📚 IB Math AI SL Paper 2 Practice | IB数学应用与解释SL试卷2练习
Paper 2 in IB Mathematics: Applications and Interpretation SL is a technology-required exam where you must tackle extended-response questions using your graphical display calculator (GDC). Success depends on both your mathematical understanding and your ability to leverage the calculator effectively. This article will guide you through essential skills, common question types, and exam techniques to help you practise effectively.
IB数学应用与解释SL的试卷2是一场要求使用技术的考试，你需要借助图形计算器（GDC）解决扩展回答题。成功既取决于你的数学理解，也取决于你有效利用计算器的能力。本文将引导你掌握关键技能、常见题型和应试技巧，帮助你高效练习。
1. Understanding Paper 2 Structure | 理解试卷2的结构
Paper 2 lasts 90 minutes and is worth 90 marks, forming 50% of your final SL grade. It contains several extended-response questions, each broken into parts (a), (b), (c) and sometimes more. Questions often integrate multiple topics such as statistics, functions, and calculus.
试卷2时长90分钟，满分90分，占SL总成绩的50%。它包含若干扩展回答题，每题分为(a)、(b)、(c)等多个小问。题目常融合统计学、函数和微积分等多个主题。
Your GDC is essential for plotting graphs, computing statistics, solving equations, and performing numerical integration. However, you must still show logical working and clear interpretation of results to earn full marks.
你的图形计算器在绘制图形、计算统计数据、求解方程和数值积分方面必不可少。然而，你仍须展示合理的解题过程和清晰的结果解释以获取满分。
Marks are allocated for method, accuracy, and interpretation; it is vital to write down the calculator functions you use (e.g. 'normCdf', 'LinReg') and state the values you input.
分数根据方法、准确性和解释来分配；务必写下你使用的计算器函数（例如 'normCdf', 'LinReg'）并说明所输入的值。
2. Mastering Your GDC | 掌握图形计算器
Before the exam, ensure you know how to store and retrieve functions, perform statistical tests (e.g. 2-Sample t-test, chi-squared test), find regression curves, calculate definite integrals, and solve equations numerically. Spend time practising menus and shortcuts for your specific GDC model.
考前确保你会存储和调用函数、执行统计检验（如双样本t检验、卡方检验）、求回归曲线、计算定积分以及数值求解方程。花时间熟悉你的GDC型号的菜单和快捷键。
Practice resetting your calculator's settings and clearing lists without losing essential modes like radian/degree depending on the question. Always check that your calculator is in the correct angle mode for trigonometric problems.
练习重置计算器设置和清除列表，同时保持弧度/角度等必要模式，视题目而定。对于三角问题，务必检查计算器处于正确的角度模式。
Learn to use the 'graph' se the intersect command to obtain the coordinates. This method is often faster than algebraic solving.
学会使用图形功能可视化函数。例如，要求 y = 2x² − 3x + 1 与 y = x + 2 的交点，绘制二者并用交点命令获取坐标。此法通常比代数求解更快。
3. Statistical Analysis Deep Dive | 深入统计分析
Paper 2 typically includes a data-based question. You may need to calculate mean, standard deviation, quartiles, and create box plots using your GDC. Understand how to read lists of data and produce summary statistics from them.
试卷2通常包含一个数据题。你可能需要用GDC计算均值、标准差、四分位数并制作箱形图。理解如何读取数据列表并生成汇总统计量。
When performing linear regression, interpret the correlation coefficient r and the coefficient of determination r². A strong r (close to 1 or −1) indicates a strong linear relationship; r² tells you the proportion of variation explained by the model.
进行线性回归时，解释相关系数r和决定系数r²。强r值（接近1或−1）表示强线性关系；r²表示模型解释的变异比例。
Be prepared to use the regression equation for prediction. If a question asks for an estimate outside the data range, comment on whether extrapolation is reliable and mention any limitations.
准备用回归方程进行预测。若题目要求数据范围外的估计，要评论外推的可靠性并指出局限性。
4. Probability Distributions in Action | 概率分布实战
The normal distribution is central: use your GDC to find probabilities P(X < a), P(X > b), and inverse normal values for a given cumulative probability. Remember that P(Z < z) functions often give left-tail areas; adjust accordingly for right-tail questions.
正态分布是核心：使用GDC求概率P(X < a)、P(X > b)以及给定累积概率的反向值。记住P(Z < z)函数通常给出左尾面积；右尾问题要相应调整。
t up Binomial PD (for exact) or CD (for cumulative) with parameters n, p, and x. Understand the difference between 'at least' and 'exactly'.
对于二项分布，问题常要求 P(X = k) 或 P(X ≤ k)。使用参数 n、p 和 x 设置二项式PD（精确）或CD（累积）。理解“至少”和“恰好”的区别。
In problems combining normal and binomial, define the random variable clearly. Example: 'Let X be the number of items that weigh more than 500 g.' This clarity helps both you and the examiner.
在正态与二项结合的问题中，明确定义随机变量。例如：“设X为重量超过500克的物品数量。”这种清晰性对答题和阅卷都有帮助。
5. Functions and Equation Solving | 函数与方程求解
(x) graphically or using the equation solver. Familiarise yourself with finding the coordinates of intersection points and checking if solutions belong to a specified domain.
题目可能要求通过图形方式或方程求解器求解 f(x) = g(x)。熟悉求交点坐标并验证解是否属于指定定义域。
Be comfortable with domain and range restrictions when modelling real-life situations. For instance, a function modelling height above ground cannot produce negative values; recognise these practical limits.
在模拟现实情景时，熟悉定义域和值域的限制。例如，模拟离地高度的函数不应产生负值；识别这些实际界限。
anding of f(x) → f(x − a) + b to sketch graphs quickly without relying solely on your GDC.
函数的变换（平移、伸缩、反射）经常出现。利用你对 f(x) → f(x − a) + b 的理解快速绘制草图，而不是仅依赖GDC。
6. Modelling Relationships with Functions | 用函数建立模型
You will often fit a linear, quadratic, exponential, or sinusoidal model to data. Use regression in your GDC and then employ the model for prediction or optimisation. Write down the regression equation with coefficients to at least 3 significant figures.
你常需要为数据拟合线性、二次、指数或正弦模型。使用GDC回归，然后用模型进行预测或优化。写下回归方程，系数至少保留3位有效数字。
Always check the reasonableness of extrapolated results. If a model predicts a population of −50 people, recognise the model's breakdown outside the given domain. Comment on the validity and suggest improvements.
始终检查外推结果的合理性。如果模型预测人口为−50，应认识到模型在给定定义域外失效。评述其有效性并提出改进。
Sinusoidal models often involve period, amplitude, and equilibrium. When given data with seasonal patterns, use the sinus regression or manually estimate parameters from the scatter plot, then refine using your GDC.
正弦模型常涉及周期、振幅和平衡值。当数据呈现季节性模式时，使用正弦回归或从散点图手动估计参数，再用GDC优化。
7. Calculus: Differentiation and Integration | 微积分：微分与积分
Paper 2 calculus emphasises applications: find the rate of change at a point (derivative), maximum/minimum points, and area under a curve (definite integral). The GDC can compute numerical derivatives and integrals when functions are complex.
试卷2微积分强调应用：求某点的变化率（导数）、最大/最小值点，以及曲线下面积（定积分）。当函数复杂时，GDC可计算数值导数和积分。
on problems, set up the function to maximise or minimise, find its derivative, and solve f' whether the critical point is a maximum or minimum by checking the sign change or using the second derivative.
对于优化问题，建立要最大或最小化的函数，求导后用计算器求解 f'(x) = 0。通过检查符号变化或使用二阶导数确认是极大还是极小。
Definite integrals represent areas. If the curve goes below the x-axis, the integral gives a negative value; use absolute values or split the interval to find the total area enclosed.
定积分表示面积。如果曲线位于x轴下方，积分为负值；使用绝对值或分割区间来求所围总面积。
8. Trigonometry and Circular Functions | 三角学与圆函数
ons to navigation, surveying, or construction. Draw a clear diagram and label all known sides and angles.
你可能会遇到非直角三角形的正弦和余弦定理、三角形面积公式½ab sinC，以及在航海、测量或建筑中的应用。画出清晰的示意图并标出所有已知边和角。
ese components and use GDC to find intersections, roots, or maximum points over a given interval.
对于圆函数，理解振幅、周期和相位偏移。对于 y = A sin(B(x − C)) + D，识别各分量并用GDC求指定区间内的交点、零点或最大值。
When solving trigonometric equations in Paper 2, use your calculator to find the primary solution, then apply the symmetry of the unit circle or the CAST diagram to locate all solutions in the required domain.
在试卷2中解三角方程时，用计算器求出主解，然后利用单位圆对称性或CAST图找到所需定义域内的所有解。
9. Exam Strategy and Time Management | 考试策略与时间管理
Read each question carefully; underline key information and what is asked. Look for command terms like 'find', 'hence', 'interpret', or 'comment on'. Do not spend more than one minute per mark as a rough guide.
仔细阅读每一题；划出关键信息和所求。留意指令词如“求”、“由此”、“解释”或“评论”。大致上，每一分不要花费超过一分钟。
If a part seems too time-consuming, move on and return later. Sometimes later parts provide hints or can be solved independently. Always have an attempt at every question — blank answers earn zero marks.
若某小问过于耗时，跳过并稍后回头。有时后面的小问会提供线索或可独立求解。每题都要尝试作答——空白必得零分。
Show all working, even if obtained from GDC, to secure method marks. Write down 'By GDC: intersection at (2.35, 4.78)', the function entered, and the window settings if relevant.
展示所有解题步骤，即使来自GDC，以获取方法分。写下“由GDC：交点 (2.35, 4.78)”，并写出输入的函数和窗口设置（如相关）。
10. Avoiding Common Pitfalls | 避免常见错误
Misinterpreting probability phrasing: 'at least 5' is P(X ≥ 5), not P(X = 5); 'more than 10' excludes the boundary. Use GDC cumulative functions with care.
误解概率表述：“至少5个”是 P(X ≥ 5)，而非 P(X = 5)；“多于10个”不包括边界。谨慎使用GDC累积函数。
Forgetting to switch to radian mode for trigonometric calculus; when differentiating sin(2x) or integrating cos(x²), radian mode is required. An incorrect mode can lead to completely wrong answers.
忘记在三角微积分中切换至弧度模式；当对 sin(2x) 求导或积分 cos(x²) 时，必须使用弧度模式。模式错误会导致答案全错。
Rounding too early in calculations. Keep intermediate values stored in calculator memory and round only the final answer to the required degree of accuracy (usually 3 significant figures).
计算中过早舍入。将中间值存储在计算器内存，仅将最终答案按要求精确度（通常3位有效数字）舍入。
11. Practice Questions and Solutions | 练习题与解析
Question 1: The weight of apples from an orchard follows a normal distribution with mean 150 g and standard deviation 20 g. Find the probability that a randomly selected apple weighs more than 170 g.
问题1：某果园苹果的重量服从正态分布，均值为150克，标准差20克。求随机选取一个苹果重量超过170克的概率。
Solution: Enter normCdf (170, 1E99, 150, 20) on GDC. The answer is approximately 0.1587. Therefore, about 15.9% of apples exceed 170 g.
解法：在GDC中输入 normCdf (170, 1E99, 150, 20)。答案约为0.1587。因此约15.9%的苹果超过170克。
+ 1.5. Use your GDC to find when the ball hits the ground and the maximum height reached.
问题2：一个球被抛出，其在t秒后的高度h（米）由 h(t) = −4.9t² + 20t + 1.5 建模。用GDC求球何时落地及达到的最大高度。
maximum, find vertex: t = −20/(2×(−4.9)) ≈ 2.04 s, h(2.04) ≈ 21.9 m. Alternately, use GDC's maximum function on the graph.
解法：使用方程求解器解 −4.9t² + 20t + 1.5 = 0 得 t > 0，t ≈ 4.16 秒。最大值：求顶点，t = −20/(2×(−4.9)) ≈ 2.04 s，h(2.04) ≈ 21.9 m。也可用GDC图形最大功能。
Published by TutorHao | IB Math AI SL Revision Series | aleveler.com
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