📚 Additional Applications and Interpretation SL Content | IB数学应用与解释SL附加内容
The Applications and Interpretation (AI) SL course extends well beyond routine calculations, focusing on the practical modelling of real-world phenomena through functions, statistics, geometry and calculus. This article revisits the key application topics that often appear in Paper 2 and the internal assessment, providing a structured summary of the concepts, formulas and problem-solving strategies every SL student should master.
应用与解释（AI）SL课程不仅注重计算，更强调通过函数、统计、几何和微积分对现实情景进行建模。本文重新梳理了经常出现在Paper 2和内部评估中的关键应用主题，为学生提供概念、公式与解题策略的结构化总结，帮助大家扎实掌握所有必要的SL应用内容。
1. Financial Mathematics: Compound Interest & Loan Repayments | 金融数学：复利与贷款还款
Compound interest lies at the heart of AI SL financial applications. An initial sum PV invested at an annual interest rate of r% compounded k times per year grows over t years to a future value given by FV = PV × (1 + r/(100k))kt. Alternatively, when interest is compounded continuously, the formula becomes FV = PV × e(r/100)t.
复利是AI SL金融应用的核心。如果本金PV以年利率r%、每年复利k次，经过t年后的终值为FV = PV × (1 + r/(100k))kt。若采用连续复利，公式变为FV = PV × e(r/100)t。
The course also requires students to work with loans and annuities. The present value of a series of equal payments PMT made at the end of each period for n periods at a rate i per period can be expressed as PV = PMT × (1 – (1 + i)–n) / i. This formula is then used to determine regular payments or the outstanding balance on a reducing‑balance loan.
课程还要求学生处理贷款和年金问题。在每期利率为i的条件下，n期期末等额支付额PMT的现值可表示为PV = PMT × (1 – (1 + i)–n) / i。该公式常用于计算定期还款额或递减贷款余额。
A typical exam question presents a scenario such as a car loan of $20,000 repaid monthly over 5 years at 6% nominal interest, asking for the monthly payment. A clear understanding of the timeline, the sign of cash flows and effective use of the finance solver is essential.
一道典型的考题会给出例如2万美元汽车贷款、名义利率6%、分5年按月偿还的情景，要求计算月供。清楚理解时间线、现金流符号并熟练运用金融求解器至关重要。
2. Trigonometric Applications: Sine & Cosine Rules | 三角学应用：正弦定理与余弦定理
In non‑right‑angled triangles, the sine rule a / sin A = b / sin B = c / sin C is used when two angles and a side are known, or when two sides and a non‑included angle are given (being cautious of the ambiguous case). The cosine rule a² = b² + c² – 2bc cos A is applied when three sides or two sides and the included angle are known.
对于非直角三角形，当已知两角一边或两边及一个非夹角（需注意多解情况）时使用正弦定理a / sin A = b / sin B = c / sin C；当已知三边或两边及其夹角时使用余弦定理a² = b² + c² – 2bc cos A。
The area of a triangle can be found using the formula Area = ½ ab sin C. AI SL problems often embed these rules in 3D contexts, such as finding the angle between a diagonal of a cuboid and its base, which requires breaking the problem into right‑angled triangles and applying the sine or cosine rule in an intermediate step.
三角形面积可用Area = ½ ab sin C计算。AI SL的题目常将这些定理融入三维情景，例如求长方体的对角线与其底面的夹角，需要将问题拆分为直角三角形，在中间步骤使用正弦或余弦定理。
Bearings, angles of elevation and depression are also tested, often combined with the sine or cosine rule. Students should draw a labelled diagram before solving and remember to convert between kilometres and metres where necessary.
方位角、仰角和俯角也常与正弦或余弦定理结合考查。学生应先画出带标注的图形，再动手计算，并根据需要进行单位换算。
3. Voronoi Diagrams: Nearest‑Neighbour Regions | Voronoi图：最近邻区域
A Voronoi diagram partitions a plane into regions based on the distance to a given set of sites. Each region consists of all points closer to its site than to any other site. The boundaries between regions are the perpendicular bisectors of the segments joining adjacent sites.
Voronoi图根据给定点集将平面划分为不同区域，每个区域包含所有距其站点比其他站点更近的点。区域间的边界是相邻站点连线的垂直平分线。
Adding a new site to an existing diagram requires identifying the site whose region will be reduced, drawing the perpendicular bisectors with the nearest neighbours, and updating the boundaries. The concept is often applied to locating services, mapping mobile phone towers or modelling the territories of animals.
在已有图上添加新站点需先确定区域将被缩减的站点，画出与相邻站点的垂直平分线，并更新边界。这一概念常用于服务选址、手机信号塔规划或动物领地建模。
Questions may ask for the equation of a boundary edge between two sites (i.e. the perpendicular bisector) or to determine which site is the nearest to a given point by comparing distances. The ability to interpret the diagram, rather than merely construct it, is strongly emphasised in AI SL.
考题可能要求给出两站点间边界的方程（即垂直平分线），或通过比较距离判断给定点属于哪个最近站点。AI SL非常强调对图形的理解能力，而非单纯的作图。
4. Bivariate Statistics: Correlation & Regression | 双变量统计分析：相关与回归
Pearson's product‑moment correlation coefficient r measures the strength and direction of a linear relationship between two quantitative variables. Its value ranges from –1 to 1. A calculation of r from a dataset using the formula r = Sxy / √Sxx·Syy (where Sxy = ∑(x – x̄)(y – ȳ)) is expected, though the GDC may be used.
皮尔逊积矩相关系数r衡量两个数值变量间线性关系的强度和方向，取值范围为 –1 至 1。需掌握利用公式r = Sxy / √Sxx·Syy（其中 Sxy = ∑(x – x̄)(y – ȳ)）从数据集计算r，也允许使用GDC。
The regression line of y on x is written as y = a + bx, where b = Sxy / Sxx and a = ȳ – b x̄. This line minimises the sum of squared vertical deviations and is used to predict values of y for given x within the range of the data. Extrapolation beyond the data should be treated with caution.
y关于x的回归直线方程为y = a + bx，其中b = Sxy / Sxx，a = ȳ – b x̄。该直线使竖直离差平方和最小，可用于在数据范围内由x预测y值。超出数据范围的外推需谨慎对待。
AI SL questions often present a table of values, ask for r, the line of best fit, and then interpret the slope or make a prediction. Students must understand that correlation does not imply causation and should be able to comment on the reliability of predictions based on the data set.
AI SL考题通常给出一张数据表，要求计算r和最佳拟合直线，进而解读斜率或进行预测。学生必须理解相关关系≠因果关系，并能基于数据集评价预测的可靠性。
5. Probability Distributions: Binomial & Normal | 概率分布：二项分布与正态分布
When a fixed number n of independent trials each has the same probability of success p, the number of successes X follows a binomial distribution: X ~ B(n, p). The probability function is P(X = k) = C(n, k) pk (1 – p)n – k, where C(n, k) is the binomial coefficient.
若进行固定次数n次独立试验，每次成功概率为p，则成功次数X服从二项分布：X ~ B(n, p)。其概率函数为P(X = k) = C(n, k) pk (1 – p)n – k，其中C(n, k)为二项系数。
The normal distribution N(μ, σ²) models continuous data with a symmetric bell‑shaped curve. To find probabilities, a value is standardised using z = (x – μ) / σ. The inverse normal function returns a data value given a cumulative probability. Both distributions are applied to quality control, examination scores and natural measurements.
正态分布N(μ, σ²)适用于连续型数据，呈对称的钟形曲线。计算概率时需将数值标准化为z = (x – μ) / σ。逆正态函数可根据累积概率反求数据值。这两个分布广泛应用于质量控制、考试成绩和自然测量。
In AI SL, problems often ask for a specific binomial probability using the GDC, or the area under a normal curve between two points. The connection between binomial and normal as an approximation for large n is also introduced, though calculation is performed via the GDC.
AI SL中常见的问题包括使用GDC求特定二项概率，或求正态曲线下两点之间的面积。当n较大时，二项分布可用正态近似，这一联系虽有所提及，但计算仍借助GDC完成。
6. Hypothesis Testing: Chi‑Squared Tests | 假设检验：卡方检验
The chi‑squared (χ²) test for independence determines whether two categorical variables are associated. Observed frequencies are compared with expected frequencies calculated under the null hypothesis of independence: expected = (row total × column total) / grand total. The test statistic is χ² = Σ (O – E)² / E.
独立性卡方检验(χ²)用来判断两个分类变量是否有关联。比较观测频数与在独立性零假设下计算的期望频数：期望值 = (行合计 × 列合计) / 总计。检验统计量为χ² = Σ (O – E)² / E。
The goodness‑of‑fit test checks whether a sample distribution follows a theoretical model, such as a given ratio or a binomial model. Degrees of freedom for the test of independence are (rows – 1) × (columns – 1); for goodness of fit, df = number of categories – number of estimated parameters – 1. The p‑value is obtained from the χ² distribution with the appropriate df.
拟合优度检验用来检验样本分布是否符合某一理论模型，如给定比例或二项模型。独立性检验的自由度为(行数–1) × (列数–1)；拟合优度检验的自由度为类别数 – 估计参数个数 – 1。p值可根据相应的自由度从χ²分布中获取。
When interpreting results, students compare the p‑value with the significance level (commonly 5%). If p < 0.05, the null hypothesis is rejected. They must also write a conclusion in the context of the problem and check that the expected frequencies are at least 5 for the test to be valid.
解释结果时，学生需将p值与显著性水平（常用5%）进行比较。若p < 0.05，则拒绝零假设。此外还必须在问题语境中写下结论，并检查期望频数是否至少为5以确保检验有效。
7. The t‑Test for Two Independent Samples | 独立双样本t检验
To compare the means of two independent groups, AI SL uses the two‑sample t‑test. The null hypothesis H₀: μ₁ = μ₂ is tested using the statistic t = (x̄₁ – x̄₂) / SE, where the standard error SE depends on whether the population variances are assumed equal. For equal variances, the pooled standard error is calculated from the combined sample data.
为了比较两个独立组的均值，AI SL使用双样本t检验。检验零假设H₀: μ₁ = μ₂所用的统计量为t = (x̄₁ – x̄₂) / SE，其中标准误SE取决于是否假定总体方差相等。若方差相等，则使用合并标准误，由两组样本数据共同计算得出。
The degrees of freedom for the pooled t‑test are n₁ + n₂ – 2. The GDC provides a p‑value directly from the sample summaries or lists. As with the chi‑squared test, the conclusion must be phrased in terms of the original question, stating whether there is sufficient evidence of a difference between the means.
合并方差t检验的自由度为n₁ + n₂ – 2。GDC可直接根据样本摘要或列表给出p值。与卡方检验一样，结论需针对原始问题表述，说明是否有足够证据表明两均值存在差异。
Exam questions often involve an additional step, such as a preliminary F‑test for equality of variances, or they simply direct students to assume equal variances. The t‑test is applied across contexts like comparing the lengths of leaves from two locations or the effect of a teaching method on test scores.
考题常常会包含额外步骤，如预先进行方差相等的F检验，或者直接要求学生假定方差相等。t检验的应用场景包括比较两地点树叶的长度或不同教学方法对考试成绩的影响。
8. Differentiation: Tangents, Normals & Optimisation | 微分：切线、法线与最优化
The derivative dy/dx of a function f(x) at a point gives the gradient of the tangent. The equation of a tangent at (x₀, y₀) is y – y₀ = f'(x₀)(x – x₀), and the normal, perpendicular to the tangent, has gradient –1 / f'(x₀) provided f'(x₀) ≠ 0.
函数f(x)在某点的导数dy/dx给出切线的斜率。点(x₀, y₀)处切线方程为y – y₀ = f'(x₀)(x – x₀)；法线与切线垂直，其斜率为–1 / f'(x₀)（假设f'(x₀) ≠ 0）。
Optimisation problems use the first derivative to find local maxima and minima. Setting f'(x) = 0 identifies stationary points; the sign of f'(x) around the point determines whether it is a maximum, minimum or inflection. AI SL problems involve maximising area, volume, or profit given a constraint, requiring students to express a quantity in terms of a single variable before differentiating.
最优化问题利用一阶导数寻找局部最大值和最小值。令f'(x) = 0可找到驻点；通过驻点两侧f'(x)的符号可判断其为极大值、极小值还是拐点。AI SL的最优化问题常涉及在给定约束下最大化面积、体积或利润，需要学生将目标量表示为单变量函数后再求导。
A classic example is finding the dimensions of a cylinder of maximal volume that can be inscribed in a sphere, or minimising the surface area of a can for a given volume. Students must always confirm that the stationary point indeed gives the required optimum.
经典例题如求内接于球体的圆柱体最大体积的尺寸，或在给定容积下使罐体表面积最小。学生务必验证所求驻点确实实现了所需的最优结果。
9. Integration: Area Under Curves & Kinematics | 积分：曲线下面积与运动学
The definite integral ∫ab f(x) dx represents the signed area between the curve y = f(x) and the x‑ Total area is computed by integrating piecewise and taking absolute values where f(x) < 0.
定积分∫ab f(x) dx表示曲线y = f(x)与x轴在x = a到x = b之间的有号面积。轴上方面积为正，下方面积为负。总面积通过分段积分并对f(x) < 0的部分取绝对值求得。
Integration is also the tool for solving simple differential equations in kinematics. Given a velocity function v(t), the displacement is ∫ v(t) dt and the total distance travelled is ∫ |v(t)| dt over the time interval. Acceleration, the derivative of velocity, connects backwards to integration as well.
积分也是求解运动学中简单微分方程的工具。已知速度函数v(t)，位移为∫ v(t) dt，总路程为时间区间内∫ |v(t)| dt。加速度作为速度的导数，也可通过积分逆向联系。
AI SL also includes volume of revolution: rotating the region under y = f(x) about the x‑axis generates a solid whose volume is π ∫ab [f(x)]² dx. Problems may combine area and volume in a single question, requiring careful selection of limits and a clear sketch.
AI SL还包括旋转体体积：将y = f(x)下区域绕x轴旋转所得立体的体积为π ∫ab [f(x)]² dx。题目可能将面积和体积组合在一起，需要仔细确定积分限并绘制清晰草图。
10. Exponential Models & Logarithmic Scales | 指数模型与对数尺度
Exponential functions of the form f(x) = k ax + c model population growth, radioactive decay, and cooling. The constant k represents the initial amount above the asymptote c, and a is the growth (a > 1) or decay factor (0 < a < 1)per unit time. Parameters are estimated from data using exponential regression on the GDC.
形如f(x) = k ax + c的指数函数用来模拟种群增长、放射性衰变和冷却过程。常数k代表高于渐近线c的初始量，a为单位时间的增长因子(a > 1)或衰减因子(0 < a < 1)。参数可通过GDC上的指数回归由数据估计。
When data display a multiplicative pattern, taking logarithms converts the exponential model into a linear one: ln(y – c) = ln k + x ln a. Students are expected to interpret semi‑log graphs, identify the linear trend, and determine the constants from the slope and intercept. This skill is vital in modelling epidemics, financial data, and sound intensity.
当数据显示乘法模式时，取对数可将指数模型转化为线性形式：ln(y – c) = ln k + x ln a。学生需要会解读半对数图，识别线性趋势，并由斜率和截距确定常数。这一技能在模拟流行病传播、金融数据和声强时至关重要。
Logarithmic scales such as the Richter scale, pH, and sound level (dB) are based on base‑10 logarithms. A difference of 1 unit corresponds to a factor of 10 in the underlying phenomenon. AI SL problems ask to compare intensities or calculate the change from a given scale reading.
里氏震级、pH值和声级分贝等对数标度以10为底。标度上1单位的差异对应于底层现象中10倍的变化。AI SL题目可能要求比较强度或根据给定标度读数计算变化量。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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