M1Maths

Modules - Calculus

. . . . .       . . . . .       . . . . .       . . . . .       . . . . .       Level 6

Level 1 is Years 5-7, Level 6 Years 11-12.

Calculus is treated fairly conventionally, though time is spent developing some of the basic concepts like how rates of change appear on graphs in practical situations.

Also, differentials are given individual meaning, rather than just their ratio. This makes things like the chain rule easier to understand.

Modules:   C6:   Calculus - Level 6

Click the link in the second column for the required format. Word documents need to be downloaded and opened in Word to display properly. The second last column is the estimated time required to work through the module; the last column is the pre-requisite Level 6 modules.

Velocity Graphically

• displacement vs time graphs: high/low, rising/falling, steeper/flatter

• gradient = velocity

• gradient of secant = average velocity

• gradient of tangent = instantaneous velocity

Velocity Algebraically

• velocity at a given time using reducing intervals

• velocity at given time using an unspecified interval, dt

• velocity at any time

Velocity by Rule

• differentiating at^n and sums of such terms by rule

Other Relations

• calculus with relations other than between time, displacement and velocity

• generic x-y relations

Applications of Derivatives

• small increments

• tangents and normals

• finding a point with a given gradient

• optimisation

• turning points and curve sketching

Chain, Product and Quotient Rules

• chain, product and quotient rules

Other Derivatives

• the derivatives of sin x, cos x, a^x, e^x, ln x

Mastering Differentiation

• differentiating all functions fluently

Differential Equations

• differential equations

Integration

• definite integrals and the fundamental theorem of calculus

• solving problems with integration

• indefinite integrals

Areas Under Curves

• area under curves

• area between curves

• trapezoidal rule

• Simpson's rule

Pert

• differential equations of the form dA/dt = rA

Higher-order Derivatives

• second and higher-order derivatives

Graph Sketching

• sketching graphs of polynomial and rational functions using knowledge of function shape, extreme behaviour, discontinuities, axis intercepts, stationary points and spot values

Optimisation

• finding the value of a control quantity which gives the optimum value of an objective quantity

Background image: https://pixabay.com/illustrations/beach-background-sea-ocean-3892386/ (cropped) (Creative Commons licence)