📚 Review set 5B - CALCULATOR | 复习题集5B - 计算器
In IB Mathematics, mastering the graphical display calculator (GDC) is a decisive factor in exam success. Review set 5B targets the advanced calculator functions required to solve equations, evaluate calculus expressions, process statistical data, and handle financial models efficiently. This article walks you through each key skill, offering step-by-step guidance and pitfall reminders to ensure you can answer calculator-active questions with confidence and speed.
在IB数学考试中，熟练驾驭图形计算器是取得高分的决定性因素。复习题集5B聚焦于高阶计算器功能，涵盖方程求解、微积分计算、统计分析以及金融模型处理。本文将逐一讲解各项核心技能，提供分步操作指南与易错点提醒，帮助你自信、快速地完成计算器相关题目。
1. Solving Equations with the GDC | 使用图形计算器解方程
To solve an equation like 3x² − 8x + 2 = 5x − 1, press Y= and enter the left-hand side as Y₁ and the right-hand side as Y₂. Adjust the window if needed, then use CALC → intersect (or zero if you rearranged the equation to equal 0) to find the x-values of the intersection points. Always check for multiple solutions by scanning the graph and using the guess cursor.
要解方程 3x² − 8x + 2 = 5x − 1，按 Y= 将左式输入为 Y₁、右式输入为 Y₂。必要时调整窗口，然后使用 CALC → intersect（若已将方程移项为 0 则用 zero）求出交点的 x 值。务必移动光标探索图形，检查是否存在多个解。
When the equation is presented in the form f(x) = 0, the Poly Root Finder app on some GDC models can return all real and complex roots at once. However, for exam purposes, you must show the setup on paper, write down the display values to the required accuracy, and verify each root by substitution or by checking the graph.
若方程为 f(x)=0 的形式，某些型号计算器上的 Poly Root Finder 程序可一次性给出所有实根与复根。但在考试中，你仍需写下计算器设置步骤，按要求的精度记录显示值，并用代入法或图形检查验证每一个根。
2. Finding Intersections of Graphs | 求函数图像的交点
e two curves meet, such as y = ln(x+1) and y = 2 − x. After plotting both functions, press CALC → intersect, select the first curve, the second curve, and move the cursor near the suspected intersection. The GDC then iterates to a solution. Repeat for each visible intersection, remembering that trigonometric or logarithmic functions may have infinitely many solutions within a given domain—always restrict the domain manually.
许多IB题目要求找出两条曲线的交点坐标，如 y = ln(x+1) 与 y = 2 − x。绘制两个函数后，按 CALC → intersect，依次选定第一条曲线、第二条曲线，并将光标移至可疑交点附近，计算器便会迭代求解。对每一个可见交点重复此操作。需注意三角函数或对数函数在给定区间内可能有无数个解——务必手动限定定义域。
A common error is forgetting to check the exact answer requirement. The GDC gives decimal approximations. If the question demands an exact value, you may need to solve algebraically or use the intersection hint to recognise a known value like √2 or π/3.
常见错误是忽略了题目对精确值的要求。计算器给出的是小数近似值。如果题目要求精确值，你可能需要进行代数求解，或借助交点提示识别出诸如 √2 或 π/3 的已知值。
3. Numerical Derivatives at a Point | 求某点处的数值导数
To find f '(2) for f(x) = x³ − cos x, go to the MATH menu, select nDeriv (or d/dx on some models), enter the expression, variable, and value. The syntax is nDeriv(expression, x, 2). The calculator returns an approximate gradient. Be aware that numerical differentiation can be unstable near vertical tangents or discontinuities; always predict the behaviour from the graph first.
求 f(x)=x³ − cos x 在 x=2 处的导数 f '(2)，进入 MATH 菜单，选择 nDeriv（某些型号为 d/dx），输入表达式、变量及数值。语法为 nDeriv(表达式, x, 2)。计算器返回近似斜率。注意在垂直切线或不连续点附近，数值微分可能不稳定，应先通过图形预判函数行为。
In paper 2, you may need to use the derivative to check monotonicity or find the slope for a tangent line equation. Record the calculator’s derivative value to at least three significant figures unless told otherwise, and avoid rounding intermediate steps when substituting into longer expressions.
在试卷二中，你可能需要利用导数判断单调性或求切线方程的斜率。除非另有要求，应将计算器的导数值记录到至少三位有效数字，并在代入长表达式时避免舍入中间步骤。
4. Evaluating Definite Integrals | 计算定积分
For the integral ∫₁³ (5x² − 2) dx, press MATH → fnInt (or the integral template), enter the integrand, variable, lower limit, and upper limit. The GDC uses an adaptive quadrature algorithm, so it handles most continuous integrands accurately. Always write the integral setup on paper, then give the calculator’s decimal result, explicitly stating that it is an approximation unless the result is exact.
对于积分 ∫₁³ (5x² − 2) dx，按 MATH → fnInt（或积分模板），输入被积函数、变量、下限和上限。计算器采用自适应求积算法，能准确处理绝大多数连续被积函数。务必在纸上写出积分式，再给出计算器的小数结果，并明确说明除非结果恰好为精确值，否则均为近似值。
When the function touches or crosses the x-axis, remember to split the integral into positive and negative parts to find the total area, as the GDC’s direct integration gives the net signed area. Use CALC → zero to locate the roots first, then sum the absolute values of the individual integrals.
当函数与x轴相切或穿越时，记得将积分分段，分别求正面积与负面积后再取绝对值相加，因为直接积分给出的是净有向面积。先用 CALC → zero 确定零点，再对每段积分求绝对值后相加。
5. One-Variable Statistics | 单变量统计分析
Enter a list of data into STAT → Edit. Then go to STAT → CALC → 1-Var Stats, specify the list (e.g., L₁), and execute. The display shows mean (x̄), sum (Σx), sum of squares (Σx²), sample standard deviation (Sx), population standard deviation (σx), and the five-number summary. Always clarify which standard deviation is required by the question—IB often asks for the sample standard deviation unless the data represents a complete population.
将数据录入 STAT → Edit，然后进入 STAT → CALC → 1-Var Stats，指定列表（如 L₁）并执行。屏幕将显示均值 (x̄)、总和 (Σx)、平方和 (Σx²)、样本标准差 (Sx)、总体标准差 (σx) 以及五数概括。务必根据题意明确使用哪种标准差——IB题目除非数据代表完整总体，通常要求样本标准差。
For grouped data, you can use the frequency list option: enter midpoints (or class centres) in L₁ and frequencies in L₂, then select 1-Var Stats with List: L₁, FreqList: L₂. The output must be interpreted correctly; remember to show the midpoints calculation on paper if not given.
对于分组数据，可使用频数列表选项：将组中值输入 L₁，频数输入 L₂，然后在 1-Var Stats 中选择 List: L₁，FreqList: L₂。务必正确解读输出结果；若组中值未直接给出，要在卷面上展示其计算过程。
6. Normal and Binomial Distributions | 正态分布与二项分布
For normal probabilities, use DISTR → normalcdf(lower, upper, μ, σ). For inverse normal, use invNorm(area, μ, σ). The area represents the left-tail probability. When the question gives a right-tail probability, subtract it from 1 first. Sketch a bell curve and label the regions to avoid confusion. The GDC does not draw standardised values, so input the actual mean and standard deviation directly.
计算正态概率用 DISTR → normalcdf(下限, 上限, μ, σ)，逆正态用 invNorm(面积, μ, σ)。面积为左尾概率。若题目给出右尾概率，先用 1 减去该值。绘制钟形曲线并标注区域，可避免混淆。计算器不会自动标准化，因此直接输入实际均值和标准差即可。
For binomial distributions, binompdf(n, p, x) gives P(X = x), while binomcdf(n, p, x) inomcdf(n, p, x−1). The calculator’s cumulative function includes the endpoint; adjust boundaries carefully in inequalities.
二项分布中，binompdf(n, p, x) 给出 P(X = x)，binomcdf(n, p, x) 给出 P(X ≤ x)。求 P(X ≥ x) 时使用 1 − binomcdf(n, p, x−1)。累积函数包含端点，处理不等式时要仔细调整边界。
7. Matrix Operations and Solving Linear Systems | 矩阵运算与解线性方程组
To solve a system of three equations, press MATRIX → EDIT olumn matrix [B] for the constants. Then on the home screen, compute [A]⁻¹ × [B] using the x⁻¹ and multiplication keys. Alternatively, use the rref( command on an augmented matrix: enter the augmented matrix [A|B], then select MATRIX → MATH → rref( and read the solution from the last column. This is particularly useful for systems that have infinitely many solutions or no solution.
解三元一次方程组，按 MATRIX → EDIT 定义系数矩阵 [A] 和常数列矩阵 [B]。回到主屏幕，用 x⁻¹ 和乘法键计算 [A]⁻¹ × [B]。也可在增广矩阵上用 rref( 命令：输入增广矩阵 [A|B]，选择 MATRIX → MATH → rref(，从最后一列读取解。该方法对有无数解或无解的系统尤其有用。
When using rref, a row of the form [0 0 0 | 1] indicates inconsistency (no solution), while a row of zeros with a zero constant indicates dependent equations (infinitely many solutions). Understand how to write the parametric solutions algebraically based on the reduced row echelon form display.
使用 rref 时，若出现 [0 0 0 | 1] 行，表示方程组矛盾（无解）；若出现全零行且常数为零，表示方程相关（无穷多解）。要理解如何根据行简化阶梯形写出含参数的代数解。
8. Finance Solver for Loans and Investments | 金融求解器：贷款与投资
ound interest, annuities, and amortisation. Key variables: N (total number of payments), I% (future value), P/Y (payments per year), C/Y (compounding periods per year). The sign convention is crucial: money received is positive, money paid out is negative.
内置金融求解器（APPS → Finance → TVM Solver）可处理复利、年金和摊还问题。关键变量：N（总付款次数）、I%（年利率）、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）。正负号规则至关重要：收到款项为正，付出款项为负。
For a loan of $5000 repaid monthly over 3 years at 6% p.a., set PV = −5000 (you receive the loan), FV = 0, P/Y = 12, C/Y = 12, I% = 6, N = 36, and place the cursor on PMT and press ALPHA → SOLVE. The payment appears positive. Always check that the compounding frequency matches the payment frequency unless stated otherwise.
一笔 5000 美元贷款，年利率 6%，分3年按月偿还，设 PV = −5000（收到贷款），FV = 0，P/Y = 12，C/Y = 12，I% = 6，N = 36，光标移至 PMT 后按 ALPHA → SOLVE，所得付款额为正。除非另有说明，务必确保复利周期与还款频率一致。
9. Sequences and Series Summation | 数列与级数求和
To evaluate Σ (n=1 to 20) (3n + 1), go to LIST → MATH → sum( and LIST → OPS → seq(. Enter sum(seq(3n+1, n, 1, 20)). The GDC generates the list of terms internally and sums them. For arithmetic and geometric series, this provides a quick check against the formula-derived answer. Always write the formula setup in your working before quoting the calculator result.
要计算 Σ (n=1 to 20) (3n + 1)，进入 LIST → MATH → sum( 和 LIST → OPS → seq(，输入 sum(seq(3n+1, n, 1, 20))。计算器内部生成各项并求和。对于等差和等比级数，这可快速检验用公式求得的答案。在给出计算器结果前，一定要先在卷面上写出公式的设定过程。
When the sequence involves factorials or recursive definitions, the calculator’s sequence mode can display individual terms and partial sums. However, recursive sequences may slow the processor if n is large; turn the step carefully and be mindful of overflow errors for extremely large values.
当数列包含阶乘或递归定义时，计算器的数列模式可显示各项及部分和。但递归数列在 n 较大时可能导致处理器变慢；请小心控制步长，并注意极大数值可能引发溢出错误。
10. Graphing and Analysing Rational Functions | 有理函数的绘图与解析
Enter a rational function such as (x² − 4)/(x − 1) into Y=. Use ZOOM → ZDecimal ity, but remember this is not part of the graph; the true vertical asymptote is x = 1. Use TRACE to examine behaviour near asymptotes and confirm limits.
将有理由函数如 (x² − 4)/(x − 1) 输入 Y=。用 ZOOM → ZDecimal 或手动设置窗口以观察渐近线和截距。计算器在间断点附近会画出一条近似垂直线，但这并非图像的一部分；真正的垂直渐近线为 x=1。使用 TRACE 检验渐近线附近的行为并确认极限。
To locate holes (removable discontinuities), examine the table of values (2ND → GRAPH) and look for undefined outputs. If the numerator and denominator share a common factor, the function has a hole at that x-value, but the calculator skips it. You must recognise this algebraically; the GDC aids by showing missing y-values.
要定位可去间断点（空洞），查看数值表（2ND → GRAPH），寻找输出为 undefined 的位置。如果分子和分母有公因式，函数在该 x 处有空洞，但计算器会跳过它。你必须通过代数识别这一点；计算器通过显示缺失的 y 值来辅助判断。
11. Using the Numeric Solver for Complex Equations | 数值求解器解复杂方程
For equations that cannot be easily rearranged to f(x) = 0, the MATH → Solver r iterate to a root within the displayed bounds. This is ideal for transcendental equations like 2ˣ = x² where algebraic manipulation fails.
对于难以简单移项为 f(x)=0 的方程，MATH → Solver（或数值求解器）允许输入 0 = (表达式) 形式的方程。键入方程，为变量指定一个猜测值，求解器将在显示边界内迭代出根。该方法特别适合处理 2ˣ = x² 这类无法用代数求解的超越方程。
Be cautious: the solver returns only one solution near the guess. To find all solutions, you must restart the solver with different guesses or use a graphical approach to identify the number of intersections. Always verify the solution by substituting back into the original equation, as rounding errors can occasionally accumulate.
注意：求解器只返回猜测值附近的一个解。要找出所有解，需用不同猜测值重新启动求解，或借助图形方法确定交点个数。务必通过代回原方程验证所得解，因为舍入误差有时可能累积。
12. Data Plotting and Regression | 数据绘图与回归
Enter bivariate data into STAT → Edit with x-values in L₁ and y-values in L₂. Turn on a scatter plot via 2ND → STAT PLOT, select a suitable marker, and set Xlist: L₁, Ylist: L₂. Press ZOOM → ZoomStat to view the data cloud. Then choose STAT → CALC and pick a regression model, e.g., LinReg(ax+b), QuadReg, ExpReg, or SinReg. The calculator will output the regression equation and the correlation coefficient r or R².
将双变量数据输入 STAT → Edit，x 值存放于 L₁，y 值存放于 L₂。通过 2ND → STAT PLOT 打开散点图，选择合适的标记，设置 Xlist: L₁，Ylist: L₂。按 ZOOM → ZoomStat 显示数据点云。然后进入 STAT → CALC 选择回归模型，如 LinReg(ax+b)、QuadReg、ExpReg 或 SinReg。计算器将输出回归方程及相关系数 r 或 R²。
When interpreting regression output, comment on the strength and direction of the correlation. If the data appears non-linear, test several models and use the R² value to justify your choice. Remember that extrapolation beyond the data range is unreliable. In IB investigations, you must plot the residual list to check the goodness of fit visually; a random residual pattern supports the chosen model.
解读回归输出时，要评论相关性的强度与方向。若数据呈非线性，可测试多个模型并用 R² 值论证选择。谨记在数据范围之外的外推不可靠。在IB探究中，必须绘制残差图以目视检验拟合优度；随机的残差模式说明所选模型合理。
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