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IB Diploma DP1 / Mathematics / Curriculum

Calculus introduction

Calculus introduction: structured theory, worked examples, answered practice, and a mastery checklist for IB Diploma DP1.

Цели обучения

Recognise, explain, and apply “Limits in route-appropriate depth”.
Recognise, explain, and apply “Differentiation”.
Recognise, explain, and apply “Tangents and normals”.
Recognise, explain, and apply “Optimisation introduction”.
Recognise, explain, and apply “Integration introduction”.

Ключевые идеи и проверки

I connect limit, derivative, and integral with a graph.
I apply rules with the correct conditions.
Every solution or explanation for “Calculus introduction” should include a method, justification, and final check.

Раздел

Основная теория

Ключевые идеи главы в понятной последовательности перед практикой.

01

Change and approximation

Calculus studies how a quantity changes and how small changes accumulate. Graphical interpretation should accompany the notation.

02

Conditions before the rule

Check domain, continuity, or differentiability where required. A correct rule cannot be applied mechanically outside its conditions.

03

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.

04

Limits in route-appropriate depth

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.

05

Differentiation

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.

06

Tangents and normals

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.

07

Optimisation introduction

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.

08

Integration introduction

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

Разобранные примеры

Следуйте методу шаг за шагом и проверяйте обоснование каждого шага.

Worked rate-of-change example

For f(x) = x³ - 5x, find the derivative and the gradient at x = 4.

1Differentiate term by term: f'(x) = 3x² - 5.
2Substitute x = 4.
3Interpret the value as the tangent gradient.

f'(x) = 3x² - 5, f'(4) = 43

Reasoning example

Before calculating, explain the key idea from “Limits in route-appropriate depth” 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 “Limits in route-appropriate depth”, but also why it is valid here.

Calculus introduction

Практика с ответами

Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.

Базовый навык4 минут

For f(x)=x²+5x, find f'(x).

Подсказка

Use the power rule term by term.

Ответ

f'(x)=2x+5.

Базовый навык5 минут

Find the gradient of f(x)=x²+8x at x=5.

Подсказка

Differentiate, then substitute.

Ответ

f'(5)=2·5+8=18.

Применение6 минут

Find ∫(4x+2)dx.

Подсказка

Increase the power by 1 and divide by the new power.

Ответ

2x²+2x+C.

Применение7 минут

Evaluate ∫₀^4 7x dx.

Подсказка

Find an antiderivative and apply upper minus lower limit.

Ответ

3.5·4² = 56.

Обоснование8 минут

Find the stationary point of f(x)=x²-12x+3.

Подсказка

Set f'(x)=0, then find y.

Ответ

x=6, y=-33=-33.

Обоснование9 минут

If ds/dt=6t+3 and s(0)=9, find s(t).

Подсказка

Integrate and use the initial condition.

Ответ

s(t)=3t²+3t+9.

Экзаменационный10 минут

Check whether F(x)=4.5x²+5x is an antiderivative of f(x)=9x+5.

Подсказка

Differentiate F.

Ответ

F'(x)=9x+5=f(x), so yes.

Экзаменационный12 минут

A learner writes d(x³)/dx=3x. Correct it and explain the rule.

Подсказка

The power rule reduces the exponent by 1.

Ответ

d(x³)/dx=3x².

Частые ошибки

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.

Проверка понимания

I connect limit, derivative, and integral with a graph.
I apply rules with the correct conditions.
I interpret the result in context.

Calculus introduction

Домашняя работа по главе

Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.

Базовый8 минут

Homework 1: For f(x)=x²+8x, find f'(x).

Подсказка

Use the power rule term by term.

Ответ

f'(x)=2x+8.

Базовый10 минут

Homework 2: Find the gradient of f(x)=x²+4x at x=6.

Подсказка

Differentiate, then substitute.

Ответ

f'(6)=2·6+4=16.

Углубленный12 минут

Homework 3: Find ∫(7x+3)dx.

Подсказка

Increase the power by 1 and divide by the new power.

Ответ

3.5x²+3x+C.

Углубленный15 минут

Homework 4: Evaluate ∫₀^5 3x dx.

Подсказка

Find an antiderivative and apply upper minus lower limit.

Ответ

1.5·5² = 37.5.

Сложный18 минут

Homework 5: Find the stationary point of f(x)=x²-4x+6.

Подсказка

Set f'(x)=0, then find y.

Ответ

x=2, y=2=2.

Сложный20 минут

Homework 6: If ds/dt=9t+4 and s(0)=13, find s(t).

Подсказка

Integrate and use the initial condition.

Ответ

s(t)=4.5t²+4t+13.

50 минут / 40 баллов

Полный тест по главе

Полный тест с таймером, баллами и образцами ответов.

Start the timer when ready, work without notes, show every step, and open model answers only after finishing.

1. For f(x)=x²+6x, find f'(x).

2 баллов

Ответ

f'(x)=2x+6.

2. Find the gradient of f(x)=x²+9x at x=6.

3 баллов

Ответ

f'(6)=2·6+9=21.

3. Find ∫(5x+3)dx.

3 баллов

Ответ

2.5x²+3x+C.

4. Evaluate ∫₀^5 8x dx.

4 баллов

Ответ

4·5² = 100.

5. Find the stationary point of f(x)=x²-4x+4.

4 баллов

Ответ

x=2, y=0=0.

6. If ds/dt=7t+4 and s(0)=11, find s(t).

4 баллов

Ответ

s(t)=3.5t²+4t+11.

7. Check whether F(x)=1.5x²+6x is an antiderivative of f(x)=3x+6.

5 баллов

Ответ

F'(x)=3x+6=f(x), so yes.

8. A learner writes d(x³)/dx=3x. Correct it and explain the rule.

5 баллов

Ответ

d(x³)/dx=3x².

9. Model a rate of change connected with “Optimisation introduction” and explain derivative units.

5 баллов

Ответ

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 “Integration introduction” and check the result.

5 баллов

Ответ

A full solution defines variables/units, uses a derivative for instantaneous rate, an integral for accumulated change, and checks their inverse relationship.

Раздел

Официальные источники

Источники программы; порядок тем уточняйте в школе и у преподавателя.

UK Department for Education - MathematicsOfficial national curriculum progression and attainment guidance.International Baccalaureate - DP curriculumOfficial DP subject structure and current subject briefs.

Mathematics

IB Diploma DP1: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.

Назад к предмету

Что покрывает глава

Структура следует учебнику и помогает организовать изучение темы.

Limits in route-appropriate depth

Differentiation

Tangents and normals

Optimisation introduction

Integration introduction

Где сфокусироваться

Темы, где чаще возникают ошибки или нужна дополнительная практика.

I connect limit, derivative, and integral with a graph.

I apply rules with the correct conditions.

I interpret the result in context.

Источники, ежедневные материалы и ресурсы

С чего начать: учебник, daily material, PDFs, videos и worked examples.

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.

Практика по подтемам

Целевая практика перед full tests, чтобы coverage было понятным.

For f(x)=x²+5x, find f'(x).

Find the gradient of f(x)=x²+8x at x=5.

Find ∫(4x+2)dx.

Evaluate ∫₀^4 7x dx.

Find the stationary point of f(x)=x²-12x+3.

If ds/dt=6t+3 and s(0)=9, find s(t).

Check whether F(x)=4.5x²+5x is an antiderivative of f(x)=9x+5.

A learner writes d(x³)/dx=3x. Correct it and explain the rule.

Mocks и проверка прогресса

Как измерять прогресс по главе и когда включать ее в общий 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.

Следующий шаг

Что делать после главы и как она связана со следующим разделом.

Complete the practice without support.

Explain the core method aloud in under two minutes.

Continue to the next chapter or request targeted tutor support.

Примечание: официальную экзаменационную программу каждого года всегда уточняйте в школе, у преподавателя и по текущим официальным источникам.