Review set 17B – CALCULATOR | 复习题集 17B – 计算器

📚 Review set 17B – CALCULATOR | 复习题集 17B – 计算器

This calculator-active review set covers a range of IB Mathematics topics where a graphical display calculator (GDC) is essential. You will practise solving equations graphically, performing numerical integration, computing descriptive statistics, working with the normal and binomial distributions, carrying out linear regression, evaluating derivatives numerically, and using the finance solver. Each section presents a worked example and the key calculator steps required to obtain the solution efficiently.

本套计算器复习题涵盖了 IB 数学中需要使用图形计算器 (GDC) 的一系列主题。你将练习通过图像解方程、执行数值积分、计算描述性统计量、处理正态分布和二项分布、进行线性回归、数值求导以及使用金融求解器。每一节都给出一个典型例题以及高效求解所需的关键计算器操作步骤。

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1. Solving Equations Graphically | 通过图像解方程

When an equation cannot be solved algebraically, we find the points of intersection of two graphs. For example, to solve eˣ = 3 − x, we graph y = eˣ and y = 3 − x and locate their intersection.

当一个方程无法用代数方法求解时，我们寻找两个图形交点的横坐标。例如，为解方程 eˣ = 3 − x，我们绘制 y = eˣ 与 y = 3 − x 并求其交点。

Calculator Step (for TI-84 family)	计算器步骤 (TI-84 系列)
Press [Y=] and enter Y₁ = e^(X), Y₂ = 3 − X.	按 [Y=]，输入 Y₁ = e^(X), Y₂ = 3 − X。
Set a suitable window, e.g. X: −2 to 2, Y: −1 to 4.	设置合适的窗口，例如 X: −2 至 2, Y: −1 至 4。
Press [2ND] [CALC] and choose 5: intersect.	按 [2ND] [CALC]，选择 5: intersect。
Follow on-screen prompts to select the two curves and provide a guess near the intersection.	按照屏幕提示选择两条曲线，并在交点附近提供猜测值。
The calculator returns x ≈ 0.79206, y ≈ 2.20794.	计算器返回 x ≈ 0.79206, y ≈ 2.20794。

Thus the solution to the equation is x ≈ 0.792 (3 s.f.). Always check that the window shows all intersections if multiple solutions may exist.

因此方程的解为 x ≈ 0.792（保留三位有效数字）。如果可能存在多个解，务必确保窗口显示了所有的交点。

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2. Numerical Integration | 数值积分

When an antiderivative is difficult to find, the definite integral can be approximated numerically. For instance, evaluate ∫₀² √(1 + sin x) dx using the calculator’s numerical integration function.

当被积函数的原函数难以求得时，可以通过数值方法近似计算定积分。例如，使用计算器的数值积分功能计算 ∫₀² √(1 + sin x) dx。

Calculator Step	计算器步骤
Press [MATH], then scroll down to 9: fnInt(	按 [MATH]，向下滚动选择 9: fnInt(
Enter the expression √(1 + sin(X)), the variable X, lower limit 0, upper limit 2.	输入表达式 √(1 + sin(X))，变量 X，下限 0，上限 2。
The full syntax: fnInt(√(1+sin(X)),X,0,2).	完整语法：fnInt(√(1+sin(X)),X,0,2)。
Press [ENTER] to get approximately 2.6525.	按 [ENTER] 得到近似值 2.6525。

The definite integral evaluates to 2.65 (3 s.f.). The GDC applies an adaptive numerical algorithm; always include the differential variable and limits to avoid syntax errors.

该定积分值为 2.65（三位有效数字）。GDC 采用自适应数值算法；务必包含微分变量与积分限以避免语法错误。

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3. Descriptive Statistics | 描述性统计

Given a data set, you can quickly obtain the mean, standard deviation, quartiles and median. Consider the values: 12, 15, 18, 22, 22, 27, 31, 35. Enter them into a list and compute the one-variable statistics.

给定一组数据，你可以快速获得均值、标准差、四分位数和中位数。考虑以下数值：12, 15, 18, 22, 22, 27, 31, 35。将它们输入列表并计算单变量统计量。

Calculator Step	计算器步骤
Press [STAT] and choose 1: Edit. Enter data into L1.	按 [STAT]，选择 1: Edit，将数据输入 L1。
Press [STAT] again, move to CALC and select 1: 1-Var Stats.	再次按 [STAT]，移至 CALC 菜单并选择 1: 1-Var Stats。
Specify the list L1 (and frequency list if needed). Press [ENTER].	指定列表 L1（若需要频数列也可指定）。按 [ENTER]。
Read output: x̄ = 22.75, sample standard deviation Sx ≈ 7.85, Q₁ = 16.5, Med = 22, Q₃ = 29.	读取输出：x̄ = 22.75，样本标准差 Sx ≈ 7.85，Q₁ = 16.5，中位数 Med = 22，Q₃ = 29。

Note that the calculator gives both σx (population standard deviation) and Sx (sample standard deviation). For IB Data Analysis questions, you usually need Sx unless the data represents the entire population.

请注意，计算器同时给出 σx（总体标准差）和 Sx（样本标准差）。在 IB 数据分析题目中，除非数据代表整个总体，否则通常使用 Sx。

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4. Normal Distribution Probabilities | 正态分布概率

Use the normal cumulative distribution function to find probabilities and the inverse function to find quantiles. Example: X ~ N(50, 5²). Find P(X < 45) and the value of k such that P(X < k) = 0.9.

利用正态累积分布函数求概率，并使用其反函数求分位数。例题：X ~ N(50, 5²)。求 P(X < 45) 以及满足 P(X < k) = 0.9 的 k 值。

Calculator Step	计算器步骤
Press [2ND] [VARS] to access DISTR.	按 [2ND] [VARS] 进入 DISTR 菜单。
For P(X < 45): select 2: normalcdf(. Enter lower: -1E99, upper: 45, μ: 50, σ: 5.	求 P(X < 45)：选择 2: normalcdf(。输入下限 -1E99，上限 45，μ: 50，σ: 5。
Result: about 0.158655. Thus P(X < 45) ≈ 0.159.	结果约为 0.158655。因此 P(X < 45) ≈ 0.159。
For the inverse: select 3: invNorm(. Enter area: 0.9, μ: 50, σ: 5.	求逆：选择 3: invNorm(。输入 area: 0.9, μ: 50, σ: 5。
Result: k ≈ 56.4077. So k ≈ 56.4 (3 s.f.).	结果：k ≈ 56.4077。因此 k ≈ 56.4（三位有效数字）。

Always remember to use a very large negative number (e.g. -1E99) for the lower bound when finding left-tail probabilities. The inverse normal gives the x-value corresponding to a given cumulative area.

始终记住，在求左尾概率时，下限用一个绝对值很大的负数（如 -1E99）。逆正态函数给出与给定累积面积相对应的 x 值。

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5. Linear Regression | 线性回归

Given bivariate data, you can find the equation of the regression line and the Pearson correlation coefficient r. Data: (1, 3.1), (2, 5.2), (3, 7.0), (4, 8.9), (5, 11.2). Determine the line of best fit y = ax + b and r.

给定双变量数据，可以求出回归直线方程以及皮尔逊相关系数 r。数据：(1, 3.1), (2, 5.2), (3, 7.0), (4, 8.9), (5, 11.2)。确定最佳拟合直线 y = ax + b 以及 r。

Calculator Step	计算器步骤
Press [STAT] > Edit, enter X values in L1 and Y values in L2.	按 [STAT] > Edit，将 X 值输入 L1，Y 值输入 L2。
Press [STAT] > CALC, choose 4: LinReg(ax+b). Ensure Xlist: L1, Ylist: L2.	按 [STAT] > CALC，选择 4: LinReg(ax+b)。确保 Xlist: L1, Ylist: L2。
Press [ENTER]. Output: a ≈ 2.02, b ≈ 1.06, r² ≈ 0.999, r ≈ 0.9995.	按 [ENTER]。输出：a ≈ 2.02, b ≈ 1.06, r² ≈ 0.999, r ≈ 0.9995。

Thus the equation is y = 2.02x + 1.06 and the correlation is r = 0.9995, indicating an extremely strong positive linear relationship. If r is not shown, check that Diagnostics are turned on via [2ND] [0] (CATALOG) > DiagnosticOn.

因此回归方程为 y = 2.02x + 1.06，相关系数 r = 0.9995，表明极强的正线性关系。如果 r 未显示，请通过 [2ND] [0] (CATALOG) > DiagnosticOn 打开诊断功能。

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6. Solving Inequalities | 解不等式

A GDC can solve inequalities by shading the region where the inequality holds true. To find where x² − 4x + 3 ≥ 0, graph the function and identify the x-values for which the graph is on or above the x-axis.

GDC 可以通过阴影区域来解不等式。为求解 x² − 4x + 3 ≥ 0，绘制函数图像并确定图形在 x 轴上或上方的 x 值。

Calculator Step	计算器步骤
Enter Y₁ = X² − 4X + 3. Use a window showing the zeros: X: −1 to 5, Y: −2 to 6.	输入 Y₁ = X² − 4X + 3。使用能显示零点的窗口：X: −1 至 5, Y: −2 至 6。
Observe zeros at X = 1 and X = 3. The parabola is above the x-axis outside these roots.	观察到零点在 X = 1 和 X = 3。抛物线在这些根的外部位于 x 轴上方。
Shade the feasible region: [2ND] [DRAW] > 7: Shade, then enter Y₂ = 0 as lower function and Y₁ as upper, but we can also read directly.	如需着色： [2ND] [DRAW] > 7: Shade，然后输入 Y₂ = 0 作为下函数，Y₁ 为上函数，但我们也可以直接读取。
Conclusion: x ≤ 1 or x ≥ 3.	结论：x ≤ 1 或 x ≥ 3。

Using the ‘Poly Root Finder’ app (or the zero function) confirms the boundaries. Always write interval notation or inequalities clearly.

使用“多项式求根”应用（或零点功能）可确认边界。务必清楚地写出区间符号或不等式。

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7. Numerical Derivative at a Point | 某点的数值导数

When the derivative function is complicated, the calculator can compute the instantaneous rate of change numerically. Example: find f'(2) for f(x) = (ln x) / (x² + 1).

当导函数很复杂时，计算器可以通过数值方法计算瞬时变化率。例题：求 f(x) = (ln x) / (x² + 1) 在 x=2 处的 f'(2)。

Calculator Step	计算器步骤
Press [MATH] and choose 8: nDeriv(	按 [MATH]，选择 8: nDeriv(
Syntax: nDeriv(expression, variable, value). Enter nDeriv(ln(X)/(X²+1), X, 2).	语法：nDeriv(表达式，变量，点)。输入 nDeriv(ln(X)/(X²+1), X, 2)。
Press [ENTER]. The result is approximately 0.0217.	按 [ENTER]。结果约为 0.0217。

The symmetric difference quotient is used; the default small increment is sufficient for most functions. You can also graph the derivative and use the [CALC] value feature at x=2.

计算器采用对称差商；默认的小增量对大多数函数足够。你也可以绘制导数图并在 x=2 处使用 [CALC] 取值功能。

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8. Polynomial Root Finder | 多项式求根

To solve cubic or higher-degree polynomial equations, use the Poly Root Finder app. Solve x³ − 4x² + x + 6 = 0.

为解三次或更高次多项式方程，可使用多项式求根应用。解方程 x³ − 4x² + x + 6 = 0。

Calculator Step	计算器步骤
Access [APPS] and select Poly Root Finder (or PlySmlt2 on TI-84 Plus CE).	打开 [APPS]，选择 Poly Root Finder（或 TI-84 Plus CE 上的 PlySmlt2）。
Set the degree to 3 and enter coefficients a₃=1, a₂=−4, a₁=1, a₀=6.	设置次数为 3，输入系数 a₃=1, a₂=−4, a₁=1, a₀=6。
Select Solve. Roots are displayed: x = −1, x = 2, x = 3.	选择 Solve。显示根：x = −1, x = 2, x = 3。

Always check if complex roots exist when the degree is even, as the app may also return complex numbers in a+bi form if that mode is active.

对于偶次多项式，务必留意是否存在复数根；如果激活了 a+bi 模式，应用可能以复数形式返回结果。

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9. Matrix Operations | 矩阵运算

For solving systems or transforming shapes, you need matrix arithmetic and determinant/inverse operations. Find the inverse of matrix A = [[3,2],[7,5]] and verify A × A⁻¹ = I.

在解线性方程组或变换图形时，你会用到矩阵运算及行列式/逆矩阵操作。求矩阵 A = [[3,2],[7,5]] 的逆并验证 A × A⁻¹ = I。

Calculator Step	计算器步骤
Press [2ND] [x⁻¹] (MATRIX), EDIT, choose matrix [A] and define 2×2, enter 3,2,7,5.	按 [2ND] [x⁻¹] (MATRIX)，EDIT 中选择矩阵 [A]，定义为 2×2，输入 3,2,7,5。
On the home screen, press [2ND] [x⁻¹] (MATRIX), choose [A], then press the inverse button [x⁻¹].	回到主屏幕，按 [2ND] [x⁻¹] (MATRIX)，选择 [A]，再按反函数键 [x⁻¹]。
Result: A⁻¹ = [[5, −2],[−7, 3]].	结果：A⁻¹ = [[5, −2],[−7, 3]]。
To verify, compute [A]*[A]⁻¹. The display shows identity matrix [[1,0],[0,1]].	为验证，计算 [A]*[A]⁻¹。屏幕显示单位矩阵 [[1,0],[0,1]]。

Be aware that fractions are shown as decimals; to obtain exact rational form, convert using the MATH > Frac command, if supported.

注意分数会以小数显示；如需精确有理数形式，可使用 MATH > Frac 命令转换（若支持）。

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10. Finance Solver (TVM) | 金融求解器（TVM）

The TVM (Time Value of Money) Solver is used for loans, annuities and investments. Example: You invest €5000 at an annual interest rate of 4.2% compounded monthly for 6 years. Find the future value if no additional payments are made.

TVM（资金时间价值）求解器用于贷款、年金和投资计算。例题：你投资 5000 欧元，年利率 4.2%，每月复利，为期 6 年。在不追加任何付款的情况下，求终值。

Calculator Step	计算器步骤
Press [APPS], choose 1: Finance, then 1: TVM Solver.	按 [APPS]，选择 1: Finance，再选 1: TVM Solver。
Set N = 6 × 12 = 72 (total months), I% = 4.2, PV = −5000 (negative because money leaves you), PMT = 0, FV = ? (cursor up then press [ALPHA] [SOLVE]), P/Y = 12, C/Y = 12, PMT: END.	设置 N = 6 × 12 = 72（总月数），I% = 4.2，PV = −5000（负号表示现金流出），PMT = 0，FV = ?（光标移至该处并按 [ALPHA] [SOLVE]），P/Y = 12，C/Y = 12，PMT: END。
The solved FV is approximately 6418.97. The investment grows to €6418.97.	计算出的 FV 约为 6418.97。投资增长至 6418.97 欧元。

Sign conventions are crucial: PV and FV usually carry opposite signs when money flows in opposite directions. Always check P/Y and C/Y match compounding frequency.

符号约定至关重要：当现金流向相反时，PV 与 FV 通常符号相反。务必确保 P/Y 和 C/Y 与计息频率一致。

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11. Binomial Distribution | 二项分布

For a fixed number of independent trials, the binomial distribution gives the probability of a given number of successes. Suppose X ~ B(20, 0.4). Find P(X = 8) and P(X ≤ 8).

对于固定次数的独立试验，二项分布给出特定成功次数的概率。设 X ~ B(20, 0.4)。求 P(X = 8) 和 P(X ≤ 8)。

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Calculator Step	计算器步骤