Calculus

• Informal ideas of limit and convergence.
• Limit notation.
• Definition of derivative from first principles as
• Derivative interpreted as gradient function and as rate of change.
• Tangents and normals, and their equations.

Not required:

• analytic methods of calculating limits.

Example: 0.3, 0.33, 0.333, ... converges to 1/3
.Technology should be used to explore ideas of limits, numerically and graphically.

Example:

• Links to 1.1, infinite geometric series; 2.5–2.7, rational and exponential functions, and asymptotes.
• Use of this definition for derivatives of simple polynomial functions only.
• Technology could be used to illustrate other derivatives.

Link to 1.3, binomial theorem.

• Use of both forms of notation, 
• for the first derivative.
• Identifying intervals on which functions are increasing or decreasing.
• Use of both analytic approaches and technology.
• Technology can be used to explore graphs and their derivatives.

• Appl: Economics 1.5 (marginal cost, marginal revenue, marginal profit).
• Appl: Chemistry 11.3.4 (interpreting the gradient of a curve).
• Aim 8: The debate over whether Newton or Leibnitz discovered certain calculus concepts.
• TOK: What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life?
• TOK: Opportunities for discussing hypothesis formation and testing, and then the formal proof can be tackled by comparing certain cases, through an investigative approach.

• Derivative of e^x and ln x .
• Differentiation of a sum and a real multiple of these functions.
• The chain rule for composite functions.
• The product and quotient rules.
• The second derivative.
• Extension to higher derivatives.

• Link to 2.1, composition of functions.
• Technology may be used to investigate the chain rule.
• Use of both forms of notation,

• Local maximum and minimum points. Testing for maximum or minimum.
• Points of inflexion with zero and non-zero gradients.
• Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ .
• Optimization.
• Applications.

Not required:

• points of inflexion where f ′′(x) is not defined: for example, y = x¹³ at (0,0).

• Using change of sign of the first derivative and using sign of the second derivative.
• Use of the terms “concave-up” for f ′′(x) > 0 ,and “concave-down” for f ′′(x) < 0 .
• At a point of inflexion , f ′′(x) = 0 and changes sign (concavity change).
• f ′′(x) = 0 is not a sufficient condition for a point of inflexion: for example, y = x4 at (0,0) .
• Both “global” (for large |x| ) and “local” behaviour.
• Technology can display the graph of a derivative without explicitly finding an expression for the derivative.
• Use of the first or second derivative test to justify maximum and/or minimum values.
• Examples include profit, area, volume.
• Link to 2.2, graphing functions.

• Indefinite integration as anti-differentiation.
• Indefinite integral of
• The composites of any of these with the linear function ax + b .
• Integration by inspection, or substitution of the form ∫ f (g(x))g '(x)dx .

• Anti-differentiation with a boundary condition to determine the constant term.
• Definite integrals, both analytically and using technology.
• Areas under curves (between the curve and the x-axis).
• Areas between curves.
• Volumes of revolution about the x-axis.

• 
• The value of some definite integrals can only be found using technology.
• Students are expected to first write a correct expression before calculating the area.
• Technology may be used to enhance understanding of area and volume.

• Int: Successful calculation of the volume of the pyramidal frustum by ancient Egyptians (Egyptian Moscow papyrus).
• Use of infinitesimals by Greek geometers.
• Accurate calculation of the volume of a cylinder by Chinese mathematician Liu Hui
• Int: Ibn Al Haytham: first mathematician to calculate the integral of a function, in order to find the volume of a paraboloid.

• Kinematic problems involving displacement s, velocity v and acceleration a.
• Total distance travelled.