IB Diploma DP2 / Mathematics / Curriculum
Advanced route content: structured theory, worked examples, answered practice, and a mastery checklist for IB Diploma DP2.
单元
练习前按清晰顺序掌握本章核心概念。
Calculus studies how a quantity changes and how small changes accumulate. Graphical interpretation should accompany the notation.
Check domain, continuity, or differentiability where required. A correct rule cannot be applied mechanically outside its conditions.
A derivative represents local rate of change and a definite integral represents net accumulation. Always reconnect the result to the original problem.
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.
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.
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.
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.
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
逐步查看方法，并理解每一步为何成立。
For f(x) = x³ - 2x, find the derivative and the gradient at x = 3.
f'(x) = 3x² - 2, f'(3) = 25
Before calculating, explain the key idea from “AA:” and which conditions must be checked.
The answer should show not only which rule is used for “AA:”, but also why it is valid here.
Advanced route content
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。
For f(x)=x²+8x, find f'(x).
Use the power rule term by term.
f'(x)=2x+8.
Find the gradient of f(x)=x²+4x at x=4.
Differentiate, then substitute.
f'(4)=2·4+4=12.
Find ∫(7x+6)dx.
Increase the power by 1 and divide by the new power.
3.5x²+6x+C.
Evaluate ∫₀^3 3x dx.
Find an antiderivative and apply upper minus lower limit.
1.5·3² = 13.5.
Find the stationary point of f(x)=x²-10x+6.
Set f'(x)=0, then find y.
x=5, y=-19=-19.
If ds/dt=9t+2 and s(0)=11, find s(t).
Integrate and use the initial condition.
s(t)=4.5t²+2t+11.
Check whether F(x)=2.5x²+4x is an antiderivative of f(x)=5x+4.
Differentiate F.
F'(x)=5x+4=f(x), so yes.
A learner writes d(x³)/dx=3x. Correct it and explain the rule.
The power rule reduces the exponent by 1.
d(x³)/dx=3x².
Advanced route content
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。
Homework 1: For f(x)=x²+4x, find f'(x).
Use the power rule term by term.
f'(x)=2x+4.
Homework 2: Find the gradient of f(x)=x²+7x at x=5.
Differentiate, then substitute.
f'(5)=2·5+7=17.
Homework 3: Find ∫(3x+2)dx.
Increase the power by 1 and divide by the new power.
1.5x²+2x+C.
Homework 4: Evaluate ∫₀^4 6x dx.
Find an antiderivative and apply upper minus lower limit.
3·4² = 48.
Homework 5: Find the stationary point of f(x)=x²-12x+9.
Set f'(x)=0, then find y.
x=6, y=-27=-27.
Homework 6: If ds/dt=5t+3 and s(0)=8, find s(t).
Integrate and use the initial condition.
s(t)=2.5t²+3t+8.
50 分钟 / 40 分
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+9x, find f'(x).
2 分f'(x)=2x+9.
2. Find the gradient of f(x)=x²+5x at x=5.
3 分f'(5)=2·5+5=15.
3. Find ∫(8x+2)dx.
3 分4x²+2x+C.
4. Evaluate ∫₀^4 4x dx.
4 分2·4² = 32.
5. Find the stationary point of f(x)=x²-12x+7.
4 分x=6, y=-29=-29.
6. If ds/dt=3t+3 and s(0)=6, find s(t).
4 分s(t)=1.5t²+3t+6.
7. Check whether F(x)=3x²+5x is an antiderivative of f(x)=6x+5.
5 分F'(x)=6x+5=f(x), so yes.
8. A learner writes d(x³)/dx=3x. Correct it and explain the rule.
5 分d(x³)/dx=3x².
9. Model a rate of change connected with “Graph theory or route-specific topics where applicable” and explain derivative units.
5 分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 “Regression and statistics” and check the result.
5 分A full solution defines variables/units, uses a derivative for instantaneous rate, an integral for accumulated change, and checks their inverse relationship.
单元
课程参考来源；教学顺序请始终与学校和老师确认。
Mathematics
IB Diploma DP2: 数学与物理的结构化路线，包含章节、练习和下一步。
结构跟随教材，用来帮助组织学习。
通常容易出错或需要更多复习的部分。
从教材、daily material、PDFs、videos 和 worked examples 开始。
进入 full tests 前先做 targeted practice，让覆盖情况更清楚。
如何衡量本章进度，以及何时加入综合 mock。
完成本章后该做什么，以及它如何连接到下一单元。
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。
