📚 Review Set 15B - CALCULATOR | 复习题集15B - 计算器
Mastering the effective use of a graphical display calculator (GDC) is essential for success in IB Mathematics. This review set explores a range of typical problems where calculator skills are not just helpful but required. From solving equations and analysing functions to performing statistical tests and matrix operations, each section reinforces techniques that save time and improve accuracy under exam conditions. Work through each example carefully and ensure you know how to access the relevant menus on your own device.
熟练掌握图形计算器 (GDC) 的有效使用是 IB 数学取得高分的必备技能。本复习题集涵盖了一系列典型问题，在这些问题中，计算器技巧不仅是辅助，更是必须掌握的核心能力。从解方程、分析函数到执行统计检验和矩阵运算，每一部分都强化了那些在考试条件下既能节省时间又能提升准确度的技术。请仔细完成每个例题，并确保你清楚如何在自己的设备上调出相关菜单。
1. Solving Equations Using the Numeric Solver | 使用数值求解器解方程
Many IB problems require solving equations that are not easily handled algebraically. The numeric solver on your GDC allows you to input an equation and obtain a solution within a specified range. Begin by accessing the equation solver from the home screen or the 'Math' menu, then type the equation exactly as it appears, using the variable X. You will usually need to provide an initial guess or bounds; choose a sensible interval based on a quick sketch or a table of values. The calculator then iterates to find the root to a high degree of accuracy.
许多 IB 问题要求解的方程并不容易用代数方法处理。你的图形计算器上的数值求解器允许你输入方程并在指定范围内得到解。首先从主屏幕或“数学”菜单进入方程求解器，然后按照原样输入方程，使用变量 X。你通常需要提供一个初始猜测值或搜索区间；根据快速草图或数值表，选择一个合理的区间。然后计算器会迭代求解，得出高精度的根。
For example, to solve 3e⁽⁻ˣ⁾ − x² = 0, access the solver, enter Y₁ = 3e−X solution makes sense by substituting back into the original equation or viewing the graph of both sides.
例如，对于方程 3e⁽⁻ˣ⁾ − x² = 0，进入求解器后，输入 Y₁ = 3e−X−X²，设定下限为 0、上限为 2。计算器会迅速返回 X ≈ 0.91。一定要将解代回原方程验算，或通过观察两边的图像来确认解的合理性。
Remember that some equations may have multiple roots. Use different initial guesses or analyse the graph to capture all solutions within the required domain. In exams, clearly record the equation you entered and the bounds used to obtain each root.
请记住，有些方程可能有多个根。使用不同的初始猜测值，或通过分析图像来捕捉所需定义域内的所有解。在考试中，要清楚地记录你输入的方程以及获取每个根所用的上下界。
2. Finding Intersections Graphically | 图形法求交点
ons but also provides a visual check of your answers. After entering both functions in the 'Y=' ust the window settings so that the point of intersection is clearly visible. Then use the 'Intersect' command under the 'Calc' (or 'G-Solve' ill prompt you to select the two curves and a guess, then it will compute the coordinates.
解形如 f(x) = g(x) 方程的一个强大技巧是绘制 Y₁ = f(x) 和 Y₂ = g(x) 的图像，并求交点。这种方法不仅展示了方程解的个数，还为你的答案提供了可视化检验。在“Y=”屏幕中输入两个函数后，调整窗口设置，使交点清晰可见。然后使用“计算”（或“G-Solve”）菜单下的“交点”命令。图形计算器会提示你选择两条曲线并给出一个猜测值，然后就会计算出交点坐标。
Consider solving 2ˣ = 4 − x. Graph Y₁ = 2X solution to the equation is approximately 1.39 (to 3 significant figures). You can also verify by computing 21.386 and 4−1.386.
以求解 2ˣ = 4 − x 为例。绘制 Y₁ = 2X 和 Y₂ = 4−X 的图像。标准窗口显示两条曲线在 X = 1 附近相交。使用交点功能得到 X ≈ 1.386，Y ≈ 2.614。这证实验证了该方程的解约为 1.39（精确到 3 位有效数字）。你也可以通过计算 21.386 和 4−1.386 来进一步验证。
When equations have more than one root, such as |x² − 4| on points. Trace along the graph and use the intersection command for each region. Always state the final solutions rounded as requested, and sketch the graph in your answer booklet if a graphical method is used.
当方程有多个根时，例如 |x² − 4| = 2，你会看到多个交点。沿图像轨迹移动，并对每个区域使用交点命令。如果你使用了图形方法，请务必按要求四舍五入给出最终解，并在答题本上画出草图。
3. Using Tables for Function Values | 使用表格计算函数值
The table feature on a GDC is an efficient way to generate function values for plotting, examining limits, or solving inequalities. After defining a function Y₁, you can set a starting value and a step (ΔTbl) for the table. This is particularly useful when you need to show that a function changes sign within an interval, confirming the existence of a root.
图形计算器上的表格功能可以高效地生成函数值，用于描点作图、考察极限或求解不等式。定义函数 Y₁ 后，你可以为表格设置起始值和步长 (ΔTbl)。当你需要证明函数在某一区间内符号改变，从而证明根的存在性时，表格功能尤其有用。
es from positive to negative between 0.2 and 0.4, a root must lie in that subinterval. You can refine the step to get a more precise estimate.
例如，要证明 f(x) = x³ − 3x + 1 在 0 和 1 之间有一个根，设置 TblStart = 0、ΔTbl = 0.2。表格显示 f(0) = 1，f(0.2) = 0.408... 继续看，f(0.4) = -0.216。由于 f 的值在 0.2 和 0.4 之间从正变成负，因此在该子区间内必定存在一个根。你可以进一步缩小步长来得到更精确的估计值。
Tables can also help identify horizontal asymptotes: if you examine values as X grows very large, the Y-values approach a constant. In exam settings, always note the start, step, and whichever rows support your conclusion.
表格还可以帮助识别水平渐近线：如果你随着 X 变得非常大来观察函数值，Y 值会趋近于某个常数。在考试情境下，始终要记明起始值、步长以及支持你结论的那些数据行。
4. Evaluating Derivatives at a Point | 在某点求导数值
compute the numerical derivative dy/dx without requiring symbolic differentiation. Use the 'nDeriv' (or 'd/dx') function, which is typically found in the 'Math' le, value). For example, nDeriv(sin(X²), X, 1) returns the derivative of sin(x²) at x = 1.
IB 题目经常要求求出函数在某特定 x 值处的切线斜率。你的图形计算器可以通过数值方式计算导数 dy/dx，而不需要符号求导。使用“nDeriv”（或“d/dx”）功能，这通常在“数学”菜单里。其语法为 nDeriv(表达式, 变量, 值)。例如，nDeriv(sin(X²), X, 1) 会返回 sin(x²) 在 x = 1 处的导数值。
Alternatively, you can graph the function, then select 'dy/dx' from the 'Calc' menu. Move the cursor to the desired x-coordinate and press enter. The calculator will display the slope. Always verify that the x-value is within the domain of the derivative, and when writing your answer, round as instructed (e.g., to 3 significant figures).
或者，你也可以先画出函数图像，然后从“计算”菜单中选择“dy/dx”。将光标移动到所需的 x 坐标上并按确认键，计算器就会显示出斜率。请务必核实该 x 值在导数的定义域内，并且在写下答案时按要求进行四舍五入（如保留三位有效数字）。
ical method is also valuable for confirming your analytical differentiation. If you find f' ly, you can check with the GDC at several points. This cross-checking habit can catch errors early.
这种数值求导方法对于验证你的解析求导结果也很有价值。如果你手动求得导数表达式，例如 2x·cos(x²) − 2x·sin(x²)，你可以用图形计算器在几个点上进行检查。这种交叉核对的习惯能够及早发现错误。
5. Definite Integrals and Area Calculations | 定积分与面积计算
lating the area under a curve or between two curves is a standard GDC application. Use the '∫f(x)dx' command from the 'Calc' , select the integration option, choose lower bound, upper bound, and the shaded region will appear with the numerical value.
计算曲线下方或两条曲线之间的面积是图形计算器的一项标准应用。使用“计算”菜单中的“∫f(x)dx”命令。你需要输入积分下限和上限，计算器就会以数值方式求出定积分。在绘制出函数 Y₁ 的图像后，选择积分选项，指定下限和上限，阴影区域就会出现并伴随着数值结果。
ed and the absolute area is found if the curves cross. For example, the area enclosed by y = x² and y = √x between their intersections at x = 0 and x = 1 is given by ∫(√x − x²)dx from 0 to 1, which yields 1/3.
对于曲线之间的面积，画出两个函数的图像并找出它们的交点。然后计算交点区间内 (上方函数 − 下方函数) 的定积分。虽然有些型号的图形计算器允许你直接对 Y₁−Y₂ 进行积分，但你必须确保使用了正确的公式，并在曲线有交叉时求出绝对面积。例如，y = x² 与 y = √x 在 x = 0 和 x = 1 的交点之间所围成的面积为 ∫₀¹ (√x − x²)dx，结果为 1/3。
Always note that the GDC gives an approximate value, so report it with the requested precision. If an exact answer is required (such as fractions or π), the calculator value helps you confirm your analytical work but may not replace showing the symbolic integration steps.
始终要注意，图形计算器给出的是近似值，因此要按照题目要求的精度进行报告。如果题目要求给出精确答案（如分数或含 π 的表达式），计算器的值有助于你验证解析计算，但不能代替展示符号积分步骤。
6. Statistical Lists and One-Variable Analysis | 统计列表与单变量分析
Handling data sets is a key part of the IB syllabus. Your GDC can store data in lists (e.g., L1, L2) and compute summary statistics instantly. After entering data into a list via the 'Stat' edit menu, choose '1-Var Stats' and specify the list. The output includes the mean (x̄), sum (Σx), sum of squares (Σx²), sample standard deviation (Sx), population standard deviation (σx), minimum, Q1, median, Q3, and maximum.
处理数据集是 IB 大纲的一个重要部分。你的图形计算器可以将数据存储在列表（如 L1、L2）中，并即刻计算出汇总统计量。通过“统计”编辑菜单将数据输入列表后，选择“单变量统计”并指定列表。输出结果包括均值 (x̄)、总和 (Σx)、平方和 (Σx²)、样本标准差 (Sx)、总体标准差 (σx)、最小值、第一四分位数 Q1、中位数、第三四分位数 Q3 和最大值。
When a frequency list is provided, you must set the frequency option to that list. For example, if values are in L1 and frequencies in L2, execute 1-Var Stats L1, L2. The calculator will then correctly weight the data. This is crucial for grouped data as well, where you may use the midpoint values in L1 and frequencies in L2.
当提供频数列表时，你必须将频数选项设置为该列表。例如，如果数值在 L1、频数在 L2 中，执行单变量统计时需指定 L1, L2。计算器随即会正确地加权数据。这对分组数据同样至关重要，此时你可以将组中值放在 L1，频数放在 L2。
Practice interpreting the output: the mean and standard deviation are used for normally distributed data, while the quartiles are essential for box-and-whisker plots. Always identify whether the question expects the sample or population standard deviation and choose accordingly.
请练习解读这些输出结果：均值和标准差用于正态分布数据，而四分位数对于箱形图至关重要。要始终判断题目期望你使用样本标准差还是总体标准差，并据此做出选择。
7. Linear Regression and Correlation | 线性回归与相关性
Bivariate data analysis often involves finding a line of best fit and the correlation coefficient. Enter the x-values in L1 and y-values in L2, then select 'LinReg(ax+b)' (or 'LinReg(a+bx)') from the statistics calculation menu. The GDC returns the coefficients a (slope) and b (y-intercept), as well as the correlation coefficient r and the coefficient of determination r². Make sure that the 'DiagnosticOn' setting is activated so r and r² are displayed.
双变量数据分析常常涉及寻找最佳拟合直线和相关系数。将 x 值输入 L1、y 值输入 L2，然后从统计计算菜单中选择“LinReg(ax+b)”（或“LinReg(a+bx)”）。图形计算器会返回系数 a（斜率）和 b（y 轴截距），以及相关系数 r 和决定系数 r²。请确保开启了“诊断开启 (DiagnosticOn)”设置，这样才能显示出 r 和 r²。
x. The r value emerges as 0.998, indicating a very strong positive linear relationship. You can also plot the scatter diagram and overlay the regression line using the 'Stat Plot' and graph functions. This visual check helps identify outliers.
例如，给定数据点 (1, 2.1)、(2, 4.0)、(3, 5.8)、(4, 8.2)，回归直线可能是 y = 0.98 + 1.78x。计算出的 r 值为 0.998，表明两者之间有着非常强的正线性关系。你还可以使用“统计绘图”和图像功能，画出散点图并叠加上回归直线。这种可视化检查有助于识别异常值。
Remember to round coefficients appropriately, and always write the equation in the context of the variables (e.g., 'y on x' regression). When questions ask for predicted values, substitute into the regression equation and comment on reliability, especially when extrapolating.
请记得对系数进行恰当的舍入，并始终结合变量的背景写出方程（如“y 对 x 回归”）。当题目要求预测值时，将其代入回归方程，并对可靠性做出评论，尤其是在外推预测的情况下。
8. Normal Distribution Calculations | 正态分布计算
IB applications of the normal distribution rely heavily on the calculator's built-in functions: normalcdf (for probabilities) and invNorm (for percentiles). To find P(a < X < b) for X ~ N(μ, σ²), use normalcdf(a, b, μ, σ). Enter lower bound, upper bound, mean, and standard deviation. If the tail extends to infinity, use a large number like 1099 or -1099 as appropriate.
IB 中对正态分布的应用很大程度上依赖于计算器的内置功能：normalcdf（用于求概率）和 invNorm（用于求百分位数）。对于 X ~ N(μ, σ²)，要求 P(a < X < b)，可使用 normalcdf(a, b, μ, σ)。依次输入下限、上限、均值、标准差。如果尾部延伸到无穷，可使用一个很大的数，例如 10⁹⁹ 或 -10⁹⁹。
, the probability that a randomly chosen student is taller than 180 cm is normalcdf(180, 1099, 170, 6) ≈ 0.0478. Conversely, to find the height above which 10% 0.9, 170, 6) ≈ 177.7 cm (since the function gives the value with a left-tail area of 0.9).
例如，假设身高服从正态分布，μ = 170 cm，σ = 6 cm，则随机挑选的学生身高高于 180 cm 的概率为 normalcdf(180, 1099, 170, 6) ≈ 0.0478。反之，要找出 10% 的学生身高超过多少厘米，可使用 invNorm(0.9, 170, 6) ≈ 177.7 cm（因为该函数给出的是左侧面积为 0.9 的值）。
Always sketch a bell curve and shade the required area before using the calculator. This reduces input errors. For 'between' probabilities, you may need to standardize first if the question requires z-scores, but the GDC can directly handle non-standard distributions.
在使用计算器之前，请务必草绘一个钟形曲线并涂黑所求区域。这将减少输入错误。对于“介于”之间的概率，如果题目要求使用 z-score，你可能需要先进行标准化，但图形计算器能够直接处理非标准分布。
9. Binomial Distribution Tools | 二项分布工具
, k); for P(X ≤ k), use binomcdf(n, p, k). Remember that binomcdf accumulates from 0 to k.
二项概率 P(X = k) 和累积概率 P(X ≤ k) 可以使用 binompdf 和 binomcdf 进行计算。这些函数要求输入试验次数 n、成功概率 p 以及值 k。对于 P(X = k)，使用 binompdf(n, p, k)；对于 P(X ≤ k)，使用 binomcdf(n, p, k)。请记住，binomcdf 是从 0 累积到 k。
) because you want the complement of P(X ≤ 9). Compute and report to a sensible precision.
假设一次多项选择题考试共有 20 题，每题有 4 个选项，且某学生完全随机猜测。则恰好答对 6 题的概率为 binompdf(20, 0.25, 6) ≈ 0.1686。通过考试（即至少答对 10 题）的概率为 1 − binomcdf(20, 0.25, 9)，因为你求的是 P(X ≤ 9) 的互补事件。进行计算并以合理精度报告答案。
When the question asks for the mode or to find the most likely number of successes, you can either use the table of values or compare binompdf for successive k. The GDC's list functions can quickly generate the full distribution for small n.
当题目要求找出众数或最可能的成功次数时，你可以使用数值表，或者比较相邻 k 值对应的 binompdf。对于较小的 n，图形计算器的列表功能可以快速生成完整的分布列。
10. Matrix Operations for HL Students | 适用于 HL 学生的矩阵运算
For IB Mathematics HL, matrix arithmetic, determinants, and inverses are often tested. After entering matrices into the GDC (using the 'Matrix' and 'Edit' menus), you can perform addition, subtraction, multiplication, and scalar multiplication. To solve a system of linear equations, you can use the inverse matrix method or the built-in simultaneous equation solver. If A is the coefficient matrix, the solution vector is A⁻¹ * B, where B is the constants matrix.
对于 IB 数学 HL 的学生来说，矩阵的算术运算、行列式和逆矩阵经常出现在考题中。在图形计算器中输入矩阵后（使用“矩阵”和“编辑”菜单），你可以执行加法、减法、乘法和标量乘法。若要解线性方程组，你可以使用逆矩阵法或内置的联立方程求解器。如果 A 是系数矩阵，则解向量为 A⁻¹ * B，其中 B 是常数项矩阵。
e the inverse, call up the matrix name and press the x⁻¹ key. For the determinant, use the 'det(' 1, 5x + 7y = 0 is [x; y] = [A]⁻¹ * [[1],[0]] = [[0.777...],[-0.555...]]. Round as needed.
要计算逆矩阵，调出矩阵名称并按下 x⁻¹ 键。对于行列式，可从矩阵数学菜单中使用“det(”命令，后跟矩阵名称。例如，若 [A] 为 [[2,1],[5,7]]，则 det([A]) = 14 − 5 = 9。方程组 2x + y = 1, 5x + 7y = 0 的解为 [x; y] = [A]⁻¹ * [[1],[0]] = [[0.777...],[-0.555...]]，根据要求进行舍入。
When dealing with transition matrices or steady-state vectors, you can use the calculator to raise a matrix to a high power (e.g., [M]20) to find long-term probabilities. Always state what matrix operations you performed and interpret the result in the context of the problem.
在处理转移矩阵或稳态向量时，你可以使用计算器将矩阵升高次幂（例如 [M]20）来求出长期概率。请始终说明你所进行的矩阵运算步骤，并结合问题背景解释结果。
11. Finance Applications: TVM Solver | 金融应用：TVM 求解器
The Time-Value-Money (TVM) solver on your GDC is built to handle compound interest, annuities, and loan repayment calculations. You will need to input the total number of compounding periods (N), annual interest rate (I%), present value (PV), payment amount (PMT), future value (FV), and number of payments per year (P/Y) and compounding periods per year (C/Y). The solver then finds the unknown variable when others are specified.
你的图形计算器中的资金时间价值 (TVM) 求解器专为处理复利、年金和贷款偿还计算而设计。你需要输入总复利期数 (N)、年利率 (I%)、现值 (PV)、每期付款额 (PMT)、终值 (FV)，以及每年的付款次数 (P/Y) 和每年的复利计算次数 (C/Y)。当指定其他变量时，求解器会求出未知变量。
It is crucial to follow the cash flow sign convention: money you receive is positive, money you pay out is negative. For a loan, PV is positive (you receive the loan), PMT is negative (you make repayments), and FV is typically 0. For an investment, PMT might be negative (your contributions) and FV positive (what you receive at the end). Always sketch a timeline to keep signs correct.
严格遵循现金流符号惯例至关重要：你收到的钱为正，你付出的钱为负。对于一笔贷款，PV 为正（你收到贷款），PMT 为负（你偿还借款），而 FV 通常为 0。对于一项投资，PMT 可能为负（你的投入），FV 为正（到期时你收到的款项）。始终画一条时间线以保持符号正确。
For example, to find the monthly payment on a $15,000 loan over 5 years at 4.5% p.a. compounded monthly, set N = 5×12 = 60, I%=4.5, PV=15000, FV=0, P/Y=12, C/Y=12. Solve for PMT, which should be about −$279.65. Interpret the negative payment as an outgoing cash flow.
例如，为了计算一笔金额为 $15,000、年利率 4.5% 按月复利、5 年期贷款的月还款额，设定 N = 5×12 = 60，I% = 4.5，PV = 15000，FV = 0，P/Y = 12，C/Y = 12。求解 PMT，应得到约 −$279.65。将负的付款额解释为资金流出。
12. Complex Numbers and Polar Conversions | 复数与极坐标转换
For HL students, complex numbers can be managed using the GDC's complex mode. Ensure the calculator is set to a+bi mode. Arithmetic with complex numbers is straightforward: enter i using the 'i' key. You can also convert between Cartesian (a+bi) and polar (r∠θ or r·eiθ) forms using commands like 'Polar' and 'Rect' (or by changing the complex output format).
对于 HL 学生，可以通过图形计算器的复数模式来处理复数。确保计算器设置为 a+bi 模式。复数运算非常直接：使用“i”键输入虚数单位 i。你还可以使用“Polar”和“Rect”等命令（或通过更改复数的输出格式），在笛卡尔形式 (a+bi) 和极坐标形式 (r∠θ 或 r·eiθ) 之间进行转换。
When finding powers or roots, De Moivre's theorem can be checked using the GDC. For example, (1 + i)⁶ can be computed directly and returns −8i. For roots of unity, you may enter the complex number in polar form and raise to a fractional power, but be careful: the calculator gives the principal root; to find all roots, you must add multiples of 2π/n to the argument before dividing.
在求幂或求根时，可用图形计算器来验证棣莫弗定理。例如，(1 + i)⁶ 可直接计算，返回值为 −8i。对于单位根问题，你可以将复数以极坐标形式输入，并求分数次幂，但需注意：计算器给出的是主值根；要找出所有根，你必须先将辐角加上 2π/n 的整数倍，再进行除法。
Many GDC models also have a polynomial root finder in the equation menu. For a quadratic with complex coefficients, the solver will display complex conjugate roots without further algebraic manipulation. This is a powerful check against your manual completion of the square or quadratic formula.
许多图形计算器型号在方程菜单中还带有多项式求根器。对于系数为复数的二次方程，求根器会直接显示共轭复根，无需进一步的代数操作。这是对你手动配方法或使用求根公式的强力检验手段。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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