学习目标
- Recognise, explain, and apply “Function properties”.
- Recognise, explain, and apply “Derivatives”.
- Recognise, explain, and apply “Curve sketching”.
- Recognise, explain, and apply “Optimisation”.
- Recognise, explain, and apply “Integrals”.
Abitur / Mathematics / Curriculum
Analysis
Analysis: structured theory, worked examples, answered practice, and a mastery checklist for Abitur.
章节计划
学习、练习并检查进度
按顺序完成步骤，或直接跳到你需要的部分。
—/5
已完成
预计主动学习时间
209 分钟
1. 理解
学习目标、核心概念与结构化理论。
打开 →2. 掌握方法
逐步说明的完整例题。
打开 →3. 练习
含提示与答案的分级练习。
打开 →4. 巩固
从基础到挑战的课后任务。
打开 →5. 检查
含计时、分值与解答的章节测试。
打开 →
进度只保存在当前浏览器中，不需要 AI credits。
核心概念与检查
- I connect limit, derivative, and integral with a graph.
- I apply rules with the correct conditions.
- Every solution or explanation for “Analysis” should include a method, justification, and final check.
单元
核心理论
练习前按清晰顺序掌握本章核心概念。
Change and approximation
Calculus studies how a quantity changes and how small changes accumulate. Graphical interpretation should accompany the notation.
Conditions before the rule
Check domain, continuity, or differentiability where required. A correct rule cannot be applied mechanically outside its conditions.
Interpret the result
A derivative represents local rate of change and a definite integral represents net accumulation. Always reconnect the result to the original problem.
Function properties
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Derivatives
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Curve sketching
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Optimisation
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Integrals
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
例题讲解
逐步查看方法，并理解每一步为何成立。
Worked rate-of-change example
For f(x) = x³ - 3x, find the derivative and the gradient at x = 3.
- 1Differentiate term by term: f'(x) = 3x² - 3.
- 2Substitute x = 3.
- 3Interpret the value as the tangent gradient.
f'(x) = 3x² - 3, f'(3) = 24
Reasoning example
Before calculating, explain the key idea from “Function properties” and which conditions must be checked.
- 1Define the idea in one clear sentence.
- 2Connect it to a representation, law, or formula.
- 3State a restriction, unit, or final check that makes the solution valid.
The answer should show not only which rule is used for “Function properties”, but also why it is valid here.
Analysis
含答案练习
八道分层练习，从基础熟练到考试应用。请先独立完成，再查看提示或答案。
基础熟练4 分钟
For f(x)=x²+6x, find f'(x).
提示
Use the power rule term by term.
答案指南
f'(x)=2x+6.
基础熟练5 分钟
Find the gradient of f(x)=x²+9x at x=4.
提示
Differentiate, then substitute.
答案指南
f'(4)=2·4+9=17.
应用6 分钟
Find ∫(5x+6)dx.
提示
Increase the power by 1 and divide by the new power.
答案指南
2.5x²+6x+C.
应用7 分钟
Evaluate ∫₀^3 8x dx.
提示
Find an antiderivative and apply upper minus lower limit.
答案指南
4·3² = 36.
推理8 分钟
Find the stationary point of f(x)=x²-10x+4.
提示
Set f'(x)=0, then find y.
答案指南
x=5, y=-21=-21.
推理9 分钟
If ds/dt=7t+2 and s(0)=9, find s(t).
提示
Integrate and use the initial condition.
答案指南
s(t)=3.5t²+2t+9.
考试题型10 分钟
Check whether F(x)=1.5x²+4x is an antiderivative of f(x)=3x+4.
提示
Differentiate F.
答案指南
F'(x)=3x+4=f(x), so yes.
考试题型12 分钟
A learner writes d(x³)/dx=3x. Correct it and explain the rule.
提示
The power rule reduces the exponent by 1.
答案指南
d(x³)/dx=3x².
常见错误
- Applying a rule without checking domain or continuity.
- Omitting the integration constant in an indefinite integral.
- Giving a numerical result without graphical or physical interpretation.
掌握度检查
- I connect limit, derivative, and integral with a graph.
- I apply rules with the correct conditions.
- I interpret the result in context.
Analysis
章节作业
六道不同作业，从基础到挑战，包含预计用时、提示和答案指南。
基础8 分钟
Homework 1: For f(x)=x²+9x, find f'(x).
提示
Use the power rule term by term.
答案指南
f'(x)=2x+9.
基础10 分钟
Homework 2: Find the gradient of f(x)=x²+5x at x=5.
提示
Differentiate, then substitute.
答案指南
f'(5)=2·5+5=15.
进阶12 分钟
Homework 3: Find ∫(8x+2)dx.
提示
Increase the power by 1 and divide by the new power.
答案指南
4x²+2x+C.
进阶15 分钟
Homework 4: Evaluate ∫₀^4 4x dx.
提示
Find an antiderivative and apply upper minus lower limit.
答案指南
2·4² = 32.
挑战18 分钟
Homework 5: Find the stationary point of f(x)=x²-12x+7.
提示
Set f'(x)=0, then find y.
答案指南
x=6, y=-29=-29.
挑战20 分钟
Homework 6: If ds/dt=3t+3 and s(0)=6, find s(t).
提示
Integrate and use the initial condition.
答案指南
s(t)=1.5t²+3t+6.
50 分钟 / 40 分
完整章节测试
带计时器、完整分值和参考答案的章节自测。
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. For f(x)=x²+7x, find f'(x).
2 分答案指南
f'(x)=2x+7.
2. Find the gradient of f(x)=x²+3x at x=5.
3 分答案指南
f'(5)=2·5+3=13.
3. Find ∫(6x+2)dx.
3 分答案指南
3x²+2x+C.
4. Evaluate ∫₀^4 9x dx.
4 分答案指南
4.5·4² = 72.
5. Find the stationary point of f(x)=x²-12x+5.
4 分答案指南
x=6, y=-31=-31.
6. If ds/dt=8t+3 and s(0)=11, find s(t).
4 分答案指南
s(t)=4t²+3t+11.
7. Check whether F(x)=2x²+5x is an antiderivative of f(x)=4x+5.
5 分答案指南
F'(x)=4x+5=f(x), so yes.
8. A learner writes d(x³)/dx=3x. Correct it and explain the rule.
5 分答案指南
d(x³)/dx=3x².
9. Model a rate of change connected with “Trigonometric models” and explain derivative units.
5 分答案指南
A full response defines independent/dependent variables, gives a function or data, and interprets the derivative as output units per input unit.
10. Combine differentiation and integration in an exam-style problem about “Differential equations where required” and check the result.
5 分答案指南
A full solution defines variables/units, uses a derivative for instantaneous rate, an integral for accumulated change, and checks their inverse relationship.
单元
官方来源与核验
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
Abitur: 数学与物理的结构化路线，包含章节、练习和下一步。
本章覆盖内容
结构跟随教材，用来帮助组织学习。
Function properties
Derivatives
Curve sketching
Optimisation
Integrals
Area/accumulation
Exponential models
Logarithmic models
Trigonometric models
Differential equations where required
重点关注
通常容易出错或需要更多复习的部分。
I connect limit, derivative, and integral with a graph.
I apply rules with the correct conditions.
I interpret the result in context.
来源、日常资料与学习资源
从教材、daily material、PDFs、videos 和 worked examples 开始。
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
按小主题练习
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
For f(x)=x²+6x, find f'(x).
Find the gradient of f(x)=x²+9x at x=4.
Find ∫(5x+6)dx.
Evaluate ∫₀^3 8x dx.
Find the stationary point of f(x)=x²-10x+4.
If ds/dt=7t+2 and s(0)=9, find s(t).
Check whether F(x)=1.5x²+4x is an antiderivative of f(x)=3x+4.
A learner writes d(x³)/dx=3x. Correct it and explain the rule.
Mocks 与进度检查
如何衡量本章进度，以及何时加入综合 mock。
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
下一步
完成本章后该做什么，以及它如何连接到下一单元。
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
说明：每学年的正式考试范围请始终以学校、老师和当前官方来源为准。