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  • Welcome to the CP Subject Centre
  • Faria Education Group
  • Core
    • Service Learning
    • Language Development
    • Reflective Project
    • Personal and Professional Skills
  • Career-related Study
  • Studies in language and literature
    • Language and Literature
      • Aims & Objectives
      • Syllabus
        • Area of Exploration - Readers, Writers and Texts
        • Area of Exploration - Time and Space
        • Area of Exploration - Intertextuality: Connecting Texts
        • Development of Linguistic Skills
        • Conceptual Understanding
        • Non-Literary Texts
      • Assessment
    • Literature
      • Aims and Objectives
      • Syllabus
        • Areas of Exploration
        • Development of Linguistic Skills
        • Conceptual Understanding
      • Assessment
    • Classical Languages (Last Assessment 2023)
      • Aims & Objectives
      • Syllabus
        • Part 1: Study of Language
        • Part 2: Study of Literature
        • Part 3: Individual study
      • Assessment
    • Classical Languages (First Assessment 2024)
    • Literature and Performance (First Assessment 2024)
  • Language acquisition
    • Language Ab Initio
      • Aims & Objectives
      • Syllabus
        • Identities
        • Experiences
        • Human Ingenuity
        • Social Organization
        • Sharing the Planet
        • Texts
      • Assessment
    • Language B
      • Aims & Objectives
      • Syllabus
        • Identities
        • Experiences
        • Human Ingenuitiy
        • Social Organization
        • Sharing the Planet
        • Texts
      • Assessment
  • Individuals and societies
    • Business Management (Last Assessment 2023)
      • Aims & Objectives
      • Syllabus
        • Unit 1: Business Organization and Environment
        • Unit 2: Human Resource Management
        • Unit 3: Finance and Accounts
        • Unit 4: Marketing
        • Unit 5: Operations Management
      • Assessment
    • Business Management (First Assessment 2024)
    • Digital Society
    • Economics
      • Aims & Objectives
      • Syllabus
        • Unit 1: Introduction to Economics
        • Unit 2: Microeconomics
          • Real-world issue 1
          • Real-world issue 2
        • Unit 3: Macroeconomics
          • Real-world issue 1
          • Real-world issue 2
        • Unit 4: The Global Economy
          • Real-world issue 1
          • Real-world issue 2
      • Assessment
    • Geography
      • Aims & Objectives
      • Syllabus
        • Part one: Geographic themes (SL and HL options)
          • Option A: Freshwater
          • Option B: Oceans and coastal margins
          • Option C: Extreme environments
          • Option D: Geophysical hazards
          • Option E: Leisure, tourism and sport
          • Option F: Food and health
          • Option G: Urban environments
        • Part two: Geographic perspectives - global change (SL and HL core)
          • Unit 1: Changing population
          • Unit 2: Global climate - vulnerability and resilience
          • Unit 3: Global resource consumption and security
        • Part two: Geographic perspectives - global change (HL core extension)
          • Unit 4: Power, places and networks
          • Unit 5: Human development and diversity
          • Unit 6: Global risks and resilience
      • Assessment
    • Global Politics
      • Aims & Objectives
      • Syllabus
        • Engagement Activity
        • Core
        • Additional Higher Level
      • Assessment
    • History
      • Aims & Objectives
      • Syllabus
        • Prescribed Subjects
        • World History Topics
        • HL Options: Depth Studies - History of Africa and the Middle East
        • HL Options: Depth Studies - History of the Americas
        • HL Options: Depth Studies - History of Asia and Oceania
        • HL Options: Depth Studies - History of Europe
      • Assessment
    • Information Technology in a Global Society
      • Aims & Objectives
      • Syllabus
        • Strand 1: Social and Ethical Significance
        • Strand 2: Application to Specific Scenarios
        • Strand 3: IT Systems
        • The Project (practical application of IT skills)
      • Assessment
    • Philosophy
      • Aims & Objectives
      • Syllabus
        • Prescribed Texts
        • Core
        • Additional Higher Level
        • Options
      • Assessment
    • Psychology
      • Aims & Objectives
      • Syllabus
        • Core
          • Biological approach to understanding behaviour
          • Cognitive approach to understanding behaviour
          • Sociocultural approach to understanding behaviour
          • Approaches to researching behaviour
        • Options
          • Abnormal psychology
          • Developmental psychology
          • Health psychology
          • Psychology of human relationships
      • Assessment
    • Social and Cultural Anthropology
      • Aims & Objectives
      • Syllabus
        • Part 1: Engaging with Anthropology
          • The language of anthropology
          • The practice of anthropology
          • Anthropological thinking
        • Part 2: Engaging with Ethnography
          • Group 1
          • Group 2
          • Group 3
        • Part 3: Engaging with Anthropological Practice
      • Assessment
    • World Religions
      • Aims & Objectives
      • Syllabus
        • Part 1: Introduction to World Religions
        • Part 2: In-depth Studies
        • Part 3: Internal Assessment
      • Assessment
  • Sciences
    • Biology (FA 2025)
    • Biology (LA 2024)
      • Aims & Objectives
      • Syllabus
        • Core
          • 1. Cell biology
          • 2. Molecular biology
          • 3. Genetics
          • 4. Ecology
          • 5. Evolution and biodiversity
          • 6. Human physiology
        • Additional Higher Level
          • 7. Nucleic acids
          • 8. Metabolism, cell respiration and photosynthesis
          • 9. Plant biology
          • 10. Genetics and evolution
          • 11. Animal physiology
        • Options
          • Option A: Neurobiology and behaviour
          • Option B: Biotechnology and bioinformatics
          • Option C: Ecology and conservation
          • Option D: Human physiology
      • Assessment
    • Chemistry (FA 2025)
    • Chemistry (LA 2024)
      • Aims & Objectives
      • Syllabus
        • Core
          • 1. Stoichiometric relationships
          • 2. Atomic structure
          • 3. Periodicity
          • 4. Chemical bonding and structure
          • 5. Energetics/thermochemistry
          • 6. Chemical kinetics
          • 7. Equilibrium
          • 8. Acids and bases
          • 9. Redox processes
          • 10. Organic chemistry
          • 11. Measurement and data processing
        • Additional Higher Level
          • 12. Atomic structure
          • 13. The periodic table - the transition metals
          • 14. Chemical bonding and structure
          • 15. Energetics/thermochemistry
          • 16. Chemical kinetics
          • 17. Equilibrium
          • 18. Acids and bases
          • 19. Redox processes
          • 20. Organic chemistry
          • 21. Measurement and analysis
        • Options
          • A. Materials
          • B. Biochemistry
          • C. Energy
          • D. Medicinal chemistry
      • Assessment
    • Computer Science (FA 2025)
    • Computer Science (LA 2024)
      • Aims & Objectives
      • Syllabus
        • Core
          • Topic 1 - System fundamentals
          • Topic 2 - Computer organization
          • Topic 3 - Networks
          • Topic 4 - Computational thinking, problem-solving and programming
        • Additional Higher Level
          • Topic 5 - Abstract data structures
          • Topic 6 - Resource management
          • Topic 7 - Control
        • Options
          • A - Databases
          • B - Modelling and simulation
          • C - Web science
          • D - Object-oriented programming
      • Assessment
    • Design Technology
      • Aims & Objectives
      • Syllabus
        • Core
          • 1. Human factors and ergonomics
          • 2. Resource management and sustainable production
          • 3. Modelling
          • 4. Final production
          • 5. Innovation and design
          • 6. Classic design
        • Additional Higher Level
          • 7. User-centred design (UCD)
          • 8. Sustainability
          • 9. Innovation and markets
          • 10. Commercial production
      • Assessment
    • Nature of Science
      • Aims & Objectives
      • Syllabus
        • Introduction
        • Part A - Concepts
        • Part B - The Quest for Understanding
        • Part C - The Impact of Science
        • Part D - Challenges and the Future
      • Assessment
    • Physics (FA 2025)
    • Physics (LA 2024)
      • Aims & Objectives
      • Syllabus
        • Core
          • 1. Measurements and uncertainties
          • 2. Mechanics
          • 3. Thermal physics
          • 4. Waves
          • 5. Electricity and magnetism
          • 6. Circular motion and gravitation
          • 7. Atomic, nuclear and particle physics
          • 8. Energy production
        • Additional Higher Level
          • 9. Wave phenomena
          • 10. Fields
          • 11. Electromagnetic induction
          • 12. Quantum and nuclear physics
        • Options
          • A. Relativity
          • B. Engineering physics
          • C. Imaging
          • D. Astrophysics
      • Assessment
    • Sports, Exercise and Health Science
      • Aims & Objectives
      • Syllabus
        • Core
          • Topic 1: Anatomy
          • Topic 2: Exercise physiology
          • Topic 3: Energy systems
          • Topic 4: Movement analysis
          • Topic 5: Skill in sports
          • Topic 6: Measurement and evaluation of human performance
        • Additional Higher Level
          • Topic 7: Further anatomy
          • Topic 8: The endocrine system
          • Topic 9: Fatigue
          • Topic 10: Friction and drag
          • Topic 11: Skill acquisition and analysis
          • Topic 12: Genetics and athletic performance
          • Topic 13: Exercise and immunity
        • Options
          • Option A: Optimizing physiological performance
          • Option B: Psychology of sports
          • Option C: Physical activity and health
          • Option D: Nutrition for sports, exercise and health
      • Assessment
  • Mathematics
    • Mathematics: analysis and approaches
      • Aims & Objectives
      • Syllabus
        • Topic 1: Numbers and Algebra
        • Topic 2: Functions
        • Topic 3: Geometry and Trigonometry
        • Topic 4: Statistics and Probability
        • Topic 5: Calculus
      • Assessment
    • Mathematics: applications and interpretation
      • Aims & Objectives
      • Syllabus
        • Topic 1: Number and Algebra
        • Topic 2: Functions
        • Topic 3: Geometry and Trigonometry
        • Topic 4: Statistics and Probability
        • Topic 5: Calculus
      • Assessment
  • The arts
    • Dance
      • Aims & Objectives
      • Syllabus
        • Composition and Analysis
        • World Dance Studies
        • Performance
      • Assessment
    • Film
      • Aims & Objectives
      • Syllabus
        • Reading Film
        • Contextualizing Film
        • Exploring Film Production Roles
        • Collaboratively Producing Film (HL only)
      • Assessment
    • Music
      • Aims & Objectives
      • Syllabus
        • Areas of Inquiry
        • Contexts
        • Musical Processes
        • Musical Roles
        • Exploring Music in Context
        • Experimenting with Music
        • Presenting Music
      • Assessment
    • Theatre (Last Assessment 2023)
      • Aims & Objectives
      • Syllabus
        • Theatre in Context
        • Theatre Processes
        • Presenting Theatre
      • Assessment
    • Theatre (First assessment 2024)
    • Visual Arts
      • Aims & Objectives
      • Syllabus
        • Visual Arts in Context
        • Visual Arts Methods
        • Communicating Visual Arts
        • The Visual Arts Journal
        • Art-making Forms
        • Research
      • Assessment
  • Interdisciplinary courses
    • Literature and Performance
      • Aims and Objectives
      • Syllabus
        • Part 1: Critical Study of Texts
        • Part 2: Exploration of the Chose Approach to the Text
        • Part 3: Realization of Texts in Performance
        • Prescribed Literature in Translation
      • Assessment
    • Environmental Systems & Societies
      • Aims & Objectives
      • Syllabus
        • Topic 1: Foundations of environmental systems and societies
        • Topic 2: Ecosystems and ecology
        • Topic 3: Biodiversity and conservation
        • Topic 4: Water and aquatic food production systems and societies
        • Topic 5: Soil systems and terrestrial food production systems and societies
        • Topic 6: Atmospheric systems and societies
        • Topic 7: Climate change and energy production
        • Topic 8: Human systems and resource use
      • Assessment
  • School-based syllabuses
    • Art History
      • Aims & Objectives
      • Syllabus
        • Topic 1: The Art and Architecture of Ancient Greece
        • Topic 2: Rome - Republic and Empire
        • Topic 3: The Middle Ages
        • Topic 4: Romanesque and Gothic Art and Architecture
        • Topic 5: The Art of the Renaissance
        • Topic 6: The Baroque Age - Art and the Architecture of 17th-century Europe
        • Topic 7: The 'Age of Reason' to 'Romanticism'
        • Topic 8: Experiments in the 19th- and 20th-century Art
      • Assessment
    • Astronomy
      • Aims & Objectives
      • Syllabus
        • Topic 1: The Stars
        • Topic 2: The Planets
        • Topic 3: Galaxies
        • Topic 4: Cosmology
      • Assessment
    • Brazilian Social Studies
      • Aims & Objectives
      • Syllabus
        • Topic 1: The Construction of Brazilian Geographical Space
        • Topic 2: Brazil in the Globalization Era - Core Topic
        • Topic 3: The Demographic and Urban Dynamics of Contemporary Brazil
        • Topic 4: Environment and Society
        • Topic 5: From Discovery to the End of the Colonial Era (1500-1822)
        • Topic 6: The Monarchical Experience (1822-1889)
        • Topic 7: Early Republican Brazil (1889-1945) - Core Topic
        • Topic 8: Contemporary Brazil (1945 - )
      • Assessment
    • Classical Greek and Roman Studies
      • Aims & Objectives
      • Syllabus
        • Part A - Two Topics from the Following Four Options
        • Part B - Two Topics from the Following Four Options
      • Assessment
    • Food Science and Technology
      • Aims & Objectives
      • Syllabus
        • 1. Nutrition
        • 2. Materials, Component and Their Application
        • 3. Food Quality and Safety
        • 4. Food Process Engineering
      • Assessment
    • Marine Science
      • Aims & Objectives
      • Syllabus
        • Core
          • Topic 1 Origin and Structure of Oceans
          • Topic 2 Dynamics of Earth's Crust
          • Topic 3 Patterns of Water Movement
          • Topic 4 Properties of Ocean Water
          • Topic 5 Life in Oceans
        • Options
          • A. Marine ecosystems and conservation
          • B. Atmosphere, ocean and climate
          • C. Geology of ocean basins
      • Assessment
    • Modern History of Kazakhstan
      • Aims & Objectives
      • Syllabus
        • Topic 1. Kazakhstan at the Beginning of the 20th Century
        • Topic 2. Kazakhstan During the Civil Confrontation (1917-1920)
        • Topic 3. The Formation of the Soviet Union and Kazakhstan (1920-1940)
        • Topic 4. The Great Patriotic War and Kazakhstan (1941-1945)
        • Topic 5. Kazakhstan and the Socialism (1946-1985)
        • Topic 6. Kazakhstan: from Perestroika to independence
      • Assessment
    • Political Thought
      • Aims & Objectives
      • Syllabus
        • Topic 1: Political Thinkers
        • Topic 2: Political Concepts
      • Assessment
    • Turkey in the 20th Century
      • Aims & Objectives
      • Syllabus
        • Turkey at the Beginning of the 20th Century
        • Topic 2: The Foundations of the Turkish Republic 1923-1945
        • Topic 3: The Global Changes Between the World Wars and Their Effect on Turkey 1918-1939
        • Topic 4: Turkey under pressure during World War II
        • Topic 5: Reconstruction, democracy and developments in the region 1945-1985
        • Topic 6: The Effects of Globalization and the Dialogue with Europe 1985-2000
      • Assessment
    • World Arts and Cultures
      • Aims & Objectives
      • Syllabus
        • Part 1: Prescribed Topics
        • Part 2: Intercultural Studies
        • Part 3: Regional Study
      • Assessment
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  • Concepts
  • SL Content
  • AHL Content
  1. Mathematics
  2. Mathematics: applications and interpretation
  3. Syllabus

Topic 5: Calculus

Concepts

  • Essential understandings:

    • Calculus describes rates of change between two variables and the accumulation of limiting areas. Understanding these rates of change allows us to model, interpret and analyze real-world problems and situations. Calculus helps us understand the behaviour of functions and allows us to interpret the features of their graphs.

  • Suggested concepts embedded in this topic:

    • Change, patterns, relationships, approximation, space, generalization.

    • AHL: Systems, quantity.

  • Content-specific conceptual understandings:

    • Students will understand the links between the derivative and the rate of change and interpret the meaning of this in context.

    • Students will understand the relationship between the integral and area and interpret the meaning of this in context.

    • Finding patterns in the derivatives of polynomials and their behavior, such as increasing or decreasing, allows a deeper appreciation of the properties of the function at any given point or instant.

    • Calculus is a concise form of communication used to approximate nature.

    • Numerical integration can be used to approximate areas in the physical world.

    • Optimization of a function allows us to find the largest or smallest value that a function can take in general and can be applied to a specific set of conditions to solve problems.

    • Maximum and minimum points help to solve optimization problems.

    • The area under a function on a graph has a meaning and has applications in space and time.

    • AHL: Kinematics allows us to describe the motion and direction of objects in closed systems in terms of displacement, velocity, and acceleration.

    • AHL: Many physical phenomena can be modelled using differential equations and analytic and numeric methods can be used to calculate optimum quantities.

    • AHL: Phase portraits enable us to visualize the behavior of dynamic systems.

SL Content

  • SL 5.1

    • Introduction to the concept of a limit.

    • Derivative interpreted as gradient function and as rate of change.

  • SL 5.2

    • Increasing and decreasing functions.

    • Graphical interpretation of f′(x)>0,f′(x)=0,f′(x)<0f'(x) > 0, f'(x)=0, f'(x) < 0f′(x)>0,f′(x)=0,f′(x)<0

  • SL 5.3

    • Derivative of f(x)=axnf(x)=ax^nf(x)=axn is f′(x)=anxn−1,n∈Zf'(x)=anx^{n-1}, n \in \mathbb{Z}f′(x)=anxn−1,n∈Z

    • The derivative of functions of the form f(x)=axn+bxn−1+−f(x)=ax^n+bx^{n-1}+-f(x)=axn+bxn−1+− where all exponents are integers.

  • SL 5.4

    • Tangents and normals at a given point, and their equations.

  • SL 5.5

    • Introduction to integration as anti-differentiation of functions of the form f(x)=axn+bxn−1+...f(x)=ax^n+bx^{n-1}+...f(x)=axn+bxn−1+... , where n∈Z,n≠−1n \in \mathbb{Z}, n \neq -1n∈Z,n=−1

    • Anti-differentiation with a boundary condition to determine the constant term.

    • Definite integrals using technology.

    • Area of a region enclosed by a curve y=f(x)y=f(x)y=f(x) and the xxx -axis, where f(x)>0f(x) > 0f(x)>0 .

  • SL 5.6

    • Values of xxx where the gradient of a curve is zero. Solution of xxx .

    • Local maximum and minimum points.

  • SL 5.7

    • Optimisation problems in context.

  • SL 5.8

    • Approximating areas using the trapezoidal rule.

AHL Content

  • AHL 5.9

    • The derivatives of sin⁡x,cos⁡x,tan⁡x,ex,lnx,xn\sin x, \cos x, \tan x, e^x, ln x, x_nsinx,cosx,tanx,ex,lnx,xn​ where n∈Qn \in \mathbf{Q}n∈Q

    • The chain rule, product rule and quotient rules.

    • Related rates of change.

  • AHL 5.10

    • The second derivative.

    • Use of second derivative test to distinguish between a maximum and a minimum point.

  • AHL 5.11

    • Definite and indefinite integration of xnx^nxn where n∈Qn \in \mathbf{Q}n∈Q , including n=−1,sin⁡x,cos⁡x,1cos⁡x2n=-1, \sin x, \cos x, \frac{1}{\cos^2_x}n=−1,sinx,cosx,cosx2​1​ and exe^xex

    • Integration by inspection, or substitution of the form ∫f(g(x))g′(x)dx\int f(g(x))g'(x)dx∫f(g(x))g′(x)dx

  • AHL 5.12

    • Area of the region enclosed by a curve and the xxx or yyy -axes in a given interval.

    • Volumes of revolution about the xxx -axis or yyy -axis.

  • AHL 5.13

    • Kinematic problems involving displacement sss , velocity vvv and acceleration aaa .

  • AHL 5.14

    • Setting up a model/differential equation from a context.

    • Solving by separation of variables.

  • AHL 5.15

    • Slope fields and their diagrams.

  • AHL 5.16

    • Euler’s method for finding the approximate solution to first order differential equations.

    • Numerical solution of dydx=f(x,y)\frac{dy}{dx}=f(x, y)dxdy​=f(x,y)

    • Numerical solution of the coupled system dxdt=f1(x,y,t)\frac{dx}{dt}=f_1(x, y, t)dtdx​=f1​(x,y,t) and dydt=f2(x,y,t)\frac{dy}{dt}=f_2(x, y, t)dtdy​=f2​(x,y,t)

  • AHL 5.17

    • Phase portrait for the solutions of coupled differential equations of the form:

    • dxdt=ax+by\frac{dx}{dt}=ax+bydtdx​=ax+by

    • dydt=cx+dy\frac{dy}{dt}=cx+dydtdy​=cx+dy

    • Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues.

    • Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points.

  • AHL 5.18

    • Solutions of d2xdt2=f(x,dxdt,t)\frac{d^2 x}{dt^2}=f(x, \frac{dx}{dt}, t)dt2d2x​=f(x,dtdx​,t) by Euler's method.

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