). The i button is accessed via SHIFT + 0. When solving polynomial equations with complex coefficients, the equation solver will return both real and complex roots if they exist.
Casio 用户可在复数模式下工作（RUN·MAT，SETUP > Complex Mode: a+bi 或 r∠θ）。i 键通过 SHIFT + 0 输入。当求解带有复系数的多项式方程时，方程求解器会返回所有实根和复根（如果存在）。
10. Sequences and Series | 数列与级数
Arithmetic and geometric sequences often require finding a specific term or the sum of the first n terms. Instead of using formulas manually, you can employ the calculator’s sequence mode or summation function.
等差与等比数列常常需要求某一项或前 n 项和。无需手动套用公式，你可以使用计算器的数列模式或求和功能。
To generate terms of a sequence explicitly, define u(n) in the sequence mode. On TI-84, press MODE, select Seq, then input u(n) in Y=. You can view a table of values. For sums, use sum(seq( from the LIST > OPS menu. For example, to sum k² from k=1 to 10: sum(seq(K², K, 1, 10)).
若要显式生成数列的项，在数列模式下定义 u(n)。在 TI-84 上，按 MODE，选择 Seq，然后在 Y= 中输入 u(n)。你可以查看数值表。对于求和，使用 LIST > OPS 菜单中的 sum(seq(。例如，对 k=1 到 10 的 k² 求和：sum(seq(K², K, 1, 10))。
Casio users can use the RECUR app or the Σ function in RUN·MAT. Press OPTN > CALC > Σ(. The syntax is Σ(expression, variable, start, end). Always ensure your calculator is not in degrees mode if there is no trigonometric part, but it is good practice to double-check mode settings.
Casio 用户可使用 RECUR 应用程序或 RUN·MAT 中的 Σ 函数。按 OPTN > CALC > Σ(，语法为 Σ(表达式, 变量, 起始, 终止)。始终确保你的计算器不处于度数模式（除非有三角函数部分），不过仔细检查模式设置是个好习惯。
11. Function Graphing and Advanced Window Tactics | 函数图像绘制与高级窗口策略
Choosing the right viewing window is half the battle. A standard ZOOM ZStandard ( –10 to 10 for both axes) often misses asymptotes, maxima, or intersections that occur at larger scales.
选择合适的观察窗口是成功的一半。标准的 ZOOM ZStandard（两轴均为 –10 到 10）常常会遗漏出现在更大尺度上的渐近线、最大值或交点。
When a function contains a vertical asymptote, the calculator may incorrectly draw a vertical line. Switch to Dot mode (instead of Connected) to avoid this. Use ZOOM Fit after entering an appropriate y-range to see the whole graph. For trigonometric graphs, set x-min and x-max in terms of π to capture complete periods.
当函数包含垂直渐近线时，计算器可能会错误地绘制一条垂直线。切换到 Dot（点）模式（而非 Connected）可避免此问题。在输入合适的 y 范围后使用 ZOOM Fit 查看完整图像。对于三角函数图像，将 x-min 和 x-max 设置为 π 的倍数以捕捉完整周期。
) = k, graph Y1 = f(x) and Y2 = k, count the intersections visually, and confirm each with CALC > intersect. Zoom in around crowded regions to avoid missing hidden roots.
如果被要求找出方程 f(x) = k 的解的个数，画出 Y1 = f(x) 和 Y2 = k，目视数出交点数量，并使用 CALC > intersect 逐一确认。在密集区域放大以避免遗漏隐藏的根。
12. Exam-Day Efficiency Tips | 考试日效率技巧
Efficiency with a calculator comes not only from knowing the menus but also from habit. Practise resetting your calculator (mem clear) before a mock exam and performing a quick diagnostic self-test.
计算器使用效率不仅来自对菜单的熟悉，也源于习惯养成。在模拟考试前练习重置计算器（内存清除）并进行快速的诊断自检。
Always store intermediate answers in memory or use the Ans feature to avoid round-off errors. If you get a syntax error, press 2 or Goto to locate the error quickly. On test day, bring spare batteries and ensure your calculator is in the exam mode or press-to-test mode required by your school.
始终将中间结果存入存储器或使用 Ans 功能以避免舍入误差。如果出现语法错误，按 2 或 Goto 可快速定位错误。考试当天，请携带备用电池，并确保你的计算器已进入学校要求的考试模式或按压测试模式。
A common mistake is to leave the calculator in radian mode for a question that requires degrees, or vice versa. Check the mode indicator at the top of the screen before every set of calculations. Finally, when the exam asks for “use your GDC”, always state the command you used in your written reasoning — this demonstrates proper method.
一个常见错误是，在需要角度的题目中将计算器留在弧度模式，反之亦然。在每一组计算之前，检查屏幕顶部的模式指示符。最后，当考题要求“使用你的 GDC”时，务必在书面推理中说明你所用到的指令——这能体现正确的方法。
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