📚 Review set 22B – CALCULATOR | 复习套题22B – 计算器部分
Review set 22B is designed to help IB Mathematics students consolidate their understanding of core topics while utilising a graphics display calculator (GDC). This article guides you through typical calculator-based problems, from solving equations to evaluating integrals, and highlights effective strategies to maximise your marks. Whether you are studying Analysis and Approaches or Applications and Interpretation, mastering GDC skills is essential for exam success.
复习套题22B旨在帮助IB数学学生巩固核心知识点，并熟练使用图形显示计算器（GDC）。本文将带你梳理典型的计算器类题目，包括解方程、求积分等，并强调高效策略以争取高分。无论你是学习分析与方法还是应用与解释，掌握GDC技能对考试成功至关重要。
1. Overview of Calculator Skills in Review Set 22B | 复习套题22B中计算器技巧概览
Review set 22B focuses on applying calculator functions to explore functions, solve equations, compute derivatives and integrals, perform statistical analysis and carry out matrix operations. The problems are structured to test both your conceptual understanding and your ability to interpret numerical outputs accurately. You will need to become fluent with the graphing, CALC, STAT and MATRIX menus on your GDC.
复习套题22B着重使用计算器功能来探索函数、解方程、计算导数和积分、进行统计分析以及执行矩阵运算。这些题目旨在考察你的概念理解能力，以及准确解读数值输出的能力。你需要熟练使用GDC上的图形、计算、统计和矩阵菜单。
Having a systematic approach saves precious time during the exam. Always begin by sketching the graph to visualise the problem, then select the appropriate tool such as ‘zero’, ‘intersect’, ‘dy/dx’ or ‘∫f(x)dx’. Record all decimal answers to three significant figures unless stated otherwise.
系统性的解题方法可以在考试中节省宝贵时间。始终从绘制草图开始，直观理解问题，然后选择合适的工具，如“零点”、“交点”、“dy/dx”或“∫f(x)dx”。除非题目另有要求，所有小数答案均保留三位有效数字。
2. Graphing Functions and Identifying Key Features | 绘制函数图像并识别关键特征
Many questions require you to graph a function such as f(x) = x³ − 2x² − 5x + 6 and determine its intercepts, turning points and asymptotes. Use the standard zoom or manually adjust the window to ensure all critical behaviour is visible. For trigonometric functions, set the angle mode to radians unless degrees are specified.
许多题目要求绘制如 f(x) = x³ − 2x² − 5x + 6 的函数图像，并确定其截距、转折点和渐近线。使用标准缩放或手动调整窗口，确保所有关键特征都可见。对于三角函数，除非指定角度制，否则设置为弧度模式。
Key features to extract using the GDC include:
使用GDC提取的关键特征包括：
- x-intercepts (roots) via the ‘zero’ function
- y-intercept by evaluating f(0)
- local maxima and minima via the ‘maximum’ and ‘minimum’ functions
- points of intersection between two curves
- horizontal and vertical asymptotes by analysing end behaviour or using the table of values
- 通过“零点”功能求出的x轴截距（根）
- 通过计算f(0)得到的y轴截距
- 通过“最大值”和“最小值”功能求出的局部极值点
- 两条曲线的交点
- 通过分析末端趋势或数值表判断水平和垂直渐近线
For example, to find the turning points of y = x⁴ − 4x³ + 3, graph the function, then select ‘minimum’. The GDC indicates a local minimum at (3, −24) and a local maximum at (0, 3). Always confirm by checking the derivative sign change if asked to justify.
例如，要找到 y = x⁴ − 4x³ + 3 的转折点，先绘制函数图像，然后选择“最小值”。GDC显示局部极小值在 (3, −24)，局部极大值在 (0, 3)。如被要求论证，务必通过导数符号变化来确认。
3. Solving Equations Numerically with the Calculator | 使用计算器数值解方程
Review set 22B includes equations that cannot be solved algebraically, such as e⁻ˣ = 2x + 1 or ln(x) + x² = 3. The GDC offers two primary numerical methods: using the graph and the ‘intersect’ function, or placing the equation in the form f(x)=0 and applying the ‘zero’ feature. Both yield approximations that are typically accurate to ±1×10⁻¹².
复习套题22B包含一些无法用代数求解的方程，例如 e⁻ˣ = 2x + 1 或 ln(x) + x² = 3。GDC提供两种主要的数值方法：使用图像和“交点”功能，或者将方程化为 f(x)=0 的形式，再应用“零点”特征。这两种方法都能得到通常精确至 ±1×10⁻¹² 的近似解。
When solving a trigonometric equation like 2sin(θ) = θ − 1 within the interval [0, 4], always restrict the domain on your calculator. Graph y = 2sin(θ) and y = θ − 1, then find the intersection. The solution θ ≈ 1.93 rad appears. Check for additional solutions by scanning the graph; there may be a second solution near 3.35 rad if the window is extended.
求解在区间 [0, 4] 内的三角方程 2sin(θ) = θ − 1 时，务必在计算器上限制定义域。绘制 y = 2sin(θ) 和 y = θ − 1，然后求交点。得到的解为 θ ≈ 1.93 rad。扫描图像检查是否有额外的解；如果扩大窗口，可能在 3.35 rad 附近出现第二个解。
| Method | GDC Steps (simplified) |
| Intersection | Graph Y1=f(x), Y2=g(x) → CALC → intersect → select curves → guess |
| Zero/Root | Set Y1=f(x)−g(x) → CALC → zero → set left bound, right bound, guess |
| 方法 | GDC步骤（简化） |
| 交点 | 绘制Y1=f(x), Y2=g(x) → CALC → intersect → 选择曲线 → 猜测值 |
| 零点/根 | 设置Y1=f(x)−g(x) → CALC → zero → 设定左边界、右边界、猜测值 |
4. Computing Derivatives at a Point | 在某点计算导数
Although you must know differentiation rules, many problems ask for the numerical derivative at a specific point, often to verify a tangent gradient. Use the ‘nDeriv’ or ‘dy/dx’ function. For instance, to find f'(2) for f(x) = x·e²ˣ, enter nDeriv(x·e^(2x), x, 2). The calculator returns approximately 273.45. This value represents the slope of the tangent at x=2.
尽管你必须掌握求导法则，许多题目要求在特定点求数值导数，通常是用来验证切线斜率。使用“nDeriv”或“dy/dx”功能。例如，求 f(x) = x·e²ˣ 在 x=2 处的 f'(2)，输入 nDeriv(x·e^(2x), x, 2)。计算器返回约 273.45。这个数值代表 x=2 处切线的斜率。
Use the derivative value to write the tangent line equation: y − f(2) = f'(2)(x − 2). First evaluate f(2) ≈ 2·e⁴ ≈ 109.20. Then the tangent is y − 109.20 = 273.45(x − 2). Simplify as needed. The GDC can plot both the function and the tangent to visually confirm accuracy.
利用导数值写出切线方程：y − f(2) = f'(2)(x − 2)。首先计算 f(2) ≈ 2·e⁴ ≈ 109.20。于是切线为 y − 109.20 = 273.45(x − 2)。根据要求化简。GDC可以同时绘制函数和切线，从视觉上确认准确性。
When questions involve implicit differentiation or parametric equations, you can still check your algebraic work by graphing the relation and using the ‘dy/dx’ feature at a point. Ensure the GDC is in function mode correctly; some curves may need to be entered in parametric form.
当问题涉及隐函数求导或参数方程时，你依然可以通过绘制关系图像并在某点使用“dy/dx”功能来检验代数结果。确保GDC处于正确的函数模式；有些曲线可能需要以参数形式输入。
5. Definite Integrals and Area Under Curves | 定积分与曲线下方面积
Evaluating a definite integral such as ∫₀² (x³ + 2x) dx on the calculator is straightforward: use the ‘∫f(x)dx’ command. Enter the function, lower limit 0, upper limit 2; the GDC gives the exact value 8. For non-polynomial functions like ∫₁³ ln(x) dx, the calculator returns a decimal 1.2958…, which is acceptable unless an exact form is required.
在计算器上计算定积分，如 ∫₀² (x³ + 2x) dx，非常简单：使用“∫f(x)dx”命令。输入函数、下限0、上限2；GDC给出精确值8。对于非多项式函数，如 ∫₁³ ln(x) dx，计算器返回小数 1.2958…，除非要求精确形式，否则可以接受。
To find the area between two curves from x=a to x=b, integrate the absolute difference: Area = ∫ₐᵇ |f(x) − g(x)| dx. The GDC can handle this if you enter Y1 = f(x) and Y2 = g(x), then integrate Y1−Y2, but carefully identify regions where curves cross. If there is an intersection at x=c, split the integral: ∫ₐᶜ (g(x)−f(x)) dx + ∫ᶜᵇ (f(x)−g(x)) dx. Numerical integration on the GDC makes this process rapid.
要求两曲线在 x=a 到 x=b 之间的面积，需对绝对差进行积分：面积 = ∫ₐᵇ |f(x) − g(x)| dx。如果输入 Y1 = f(x) 和 Y2 = g(x)，然后对 Y1−Y2 进行积分，GDC可以处理，但要仔细识别曲线相交的区域。如果在 x=c 有交点，则需拆分积分：∫ₐᶜ (g(x)−f(x)) dx + ∫ᶜᵇ (f(x)−g(x)) dx。GDC上的数值积分使该过程十分迅速。
A typical problem: Find the area enclosed by y = x² − 4 and y = 2x − 1. Plot both, observe intersections at x = −1 and x = 3. Compute ∫₋₁³ [(2x−1) − (x²−4)] dx. The GDC yields 32/3 ≈ 10.667. Always shade the region mentally to ensure correct setup.
一个典型题目：求由 y = x² − 4 和 y = 2x − 1 围成的面积。同时绘制两图，观察到交点位于 x = −1 和 x = 3。计算 ∫₋₁³ [(2x−1) − (x²−4)] dx。GDC得出 32/3 ≈ 10.667。务必在脑中对区域上色，以确保列式正确。
6. Optimisation Problems with Calculator Assistance | 使用计算器辅助优化问题
Optimisation tasks often model a real-world scenario where you must maximise volume or minimise surface area. After deriving the objective function in one variable, enter it into the GDC and graph it over a feasible domain. Use the ‘maximum’ or ‘minimum’ CALC function to locate the optimal x-value. For instance, the function V(x) = x(15 − 2x)(10 − 2x) for a box might be graphed in the window 0 < x < 5. The GDC finds a maximum volume at x ≈ 2.13.
优化题目通常模拟现实情境，要求最大化体积或最小化表面积。在推导出单变量目标函数后，将其输入GDC并在可行域上绘制图像。使用CALC中的“最大值”或“最小值”功能定位最优的 x 值。例如，盒子的函数 V(x) = x(15 − 2x)(10 − 2x) 可以在 0 < x < 5 窗口内绘制，GDC找到最大体积在 x ≈ 2.13 处。
Sometimes the derivative is complex and solving algebraically is messy. The GDC allows you to find the stationary point directly without differentiating. However, you must still show the derivative set to zero in your working for method marks. Use the calculator to confirm the critical number and then evaluate the function there.
有时导数很复杂，代数求解很繁琐。GDC允许你直接找到驻点而无需求导。然而，你仍须在解题过程中写出导数为零的表达式以获得方法分。用计算器确认临界值，然后在该处计算函数值。
For constrained optimisation, for example maximising area with a given perimeter, you can express the function and graph it. The GDC’s ability to compute the maximum on an interval reduces algebraic errors, but always round your final answer according to the context—to the nearest cm, for instance.
对于有约束的优化，例如在给定周长下最大化面积，你可以表达出函数并绘制图像。GDC在区间上计算最大值的能力可以减少代数错误，但务必根据上下文舍入最终答案，例如精确到最接近的厘米。
7. Numerical Integration for Data and Non-Analytical Functions | 对数据和非解析函数进行数值积分
Review set 22B may present a table of values for velocity or a physical rate, requiring you to estimate the total accumulation using numerical integration. The GDC can perform a numerical integration if the function is entered, but with raw data you need to use trapezoidal rule or Simpson’s rule manually. However, you can first regress the data to a function, then integrate. For example, given data points (0, 2), (1, 3.5), (2, 5.2), (3, 7.1), assume a linear model v(t) = 1.7t + 2.0, then integrate from 0 to 3.
复习套题22B可能会给出速度或物理速率的数据表格，要求你使用数值积分估算总累积量。如果输入了函数，GDC可进行数值积分，但对于原始数据，你需要手动使用梯形法则或辛普森法则。然而，你可以先将数据回归为一个函数，然后积分。例如，给定数据点 (0, 2), (1, 3.5), (2, 5.2), (3, 7.1)，假设线性模型 v(t) = 1.7t + 2.0，然后从0到3积分。
The built-in integral function on a GDC can handle piecewise functions if defined properly. When using data-generated functions, be cautious of extrapolation beyond the given range. The area under the curve represents total distance if v(t) is velocity, or total growth if it is a rate of change.
如果正确定义，GDC内置的积分功能可处理分段函数。在使用由数据生成的函数时，注意不要在外推超过给定范围。如果 v(t) 是速度，曲线下面积代表总距离；如果是变化率，则代表总增长。
8. Statistics and Probability Calculations | 统计与概率计算
Many calculator problems in this review set involve single-variable statistics, linear regression, or probability distributions. Input data into lists L1 and L2, then use the 1-Var Stats command to find mean, standard deviation, median, and quartiles. For grouped data, enter midpoints in L1 and frequencies in L2. This is essential for questions on descriptive statistics.
本复习套题中的许多计算器问题涉及单变量统计、线性回归或概率分布。将数据输入列表 L1 和 L2，然后使用 1-Var Stats 命令求出均值、标准差、中位数和四分位数。对于分组数据，将组中值输入 L1，频数输入 L2。这对描述统计问题至关重要。
To perform linear regression on bivariate data, choose LinReg(ax+b) from the STAT CALC menu. The GDC outputs the correlation coefficient r and the regression line y = ax + b. Use this to predict values and assess the strength of the linear relationship. Always store the regression equation in a Y-variable for graphing the line on the scatter plot.
要对双变量数据进行线性回归，请从 STAT CALC 菜单中选择 LinReg(ax+b)。GDC输出相关系数 r 和回归直线 y = ax + b。利用此进行预测并评估线性关系的强度。始终将回归方程存储到 Y 变量中，以便在散点图上绘制直线。
For probability distributions such as normal or binomial, use the DISTR menu. For example, to find P(X > 12) for X ~ B(25, 0.4), use the binomcdf function to compute cumulative probability. Input upper bound 25, lower bound 12, then subtract from 1, or directly use the appropriate complementary option. Always sketch a graph to avoid direction errors.
对于正态或二项分布等概率分布，使用 DISTR 菜单。例如，要求 X ~ B(25, 0.4) 的 P(X > 12)，可使用 binomcdf 函数计算累积概率。输入上限25、下限12，然后从1中减去，或者直接使用适当的互补选项。务必画出草图以避免方向错误。
9. Matrix Operations on the GDC | GDC上的矩阵运算
If your IB course includes matrices (typically in Applications and Interpretation HL), you will need to input and manipulate matrices for solving systems of equations, finding determinants, and computing inverses. Access the MATRIX menu, create a matrix with the required dimensions, and fill in the entries. The calculator can then compute det([A]), [A]⁻¹, or perform multiplication.
如果你的IB课程包含矩阵（通常在应用与解释HL中），你需要输入并操作矩阵，以解方程组、求行列式和计算逆矩阵。进入 MATRIX 菜单，创建所需维度的矩阵，并填入各项。然后计算器可以计算 det([A])、[A]⁻¹ 或执行乘法。
For a system like 2x + 3y = 5, 4x − y = 8, you can set up the coefficient matrix A and constant matrix B, then solve by evaluating [A]⁻¹·[B]. The GDC returns x = 2.071, y = 0.2857. Always check the determinant is non-zero first; if det(A) = 0, the system has no unique solution. Matrix operations are error-prone by hand, so the GDC is a reliable check.
对于如 2x + 3y = 5, 4x − y = 8 的方程组，你可以设置系数矩阵 A 和常数矩阵 B，然后通过计算 [A]⁻¹·[B] 求解。GDC返回 x = 2.071，y = 0.2857。首先检查行列式是否非零；若 det(A) = 0，方程组无唯一解。矩阵运算容易手算出错，因此GDC是可靠的检验工具。
10. Avoiding Common Calculator Errors and Checking Your Work | 避免常见计算器错误并检查工作
Even a small slip in GDC settings can cost valuable marks. Always check that the angle mode (radian/degree) matches the problem context. Ensure parentheses are correctly placed, especially when typing fractions like (x+1)/(x−2). A missing bracket leads to a syntax error or incorrect graph. Use the ‘TRACE’ function to verify coordinates on a graph.
即便GDC设置中出现一个微小的失误，也可能导致丢分。务必检查角度模式（弧度/度）是否与题目背景匹配。确保括号位置正确，尤其在输入 (x+1)/(x−2) 这类分式时。漏掉括号会导致语法错误或图形错误。使用“TRACE”功能验证图形上的坐标。
Misinterpretation of scientific notation is another typical mistake. The GDC may display 1.5E-3, which means 1.5 × 10⁻³ = 0.0015. When recording answers, write them in standard decimal form as required. Also, if the question asks for an exact answer, do not leave a decimal approximation; instead, copy the fraction or radical if the GDC provides it.
错误解读科学计数法是另一个典型错误。GDC可能显示 1.5E-3，这代表 1.5 × 10⁻³ = 0.0015。在记录答案时，按题目要求写作标准小数形式。另外，若题目要求精确答案，不要保留小数近似值；如果GDC给出分数或根式，请照抄。
After obtaining a solution, perform a sanity check. Substitute the value back into the original equation using the calculator’s store and recall functions. For integration, compare the numerical result with an estimate from a simple geometric shape if possible. For statistics, quickly verify the mean lies between the minimum and maximum values. These habits build confidence and reduce careless errors.
在得到解答后，进行合理性检查。利用计算器的存储和调用功能，将数值代回原方程。对于积分，可尽量与简单几何形状的估算值比较。对于统计，快速验证均值介于最小值和最大值之间。这些习惯能建立信心，减少粗心错误。
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