Panhellenic Exams / Mathematics / Curriculum
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
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 = 5.
f'(x) = 3x² - 2, f'(5) = 73
Before calculating, explain the key idea from “Αρχική συνάρτηση στο τμήμα που προβλέπεται” and which conditions must be checked.
The answer should show not only which rule is used for “Αρχική συνάρτηση στο τμήμα που προβλέπεται”, but also why it is valid here.
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός
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+4x, find f'(x).
Use the power rule term by term.
f'(x)=2x+4.
Find the gradient of f(x)=x²+7x at x=6.
Differentiate, then substitute.
f'(6)=2·6+7=19.
Find ∫(3x+3)dx.
Increase the power by 1 and divide by the new power.
1.5x²+3x+C.
Evaluate ∫₀^5 6x dx.
Find an antiderivative and apply upper minus lower limit.
3·5² = 75.
Find the stationary point of f(x)=x²-4x+9.
Set f'(x)=0, then find y.
x=2, y=5=5.
If ds/dt=5t+4 and s(0)=9, find s(t).
Integrate and use the initial condition.
s(t)=2.5t²+4t+9.
Check whether F(x)=4x²+6x is an antiderivative of f(x)=8x+6.
Differentiate F.
F'(x)=8x+6=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.
Κεφάλαιο 3: Ολοκληρωτικός Λογισμός
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+7x, find f'(x).
Use the power rule term by term.
f'(x)=2x+7.
Homework 2: Find the gradient of f(x)=x²+3x at x=2.
Differentiate, then substitute.
f'(2)=2·2+3=7.
Homework 3: Find ∫(6x+4)dx.
Increase the power by 1 and divide by the new power.
3x²+4x+C.
Homework 4: Evaluate ∫₀^6 9x dx.
Find an antiderivative and apply upper minus lower limit.
4.5·6² = 162.
Homework 5: Find the stationary point of f(x)=x²-6x+5.
Set f'(x)=0, then find y.
x=3, y=-4=-4.
Homework 6: If ds/dt=8t+5 and s(0)=13, find s(t).
Integrate and use the initial condition.
s(t)=4t²+5t+13.
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+5x, find f'(x).
2 分f'(x)=2x+5.
2. Find the gradient of f(x)=x²+8x at x=2.
3 分f'(2)=2·2+8=12.
3. Find ∫(4x+4)dx.
3 分2x²+4x+C.
4. Evaluate ∫₀^6 7x dx.
4 分3.5·6² = 126.
5. Find the stationary point of f(x)=x²-6x+3.
4 分x=3, y=-6=-6.
6. If ds/dt=6t+5 and s(0)=11, find s(t).
4 分s(t)=3t²+5t+11.
7. Check whether F(x)=4.5x²+2x is an antiderivative of f(x)=9x+2.
5 分F'(x)=9x+2=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 “Ορισμένο ολοκλήρωμα” 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 “Ιδιότητες ολοκληρώματος” 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
Panhellenic Exams: 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。
