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) < 0.

SL 5.3

• • Derivative of f(x) = axn is f ′(x) = anxn − 1 , n ∈ ℤ

  • The derivative of functions of the form 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 + ...., where n ∈ 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) and the x -axis, where f(x) > 0.

SL 5.6

• • Derivative of x n (n ∈ ℚ), sinx, cosx, e x and lnx.

  • Differentiation of a sum and a multiple of these functions.

  • The chain rule for composite functions.

  • The product and quotient rules.

SL 5.7

• • The second derivative.

  • Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ″.

SL 5.8

• • Local maximum and minimum points.

  • Testing for maximum and minimum.

  • Optimization.

  • Points of inflexion with zero and non-zero gradients.

SL 5.9

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

SL 5.10

• • Indefinite integral

  • The composites of any of these with the linear function ax + b.

  • Integration by inspection (reverse chain rule) or by substitution  

SL 5.11

• • Definite integrals, including analytical approach.

  • Areas of a region enclosed by a curve y = f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology.

  • Areas between curves.

AHL 5.12

• • Informal understanding of continuity and differentiability of a function at a point.

  • Understanding of limits (convergence and divergence).

  • Definition of derivative from first principles

  • Higher derivatives.

AHL 5.13

• • The evaluation of limits using l’Hôpital’s rule or the Maclaurin series.

  • Repeated use of l’Hôpital’s rule

AHL 5.14

• • Implicit differentiation.

  • Related rates of change.

  • Optimisation problems.

AHL 5.15

• • Derivatives of tanx, secx, cosecx, cotx, ax , logax, arcsinx, arccosx, arctanx.

  • The composites of any of these with a linear function.

  • Use of partial fractions to rearrange the integrand.

AHL 5.16

• • Integration by substitution.

  • Integration by parts.

  • Repeated integration by parts.

AHL 5.17

• • Area of the region enclosed by a curve and the y-axis in a given interval. Volumes of revolution about the x-axis or y-axis.

AHL 5.18

• • First order differential equations.

  • Numerical solution of dy / dx = f(x, y) using Euler’s method.

  • Variables separable.

  • Homogeneous differential equation dy / dx = f( y/ x ) using the substitution y = vx.

  • Solution of y′ + P(x)y = Q(x), using the integrating factor.

AHL 5.19

• • Maclaurin series to obtain expansions for ex , sinx, cosx, ln(1 + x), (1 + x) p , p ∈ ℚ.

  • Use of simple substitution, products, integration and differentiation to obtain other series.

  • Maclaurin series developed from differential equations.