📚 Review Set 10B – Calculator | 复习集10B – 计算器篇
Review Set 10B focuses on sharpening the skills you need to solve IB Mathematics problems using a graphical display calculator (GDC). Whether you are working through polynomial equations, statistical analysis, or financial models, this review brings together typical calculator-allowed question types that appear in Paper 2 (and Paper 3 for HL) across both Analysis & Approaches and Applications & Interpretation. Mastering these techniques not only saves time but also reduces algebraic mistakes, letting you tackle more complex reasoning tasks with confidence.
复习集10B旨在帮助你提升使用图形计算器解决IB数学问题的能力。不论你是面对多项式方程、统计分析还是金融模型，本复习涵盖了试卷二（以及高级别试卷三）中典型的允许使用计算器的题型，适用于分析与方法和应用与解释两个方向。熟练掌握这些技巧不仅能节省时间，还能减少代数运算错误，让你有信心应对更复杂的推理题。
1. Solving Equations with the Solver | 使用求解器解方程
finder (often found under the equation menu) to obtain all real roots directly. For a non‑ ted solution. The solver will quickly return an approximate value, for instance x ≈ 0.511.
IB试题经常要求求解无法轻易手算的方程。图形计算器的方程求解器可以处理多项式、三角方程、指数方程和混合形式。对于 x³ − 4x + 1 = 0 这样的多项式，可利用多项式求根器（通常在方程菜单下）直接得到所有实数根。对于非多项式方程，例如在区间 0 < x < π 上求解 2 sin x = 1 − x，则切换到数值求解器，输入方程并给出一个靠近预期解的初始猜测值，求解器会快速返回近似值，例如 x ≈ 0.511。
s be aware that the solution displayed depends on the initial guess. If the equation has multiple roots, you must supply different starting values to find each one. Some calculators also allow you to set bounds: for example, solve 3ˣ rch interval. After obtaining a solution, it is good practice to store it in a variable and plug it back into the original equation to confirm that both sides match to a high degree of precision.
使用数值求解器时，务必要注意显示的答案取决于初始猜测值。如果方程有多个根，必须提供不同的起始值才能分别求出。一些计算器还允许设置求解区间，例如在 [0,2] 上求解 3ˣ = 7 − x² 时限制搜索范围。得到解之后，最好将它存入变量并代回原方程，以验证等号两侧在高精度下是否相等。
2. Graphical Analysis: Intersections and Roots | 图形分析：交点与零点
on the calculator. By entering two functions, say y₁ = x² and y₂ = 2 − x, you can use the ‘intersection’ command to find the x‑ solving simultaneous equations that do not have a neat algebraic solution. Similarly, the ‘zero’ or ‘root’ tool locates the x‑intercepts of a single function, which correspond to the solutions of f(x) = 0.
许多IB问题要求直接在计算器上处理函数图像。输入两个函数，比如 y₁ = x² 和 y₂ = 2 − x，可以使用“交点”命令求出它们相交的 x 坐标。这种可视化方法对于求解没有优美代数解的联立方程特别有用。类似地，“零点”或“根”工具可以找出单个函数的 x 轴截距，这些截距正是 f(x)=0 的解。
Make full use of the zoom and window adjustments to capture all relevant points. When a function oscillates rapidly or has asymptotes, choose an appropriate viewing window to avoid missing intersections. You can also trace along the curve to read off coordinates of special points, but for precise answers always rely on the intersection or zero functions, which give numerical values to several decimal places. In exam settings, you may be asked to write the coordinates correct to three significant figures.
充分利用缩放和窗口调整来捕捉所有相关的点。当函数振荡剧烈或有渐近线时，选择合适的观察窗口以免遗漏交点。你也可以沿曲线移动光标读取特殊点的坐标，但若要精确答案，务必依赖交点或零点功能，它们能给出多位小数的数值。在考试中，通常要求将坐标保留三位有效数字。
nique is also valuable for inequality questions: after finding the boundaries, you can visually determine the intervals where f(x) > g(x). A quick sketch on the calculator screen, combined with correct logical signs, guarantees full marks on such questions.
有时你需要找出 f(x) 等于常数的点，例如 f(x)=3。此时不用求解 f(x)−3=0，只需绘制 y=f(x) 和 y=3 并求交点即可。该方法对于不等式题也很有价值：找出分界点后，你可以直观地确定 f(x)>g(x) 的区间。在计算器屏幕上画出简图，再结合正确的逻辑符号，就能确保此类题目拿到满分。
3. Finding Maximum and Minimum Values | 寻找最大值与最小值
he calculator compute the coordinates. For instance, to find the maximum value of f(x) = x⁴ − 4x³ + 2x on the interval [−1, 3], you would graph the function, activate the maximum command, and place the bounds near the visible hump. The calculator returns both the x‑coordinate and the function value.
优化问题经常在IB考试中出现，计算器上的最大值/最小值功能是寻找局部极值点的可靠工具。在合适的窗口画出函数图像后，选择最大值（或最小值）选项，然后在观察到的峰附近设定左、右边界，计算器便会计算出坐标。例如，要找出 f(x)=x⁴−4x³+2x 在区间 [−1,3] 上的最大值，先画图，启动最大值命令，并在可见的峰附近放置边界，计算器会返回 x 坐标和函数值。
When a question asks for the absolute maximum (or minimum) on a closed interval, remember to also evaluate the function at the interval endpoints, because the extreme value might occur at the boundary rather than at a turning point. Your calculator can store these endpoint values quickly by using the table feature or by evaluating y₁(−1) and y₁(3) on the home screen. Compare the endpoint values with any local extrema found by the GDC, then declare the absolute extreme accordingly.
当题目要求在闭区间上求绝对最大值（或最小值）时，记得还要计算函数在区间端点处的值，因为极值可能出现在边界上而非转折点。你可以利用表格功能或直接在主屏幕上计算 y₁(−1) 和 y₁(3)，快速存储端点值。将端点值与计算器找到的局部极值进行比较，然后相应给出绝对极值。
Be cautious with functions that have multiple peaks: you need to check each one using the maximum/minimum tool. If your window is too wide, you might inadvertently skip a local extremum. A good habit is to start with a panoramic view, then zoom in on regions where the gradient appears to change sign. Always round your final coordinates to the accuracy requested in the problem, usually three significant figures or two decimal places as specified.
对于有多个峰的函数要格外小心：你需要用最大值/最小值工具逐一检查每个峰。如果窗口过大，可能会无意中跳过一个局部极值点。好习惯是从全景开始观察，然后在梯度似乎改变符号的区域放大。最后答案的坐标应按照题目要求的精度进行舍入，通常是三位有效数字或指定的小数位数。
4. Numerical Derivatives and Integrals | 数值导数与积分
rivative of a function at a given point. For example, to find the rate of change of f(x) = √x² + 1 at x = 2, enter nDeriv(√X²+1, X, 2). The result is an approximation of the instantaneous gradient. This is particularly useful when the algebraic derivative would be messy or when you need to verify your own differentiation.
即使可以进行解析求导或积分，IB考试的计算器部分也常常考查学生应用数值近似的本领。nDeriv 命令可以计算函数在给定点的导数。例如，要找出 f(x)=√x²+1 在 x=2 处的变化率，可输入 nDeriv(√X²+1, X, 2)。结果是瞬时梯度的近似值。当解析求导较为繁琐时，或者当你需要验证手算的导数时，这一功能特别有用。
, X, 0, 1). The calculator uses a sophisticated algorithm, so the answer is reliable even when the antiderivative cannot be expressed in elementary functions. Remember that the answer is a numerical approximation, so rounding to three significant figures is standard unless otherwise instructed. In context questions, always state the area or accumulated value with appropriate units.
定积分功能通常称为 fnInt，用于数值计算曲线下的面积。要计算 ∫₀¹ sin(x²) dx，输入 fnInt(sin(X²), X, 0, 1)。计算器使用复杂的算法，因此即使原函数无法用初等函数表示，答案也十分可靠。记住这个结果是数值近似，除非另有说明，通常保留三位有效数字。在应用题中，务必附上正确的单位来表示面积或累积量。
When you need the area between two curves, do not manually subtract one function from the other on paper; simply graph both and use the integration function to integrate the upper curve minus the lower curve on the appropriate interval. Alternatively, many calculators allow you to shade the region and compute the integral directly. Always check whether the problem calls for signed area or absolute area, and adjust your limits if the curves cross.
当你需要求两条曲线之间的面积时，无需在纸上手工将一个函数减去另一个，只需画出两条曲线，然后在适当区间上对“上曲线减下曲线”进行积分。很多计算器还允许对区域进行阴影填充并直接计算积分。要始终判断题目要求的是带符号面积还是绝对面积，若曲线相交则需调整积分限。
5. Statistical Calculations: One‑Variable and Two‑Variable | 统计计算：单变量与双变量
IB data‑handling questions expect you to enter lists of numbers and quickly obtain summary statistics. After typing data into, say, List 1, use 1‑Var Stats to display the mean (x̄), sum, sum of squares, standard deviation (σₙ or sₙ₋₁), median, and quartiles. Be aware that IB typically uses the sample standard deviation sₙ₋₁ when analysing a sample, so always check your syllabus or question wording. If you need to find the variance, simply square the standard deviation value shown.
IB数据处理题目期望你能输入数字列表并迅速获得摘要统计量。将数据输入到列表1后，使用单变量统计功能即可显示均值 x̄、总和、平方和、标准差（σₙ 或 sₙ₋₁）、中位数和四分位数。注意，IB在分析样本时通常使用样本标准差 sₙ₋₁，因此务必核对教学大纲或题目措辞。如果需要方差，只需将显示的标准差平方即可。
For two‑ dependent variable in L2. Run linear regression (LinReg or y=mx+b) to obtain the slope, y‑ oefficients to three significant figures. Always examine the r value to comment on the strength of the linear relationship.
对于双变量数据，将自变量输入L1，因变量输入L2。执行线性回归（LinReg 或 y=mx+b）得到斜率、y截距和相关系数 r。计算器还会给出决定系数 r²。当IB题目要求求出回归直线方程并用于预测时，以 y=ax+b 的形式报告方程，系数保留三位有效数字。要始终根据 r 值判断线性关系的强弱。
Advanced calculators also support non‑linear regressions, such as quadratic, exponential, and logarithmic models. In an IA or exam investigation, you may be asked to choose the most appropriate model by comparing r² values. Once the model is stored, you can use the trace or table feature to interpolate or extrapolate values. Remember to state clearly the limitations of extrapolation, particularly when the x‑value lies far outside the original range.
高级计算器还支持非线性回归，例如二次、指数和对数模型。在内部评估或考试探究中，你可能需要比较 r² 值来选择最合适的模型。模型储存后，你可以利用跟踪或表格功能进行内插或外推。记住要明确说明外推的局限性，尤其是当 x 值远超出原始数据范围时。
6. Normal Distribution and Inverse Normal | 正态分布与逆正态计算
Many IB probability problems involve the normal distribution, and the GDC provides fast-access functions. For a normal variable X ~ N(μ, σ²), use normalcdf(lower bound, upper bound, μ, σ) to calculate P(a < X < b). For instance, if IQ scores follow N(100, 15²), then P(85 < X < 115) is obtained by normalcdf(85, 115, 100, 15). The calculator returns the probability as a decimal; you should round to three significant figures unless told otherwise.
许多IB概率问题涉及正态分布，图形计算器提供了便捷的函数。对于正态变量 X ~ N(μ, σ²)，使用 normalcdf(下限, 上限, μ, σ) 计算 P(a<X<b)。例如，设智商分数服从 N(100, 15²)，则 P(85<X<115) 可通过 normalcdf(85, 115, 100, 15) 求出。计算器以小数形式返回概率；除非另有说明，应舍入至三位有效数字。
erse normal function invNorm(area, μ, σ). Suppose you need the score c such that the top 5% of students exceed it: P(X > c) = 0.05. Because invNorm works with left‑ , 100, 15) ≈ 124.7. Always draw a sketch and identify whether the question requires a left‑tail, right‑tail, or central region, and adjust the area accordingly.
当需要求解未知边界时，可使用逆正态函数 invNorm(面积, μ, σ)。假设要求出 c 使得得分最高的5%学生超过它：P(X>c)=0.05。由于 invNorm 作用于左尾面积，需要用 1−0.05=0.95 作为面积参数：invNorm(0.95, 100, 15)≈124.7。务必画出示意图，判断题目需要的是左尾、右尾还是中间区域，并相应调整面积。
In some syllabus topics (such as the t‑distribution for smaller samples in AI HL), similar commands like tcdf and invT are available. The logic is analogous: input the degrees of freedom along with the required area or boundaries. Regardless of the distribution, always confirm that you have entered the standard deviation (σ) and not the variance. Mismatching μ and σ² is one of the most common calculator errors in normal distribution problems.
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