Topic plans for AS/A level Mathematics
Differentiation
G1 Understand and use the second derivative in connection to convex and concave sections of curves and points of inflection.
G3 Apply differentiation to find points of inflection.
G4 Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.
For a brief commentary on this content go to the MEI outline SoW.
Pre-requisites
Common student errors
▪ AS Differentiation: Using differentiation techniques in problems.
▪ Failing to be clear which variable is being differentiated and with respect to which other variable
d𝑦
and generally using
as a universal notation for “the differential coefficient of”.
▪ A level Functions: Composite functions.
d𝑥
▪ Assuming a more simple model for product and chain rule (e.g. just the product of differentials).
▪ Difficulty recognising products and composite functions.
Teaching it!
Getting them thinking
▪ Tangents and normals: A card sort using chain, product and quotient rules to solve two problems.
▪ Make up one question in which you need to use both the product rule and the chain rule.
▪ Types of differentiation: Categorising functions according to how to differentiate them.
▪ Why is the chain rule sometimes called the function of a function rule?
▪ I can see u!: An Underground Mathematics resource for developing fluency with the chain rule.
▪ Prove that every cubic curve has rotational symmetry about its point of inflection.
𝑥2
𝑦2
▪ Rectangles in an ellipse: Maximising the area of a rectangle contained within + = 1.
16
9
▪ Using
𝑢+𝛿𝑣 𝑣+𝛿𝑢 −𝑢𝑣
, prove the product rule from first principles.
𝛿𝑥
d 𝑢
▪ Using the product rule on 𝑦 = 𝑢𝑣, prove the quotient rule formula for
.
d𝑥 𝑣
▪ Points of inflection (student task): Autograph, Casio, GeoGebra
𝛿 𝑢𝑣
𝛿𝑥
=
SC 08/08/22
Year 2 Pure:
Proof
Trig
Seq & Ser
Functions
Diff
Trig fns
Algebra
Trig ids
Furth diff
Int
Para eqn
Vectors
Diff eqns
Num meth
Version 2.0