📚 Review set 16B - CALCULATOR | 复习题集 16B - 计算器
This review set is designed to strengthen your skills in using a graphing calculator for IB Mathematics. The problems cover a broad range of topics where technology is essential: solving equations numerically, analysing functions, performing calculus operations, and handling statistical data. Work through each question with your calculator, paying close attention to the settings and syntax required by the specific model you are using (such as TI-Nspire or Casio fx-CG50). Mastery of these techniques will save time and improve accuracy in both Paper 2 and the exploration.
本复习题集旨在强化您在 IB 数学中运用图形计算器的技能。题目涵盖计算器必备的多个主题：数值求解方程、分析函数、执行微积分运算以及处理统计数据。请使用您的计算器逐一完成练习，并密切注意您所使用的特定型号（如TI-Nspire或Casio fx-CG50）的设置与语法。熟练掌握这些技巧将助力您在试卷二和内部探索中节省时间并提升准确性。
1. Solving polynomial equations | 求解多项式方程
A common task is to find the roots of a polynomial such as x³ – or polynomial root-finder on your calculator. Enter the coefficients: 1 for x³, 0 for x², – alculator should return three real roots. Round your answers to three significant figures.
常见任务是求解多项式的根，例如 x³ – 4x + 1 = 0。使用计算器上的方程求解器或多项式寻根功能。依次输入系数：x³ 的系数 1，x² 的系数 0，x 的系数 –4，常数项 1。计算器应返回三个实根。请将答案四舍五入至三位有效数字。
Check that you obtain approximately x ≈ – evaluating the polynomial to see if it is close to zero. This habit prevents sign errors.
请核对您是否得到近似解 x ≈ –2.11、0.254 和 1.86。始终通过存储每个根并计算多项式的值来验证其是否接近于零。这个习惯能有效避免符号错误。
2. Finding intersection points | 求交点
Consider the functions f(x) = 2ˣ and g(x) = x² + 2. Graph both and use the 'intersect' tool to find the x‑ eed to set a lower and upper bound near each intersection. The two curves intersect at x ≈ –0.585 and x ≈ 2.26. The corresponding y‑values can be found by evaluating either function at these x‑values.
考虑函数 f(x) = 2ˣ 和 g(x) = x² + 2。绘制两者的图像，并使用“交点”工具求解它们交点的 x 坐标。在典型的 IB 计算器上，您需要为每个交点设定下界和上界。这两条曲线相交于 x ≈ –0.585 和 x ≈ 2.26。相应的 y 值可通过将 x 代入任一函数求得。
Be careful with the viewing window. Start with a standard window of –5 ≤ x ≤ 5 and – h can be applied to any pair of functions, including trigonometric and logarithmic curves.
请留意图形的观察窗。从标准窗口 –5 ≤ x ≤ 5 和 –2 ≤ y ≤ 10 开始，必要时再进行调整。该方法适用于任意函数组合，包括三角函数与对数函数的曲线。
3. Numerical derivatives | 数值导数
l derivative command (nDeriv or d/dx). To find f ′(2), input nDeriv(ln(x²+1), x, 2). The result is about 0.800. Remember to check whether your calculator expects an expression or a function name.
IB 考试中经常要求您计算某一点的导数，而无需进行代数求导。对于 f(x) = ln(x² + 1)，请使用数值导数指令（nDeriv 或 d/dx）。要计算 f ′(2)，输入 nDeriv(ln(x²+1), x, 2)。结果约为 0.800。请记住检查您的计算器是需要输入表达式还是函数名称。
This feature is particularly useful for verifying the gradient of a tangent in an unfamiliar function. You can also use it to plot the derivative function and examine its behaviour, helping you understand increasing/decreasing intervals.
该功能对于验证陌生函数切线的斜率尤为有用。您还可以借此绘制导函数图像并分析其性态，从而理解函数的递增与递减区间。
4. Definite integrals and area | 定积分与面积
in(2x), x, 0, π/2) yields 1. For an area bounded by two curves, compute the integral of the upper minus the lower function. Always sketch the region first to decide the limits and which function is on top.
要计算 f(x) = sin(2x) 在 0 到 π/2 区间内的精确面积，请使用定积分功能。指令 ∫(sin(2x), x, 0, π/2) 得到的结果为 1。对于由两条曲线围成的区域，应计算上函数减去下函数的积分。务必先绘制草图，以确定积分限和上下函数关系。
If your calculator gives a numerical approximation, check the settings to switch to exact mode if needed. For problems involving parametric equations or arc lengths, the calculator can still evaluate the resulting definite integral numerically.
如果您的计算器给出近似数值，请检查设置并在需要时切换为精确模式。对于涉及参数方程或弧长的问题，计算器仍能对所得定积分进行数值求解。
5. Statistics: one‑variable analysis | 统计：单变量分析
Enter the following data set into list L₁: 12, 15, 14, 10, 13, 16, 18, 17, 14, 15. Use one‑ istics to find the mean, sample standard deviation sx, and the median. You should obtain x̄ edian = 14.5. Your calculator will also display Q₁ and Q₃, allowing you to construct a box‑and‑whisker plot.
将下列数据集输入列表 L₁：12, 15, 14, 10, 13, 16, 18, 17, 14, 15。使用单变量统计功能计算均值、样本标准差 sx 以及中位数。您应得到 x̄ = 14.4，sx ≈ 2.46，中位数 = 14.5。您的计算器还会显示 Q₁ 和 Q₃，以便您构建箱线图。
Be absolutely sure to select the sample standard deviation sx and not the population σx unless the data represent the whole population. In IB, data from a sample is far more common.
务必选择样本标准差 sx 而不是总体标准差 σx，除非数据代表整个总体。在 IB 中，样本数据要常见得多。
6. Linear regression and correlation | 线性回归与相关性
linear regression (y = ax + b). The calculator returns a ≈ 2.14, b ≈ 0.92, and the product‑ r ≈ 0.998. This strong positive correlation suggests a nearly perfect linear relationship.
给定双变量数据 (x, y)： (1, 3.1), (2, 5.4), (3, 7.2), (4, 9.6), (5, 11.8)，执行线性回归 (y = ax + b)。计算器给出 a ≈ 2.14，b ≈ 0.92，以及积矩相关系数 r ≈ 0.998。如此强的正相关表明几乎完美的线性关系。
3.8. Also check the residual plot to confirm that a linear model is appropriate. The high r² value (about 0.997) further indicates that 99.7% of the variability is explained by the model.
使用回归方程预测 x = 6 时的 y 值。代入后得到 y ≈ 13.8。同时检查残差图以确认线性模型是合适的。较高的 r² 值（约 0.997）进一步表明模型解释了 99.7% 的变异性。
7. Normal probability distributions | 正态概率分布
A random variable X follows N(100, 15²). Find P(85 < X < 115). Use the normal cumulative distribution function (normCdf) with lower bound 85, upper bound 115, mean 100, and standard deviation 15. The result is approximately 0.683. This matches the well‑known 68% rule for data within one standard deviation of the mean.
随机变量 X 服从 N(100, 15²)。求 P(85 < X < 115)。使用正态累积分布函数（normCdf），下限为 85，上限为 115，均值为 100，标准差为 15。结果约为 0.683。这符合众所周知的“68% 规则”，即数据落在均值一个标准差范围内的概率。
rse normal calculations are equally important. To find the score that separates the top 10% tical value is about 119.2. Always draw a quick sketch to confirm the direction of the tail.
逆正态计算同样重要。要找出分布前 10% 的分界线，使用 invNorm，面积 = 0.9，均值 = 100，标准差 = 15。临界值约为 119.2。请始终快速绘制简图以确认尾部方向。
8. Financial mathematics: TVM solver | 金融数学：TVM 求解器
A loan of $5000 is repaid with monthly payments over 4 years at an annual interest rate of 6.5%, compounded monthly. Use the TVM solver to find the monthly payment. Set N = 48, I% = 6.5, PV = 5000, FV = 0, P/Y = 12, C/Y = 12. Solve for PMT; the payment is approximately $118.81.
一笔 5000 美元的贷款，在年利率 6.5%、每月复利的条件下，分 4 年按月偿还。使用 TVM 求解器计算每月还款额。设置 N = 48，I% = 6.5，PV = 5000，FV = 0，P/Y = 12，C/Y = 12。求解 PMT，得到每月还款额约为 118.81 美元。
Be mindful of the sign convention: money you receive (the loan) is positive, while payments are negative. If your calculator displays a negative PMT, that simply means it is a cash outflow from your perspective. The TVM solver can also handle savings plans, mortgages, and sinking funds.
请注意符号约定：您收到的资金（贷款）为正，而还款为负。如果计算器显示负的 PMT，那仅意味着从您的角度看是现金流出。TVM 求解器还可处理储蓄计划、抵押贷款和偿债基金。
9. Matrix operations | 矩阵运算
minant of A. The product A × B should be [[8, -6], [9, 3]]. The determinant of A is (3)(1) – (–5) = 13. Your calculator can verify these quickly and also find the inverse, if it exists.
输入矩阵 A = [[3, -2], [5, 1]] 和 B = [[2, 0], [-1, 3]]。计算 A × B 以及 A 的行列式。乘积 A × B 应为 [[8, -6], [9, 3]]。A 的行列式为 (3)(1) – (–2)(5) = 13。您的计算器可以快速验证这些结果，并在存在逆矩阵时求出它。
X = A⁻¹C. Use the calculator to find the inverse and multiply. This method is a huge time‑saver in Paper 2, especially for 3×3 systems.
对于写成矩阵形式 AX = C 的线性方程组，其解为 X = A⁻¹C。使用计算器求逆矩阵并执行乘法。该方法在试卷二中能极大节省时间，尤其适用于 3×3 的方程组。
10. Exploring functions with table of values | 利用数值表探究函数
Generate a table of values for f(x) = (sin x)/x from x = – spot asymptotes, turning points, and sign changes without manually evaluating many points.
为 f(x) = (sin x)/x 生成一个从 x = –2π 到 2π，步长为 π/4 的数值表。观察 x = 0 附近的性态；函数值趋近于 1，展示了一个经典极限。数值表功能有助于您发现渐近线、极值点和符号变化，而无需手动计算大量点值。
Many IB problems ask you to 'find the smallest positive x for which f(x) = 0.5' numeric approach is acceptable when an algebraic solution is impossible or overly complex.
许多 IB 题目要求使用表格“寻找使得 f(x) = 0.5 的最小正 x 值”。通过调整表格设置不断缩小区间，直至达到所需精度。当无法获得代数解或其过于复杂时，这种数值方法是可接受的。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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