Panhellenic Exams / Mathematics / Curriculum
Κεφάλαιο 2: Διαφορικός Λογισμός: structured theory, worked examples, answered practice, and a mastery checklist for Panhellenic Exams.
CHAPTER PLAN
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Estimated active study time
209 minutes
Objectives, key ideas, and structured theory.
Open →Worked examples that explain every step.
Open →Graded tasks with hints and answer guidance.
Open →Assignments from core work to challenge.
Open →A timed, marked test with solution guidance.
Open →Progress is stored only in this browser and does not require AI credits.
Unit
The essential chapter ideas in a clear sequence before practice.
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
Follow the method step by step and check why every step is valid.
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 “Έννοια παραγώγου” and which conditions must be checked.
The answer should show not only which rule is used for “Έννοια παραγώγου”, but also why it is valid here.
Κεφάλαιο 2: Διαφορικός Λογισμός
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
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=6.
Differentiate, then substitute.
f'(6)=2·6+4=16.
Find ∫(7x+3)dx.
Increase the power by 1 and divide by the new power.
3.5x²+3x+C.
Evaluate ∫₀^5 3x dx.
Find an antiderivative and apply upper minus lower limit.
1.5·5² = 37.5.
Find the stationary point of f(x)=x²-4x+6.
Set f'(x)=0, then find y.
x=2, y=2=2.
If ds/dt=9t+4 and s(0)=13, find s(t).
Integrate and use the initial condition.
s(t)=4.5t²+4t+13.
Check whether F(x)=2.5x²+6x is an antiderivative of f(x)=5x+6.
Differentiate F.
F'(x)=5x+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 2: Διαφορικός Λογισμός
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
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=2.
Differentiate, then substitute.
f'(2)=2·2+7=11.
Homework 3: Find ∫(3x+4)dx.
Increase the power by 1 and divide by the new power.
1.5x²+4x+C.
Homework 4: Evaluate ∫₀^6 6x dx.
Find an antiderivative and apply upper minus lower limit.
3·6² = 108.
Homework 5: Find the stationary point of f(x)=x²-6x+9.
Set f'(x)=0, then find y.
x=3, y=0=0.
Homework 6: If ds/dt=5t+5 and s(0)=10, find s(t).
Integrate and use the initial condition.
s(t)=2.5t²+5t+10.
50 minutes / 40 marks
A timed, full-mark self-assessment with model-answer guidance.
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 marksf'(x)=2x+9.
2. Find the gradient of f(x)=x²+5x at x=2.
3 marksf'(2)=2·2+5=9.
3. Find ∫(8x+4)dx.
3 marks4x²+4x+C.
4. Evaluate ∫₀^6 4x dx.
4 marks2·6² = 72.
5. Find the stationary point of f(x)=x²-6x+7.
4 marksx=3, y=-2=-2.
6. If ds/dt=3t+5 and s(0)=8, find s(t).
4 markss(t)=1.5t²+5t+8.
7. Check whether F(x)=3x²+2x is an antiderivative of f(x)=6x+2.
5 marksF'(x)=6x+2=f(x), so yes.
8. A learner writes d(x³)/dx=3x. Correct it and explain the rule.
5 marksd(x³)/dx=3x².
9. Model a rate of change connected with “Μονοτονία” and explain derivative units.
5 marksA 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 “Τοπικά ακρότατα” and check the result.
5 marksA full solution defines variables/units, uses a derivative for instantaneous rate, an integral for accumulated change, and checks their inverse relationship.
Unit
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Panhellenic strategy: theory, methodology, timing, grading, and continuous correction.
The structure follows the official textbook layout and is used to organise study.
The areas that usually create mistakes or need extra revision.
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Targeted practice before full tests so coverage is clear.
How to measure progress in this chapter and when it enters a cumulative mock.
What to do after finishing the chapter and how it connects to the next unit.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.
