Review Set 21B – Mastering Calculator Skills | 复习集21B – 精通计算器技巧

📚 Review Set 21B – Mastering Calculator Skills | 复习集21B – 精通计算器技巧

In IB Mathematics, effective use of a graphing calculator is not just a convenience – it is a strategic necessity. Review Set 21B is designed to sharpen your calculator skills across essential topics: solving equations numerically, analysing functions, performing calculus operations, handling statistical data, and working with matrices. Mastery of these skills will save time in exams and reduce algebraic errors.

在IB数学中，熟练使用图形计算器不仅是便利，更是策略性必需品。复习集21B旨在强化你在各核心主题上的计算器技能，包括数值求解方程、分析函数、执行微积分运算、处理统计数据以及矩阵操作。掌握这些技能将在考试中节省时间并减少代数错误。

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1. Solving Equations Numerically | 数值求解方程

Many equations in IB problems cannot be solved algebraically. Use your calculator’s built-in equation solver (often labelled ‘nSolve’ or ‘Equation Solver’) to find real roots.

许多IB题目中的方程无法用代数求解。可以使用计算器内置的方程求解器（通常标记为 ‘nSolve’ 或 ‘Equation Solver’）来求实根。

Example: Solve 2x³ – 5x² + x – 7 = 0.

示例：求解 2x³ – 5x² + x – 7 = 0。

Enter the equation, set an initial guess near x = 2. The calculator returns x ≈ 2.913.

输入方程，设置初始猜测值在 x = 2 附近。计算器返回 x ≈ 2.913。

For transcendental equations like eˣ = 5 – x, you can rearrange to eˣ + x – 5 = 0 and solve numerically. The solution is x ≈ 1.306.

对于超越方程如 eˣ = 5 – x，可重排为 eˣ + x – 5 = 0 然后数值求解。解为 x ≈ 1.306。

Always check that your answer satisfies the original equation, and be aware that some equations may have multiple solutions – you can find others by using different initial guesses.

始终检验答案是否满足原方程，并留意某些方程可能有多个解——可以通过使用不同的初始猜测值来找到其他解。

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2. Graphing and Finding Intersections | 绘图与求交点

When comparing two functions, graphing is the quickest way to locate intersection points. This is essentially solving f(x) = g(x) visually and numerically.

当比较两个函数时，绘图是定位交点的最快方法。这本质上就是通过图形和数值方式求解 f(x) = g(x)。

Example: Find the intersection of y = ln(x) and y = 2 – x.

示例：求 y = ln(x) 与 y = 2 – x 的交点。

Enter Y₁ = ln(X) and Y₂ = 2 – X. Adjust the window to X: 0 to 3, Y: 0 to 2. Use the ‘Intersect’ command from the CALC menu. The x-coordinate of the intersection is approximately 1.557.

输入 Y₁ = ln(X) 和 Y₂ = 2 – X。调整窗口为 X: 0 至 3，Y: 0 至 2。使用 CALC 菜单中的 ‘Intersect’ 命令。交点的 x 坐标约为 1.557。

This technique is invaluable when dealing with composite functions or when you need to confirm the number of solutions an equation has.

当处理复合函数或需要确认方程解的个数时，此技巧非常有用。

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3. Evaluating Derivatives at a Point | 在某点计算导数值

Your calculator can compute the numerical derivative of a function at a given point without the need for analytic differentiation. This is extremely helpful for checking your manual work.

你的计算器可以在给定点计算函数的数值导数，无需解析求导。这对于检查手动计算非常有帮助。

Example: If f(x) = x² sin(x), find f'(π/3).

示例：若 f(x) = x² sin(x)，求 f'(π/3)。

Use the numerical derivative function (often accessed via ‘nDeriv’ or ‘d/dx’). Input the function, variable x, and value π/3 ≈ 1.0472. The result is approximately 2.362.

使用数值导数功能（通常通过 ‘nDeriv’ 或 ‘d/dx’ 访问）。输入函数、变量 x 和值 π/3 ≈ 1.0472。结果约为 2.362。

Remember to set your calculator to radian mode when dealing with trigonometric functions.

处理三角函数时记得将计算器设为弧度模式。

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4. Definite Integration | 定积分计算

Evaluating definite integrals accurately is a staple in calculus questions. The calculator can handle integrals of complex functions, including those without elementary antiderivatives.

精确计算定积分是微积分题目的基本要求。计算器能够处理复杂函数的积分，包括那些没有初等原函数的积分。

Example: Compute ∫₁³ (1/x) dx.

示例：计算 ∫₁³ (1/x) dx。

Input the integral using the ‘∫dx’ function, with lower limit 1 and upper limit 3. The calculator gives 1.098612…, which is ln(3).

使用 ‘∫dx’ 功能输入积分，下限为1，上限为3。计算器给出 1.098612…，即 ln(3)。

For a more complex case, ∫₀¹ e^(–x²) dx yields approximately 0.7468. Such integrals frequently appear in probability contexts.

对于更复杂的情况，∫₀¹ e^(–x²) dx 约等于 0.7468。这类积分常出现在概率问题中。

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5. Areas Between Curves | 曲线间面积

Finding the area enclosed by two curves involves integrating the difference of the functions over the intersection interval. The calculator can perform this directly once intersection points are known.

求两条曲线围成的面积需要对函数差值在交点区间内积分。一旦知道交点，计算器可直接完成。

Example: Find the area between y = x² and y = √x.

示例：求 y = x² 与 y = √x 之间的面积。

The curves intersect at x = 0 and x = 1. The area is ∫₀¹ (√x – x²) dx. Using the integration function, the calculator returns 0.333333, which equals 1/3.

曲线相交于 x = 0 和 x = 1。面积 = ∫₀¹ (√x – x²) dx。使用积分功能，计算器返回 0.333333，即 1/3。

Always sketch the graphs first to determine which function is the upper boundary; an incorrect order will yield a negative sign.

务必先绘制草图以确定哪个函数为上边界；顺序错误会导致出现负号。

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6. One-Variable Statistics | 单变量统计

For data analysis tasks, quickly entering a list of numbers into the statistics editor provides key summary metrics: mean, median, standard deviation, and quartiles.

对于数据分析任务，快速将一组数字输入统计编辑器即可获得关键汇总指标：均值、中位数、标准差和四分位数。

Example: Data set {7, 9, 8, 12, 10}. Find the mean and sample standard deviation.

示例：数据集 {7, 9, 8, 12, 10}。求均值和样本标准差。

Enter the values into List 1. Run 1‑Var Stats. The calculator displays x̄ = 9.2, and the sample standard deviation Sx ≈ 1.9235.

将数值输入列表1。运行单变量统计。计算器显示 x̄ = 9.2，样本标准差 Sx ≈ 1.9235。

Make sure you select the correct standard deviation: σx for population, Sx for sample – IB questions usually require the sample standard deviation.

确保选择正确的标准差：σx 表示总体，Sx 表示样本——IB题目通常要求样本标准差。

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7. Linear Regression and Correlation | 线性回归与相关系数

When investigating relationships between two variables, the calculator can fit a least-squares regression line and compute Pearson’s correlation coefficient r.

在研究两个变量之间的关系时，计算器可以拟合最小二乘回归直线并计算皮尔逊相关系数 r。

Example: Bivariate data (1,2), (2,4), (3,5), (4,7). Find the line of best fit y = a + bx and the correlation coefficient.

示例：双变量数据 (1,2), (2,4), (3,5), (4,7)。求最佳拟合直线 y = a + bx 及相关系数。

Enter x-values in List 1 and y-values in List 2. Run LinReg (a + bx). The output shows a = 0.5, b = 1.6, so the equation is y = 0.5 + 1.6x. The correlation coefficient r ≈ 0.9922, indicating a very strong positive linear association.

将 x 值输入列表1，y 值输入列表2。运行线性回归 (a + bx)。输出显示 a = 0.5, b = 1.6，因此方程为 y = 0.5 + 1.6x。相关系数 r ≈ 0.9922，表明有很强的正线性关联。

Remember to turn on diagnostic r² or r if your calculator does not display it by default.

如果计算器默认不显示 r² 或 r，记得开启诊断功能。

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8. Binomial Probability Distribution | 二项式概率分布

The binomial probability functions – binompdf for exact probabilities and binomcdf for cumulative probabilities – are essential for handling discrete probability distributions efficiently.

二项式概率函数——binompdf 用于精确概率，binomcdf 用于累积概率——对于高效处理离散概率分布至关重要。

Example: A fair coin is tossed 10 times. Find the probability of exactly 4 heads (n=10, p=0.5, X=4).

示例：一枚公平硬币抛掷10次。求恰好出现4次正面的概率（n=10, p=0.5, X=4）。

Use binompdf(10, 0.5, 4). The calculator returns approximately 0.2051.

使用 binompdf(10, 0.5, 4)。计算器返回约 0.2051。

For cumulative probabilities, such as P(X ≤ 4), use binomcdf(10, 0.5, 4), which gives 0.3770. To find P(X ≥ 7), use 1 – binomcdf(10, 0.5, 6).

对于累积概率，如 P(X ≤ 4)，使用 binomcdf(10, 0.5, 4)，结果为 0.3770。要求 P(X ≥ 7)，则用 1 – binomcdf(10, 0.5, 6)。

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9. Normal Distribution Calculations | 正态分布计算

Normal probability calculations, including inverse normal (finding the value corresponding to a given area), are frequently tested. The calculator functions normalcdf and invNorm streamline these tasks.

正态概率计算，包括逆正态（求给定面积对应的值），是常考内容。计算器函数 normalcdf 和 invNorm 简化了这些任务。

Example: A variable X is normally distributed with mean μ = 100 and standard deviation σ = 15. Find P(X > 115).

示例：变量 X 服从正态分布，均值 μ = 100，标准差 σ = 15。求 P(X > 115)。

Use normalcdf(115, 1E99, 100, 15). The area is approximately 0.1587. (1E99 is used to represent positive infinity.)

使用 normalcdf(115, 1E99, 100, 15)。面积约为 0.1587。（1E99 用于表示正无穷大。）

To find the 90th percentile, use invNorm(0.9, 100, 15), which yields about 119.2.

要求第90百分位数，使用 invNorm(0.9, 100, 15)，结果约为 119.2。

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10. Matrix Operations | 矩阵运算

Matrices appear in linear algebra and transformation geometry. Your calculator can store matrices and compute determinants, inverses, and transposes effortlessly.

矩阵出现在线性代数和变换几何中。你的计算器可以存储矩阵并轻松计算行列式、逆矩阵和转置矩阵。

Example: Let A = [[2,3],[1,4]]. Find det(A) and A⁻¹.

示例：设 A = [[2,3],[1,4]]。求 det(A) 和 A⁻¹。

Input the 2×2 matrix into matrix [A]. The determinant function returns 5. The inverse of A is [[0.8, –0.6],[–0.2, 0.4]] (displayed as fractions on some models).

将 2×2 矩阵输入矩阵 [A]。行列式函数返回 5。A 的逆矩阵为 [[0.8, –0.6],[–0.2, 0.4]]（某些型号可显示为分数）。

Make sure the determinant is non-zero; otherwise, the matrix is singular and has no inverse.

确保行列式不为零；否则矩阵是奇异矩阵，没有逆矩阵。

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11. Solving Linear Systems with Matrices | 用矩阵求解线性方程组

A powerful application of matrices is solving systems of linear equations via reduced row-echelon form (rref). This method handles large systems with speed and accuracy.

矩阵的一个重要应用是通过简化行阶梯形 (rref) 求解线性

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