Autumn Core 3

• The Number 'e'
• Differentiation and the Chain Rule
• Differentiation, the Product and the Quotient Rules
• Integration by Inspection and Substitution
• Integration by Parts and Standard Integrals
• Volumes of Revolution
• Functions
• Transformations
• Inverse Trigonometric Functions
• Numerical Methods
• Numerical Integration
• Revision

Spring Core 4

Core 3 Examination 17th January 2008

• Implicit Differentiation
• Parametric Equations
• Further Trigonometry
• Exponential Growth and Decay
• Differential Equations
• Binomial Series and Expansion
• Rational Functions and the Division of Polynomials
• Partial Fractions

Summer

• Exponential Growth and Decay
• Differential Equations
• Vectors
• Revision

Core 4 Examination 12th June 2008

CORE 3

Topic and Chapter

Time (weeks)

Objectives

These objectives relate to the main text objectives at the start of each chapter from the Heinemann text books.

Worthy of Note

These 'worthy of notes' are teacher observations that are worthy of a moment's thought.

Tasks

Any additions that teachers may feel are worthy of inclusion into the main scheme of work.

• Recognise the number 'e'
• Differentiate and integrate the number 'e'
• Understand what is meant by a natural logarithm
• Realise that the inverse of e^x is ln x
• Find the integral of 1/x

• The function e^x and its graph.
• The function ln x and its graph; ln x as the inverse function of e^x.
• Differentiation of e^x, ln x, sin x, cos x, tan x and their sums and differences.

• Find and use the derivatives of sine and cosine
• Differentiate composite functions using the chain rule

• The use of dy/dx = 1/(dx/dy)
• Use dy/dx = dy/dt . dt/dx for composite functions.

• Find the derivatives of tangent, cotangent, secant and cosecant
• Use the product rule
• Use the quotient rule

• Integrate expressions using the reverse idea to the chain rule
• Integrate trigonometric functions
• Integrate using suitable substitutions

• Except in the simplest of cases the substitution will be given.
  The integral ∫ln x dx is required.
• More than one application of integration by parts may be required, for example
  ∫ x² e^x dx.

• Integrate expressions using integration by parts
• Use relevant standard integrals quoted in the course formulae books

• Evaluate volumes of revolution
• Use the mid ordinate rule to find areas bounded by curves
• Use Simpsons' rule to find the area bounded by curves

• ∏ ∫ y² dx is required but not
  ∏ ∫ x² dy.
• Candidates should be able to find a volume of revolution, given parametric equations.

• Use one-one mappings
• Use the terms range and domain
• Be able to find the range of a function
• To be able to form composite functions
• To understand the conditions for the inverse of a function to exist

• Definition of a function. Domian and range of functions. Composition of functions. Inverse functions and their graphs
• This includes odd, even, inverse and composite functions and their graphical representations.
• The concept of a function as a one-one or many-one mapping from R (or a subset of R) to R.
• The notation f:x →... and f(x) will be used.
• Candidates should know that fg will mean 'do g first, then f'.
• Candidates should know that if f^-1 exists,
  then f^-1f(x) = ff^-1(x) = x.

• Transform graphs to produce other graphs
• Understand the effect of composite transformations
• Understand what is meant by the modulus function
• Sketch graphs of the modulus function
• Solve equations and inequalities involving the modulus function

• Combinations of the transformations y = f(x) as represented by
  y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
• Candidates should eb able to sketch the graph of e.g. y = 2f(3x), y = f(-x) + 1, given the graph of y = f(x) or the graph of,
  e.g. y = 3 + sin 2x, y = -cos (x + π/4).
• Candidates should be able to sketch the graphs of y = I ax + b I and the graphs of y = I f(x) I and y = f ( IxI ), given the graph of y = f(x).
• Knowledge of the graph of |ax+b| .
• Cover modulus equations. Solve graphically and by 'squaring' both sides.
• This includes transformations of graphs (shifts, stretches and reflections).

• Work with inverse trig functions and be able to draw their graphs
• Understand secant, cosecant and cotangent functions
• Be able to sketch inverse functions
• Use more complex (than core 2 ) trig identities

• Knowledge of secant, cosecant and cotangent and of arcsin, arcos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains.

• Locate roots by change of sign
• Use numerical methods to find the solution of equations
• Understand the principle of iteration
• Appreciate the need for convergence in iterations
• Use and understand cobweb and staircase diagrams

• Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous.
• Approximate solution of equations using simple iterative methods, including recurrence relations of the form x_n+1 = f( x_n)

CORE 4

• Differentiate functions implicitly
• Find equations of tangents and normals to curves specified implicitly

• Sketch curves of functions expressed in parametric form
• Find gradients of curves expressed in parametric form
• Find the Cartesian form of functions expressed in parametric form

• Parametric equations of curves and conversion between Cartesian and parametric forms.

• Use compound angle identities in trigonometry to prove other identities and solve equations
• Use double angle identities in integration
• Express additative forms of sine and cosine as a single sine or cosine

• Knowledge and use of
• sec²θ = 1 + tan²θ and
• cosec²θ = 1 + cot²θ
• Knowledge and use of double angle formulae;
• use of formulae for sin (A ± B), cos (A ± B) and tan (A ± B)
• and of expressions for acos θ + bsinθ in the equivalent forms of rcos ( θ ± α) or rsin ( θ±α).

• Solve equations with the variable as an exponent
• Understand what is meant by exponential growth and decay
• Solve problems involving growth and decay

• Formulate first order differential equations
• Solve analytically first order differential equations with seperable variables

• Use geometric series to expand functions
• Extend the earlier binomial expansions

• For I x I < b/a, candidates should be able to obtain the expansion of (ax + b)ⁿ, and the expansion of rational functions by decomposition into partial fractions.

• Simplify rational expressions
• Multiply and divide rational expressions
• Add and subtract rational expressions
• Divide a polynomial by a linear expression
• Recall and use the Factor Theorem
• Recall and use the Remainder Theorem

• Split a rational expression into a partial fraction
• Use partial fractions to write rational functions as a series expansion
• Use partial fractions to find and evaluate integrals

• Partial fractions (denominators not more complicated than repeated linear terms)
• Partial fractions to include denominators such as:
  (ax + b)(cx + d)( ex + f) and
  (ax + b)(cx + d)²
• The degree of the numerator may equal or exceed the degree of the denominator.
• Applications to integration, differentiation and series expansions.
• Integration of rational expressions such as those arising from partial fractions,
  e.g. 2 / (3x + 5) , 3/(x - 1)²
• Note that the integration of other rational expressions, such as:
  x /(x² + 5 ) and 2/(2x - 1)⁴ is also required

• Use vector notation
• Find the magnitude of a vector
• Understand the term 'unit vector'
• Find the vector equation of a line
• Determine whether two lines intersect
• Calculate the scalar product and use it to find the angle between two vectors
• Calculate the perpendicular distance from a point to a line

• Candidates should be able to find a unit vector in the direction of a, and be familiar with I a I

• →		→		→
  OB	-	OA	=	AB	=b-a

• The distance d between two points,
  ( x₁, y₁, z₁ ) and ( x₂, y₂, z₂ )
  is given by:
  d² = ( x₁-x₂ )²+( y₁-y₂ )²+( z₁-z₂ )²
• To include the forms r = a + rb and
  r = c + t(d - c)
• Intersection, or otherwise of two lines.
• Candidates should know that for:
  

  →
  OA	=a = a1i+a2j+a3k

• and
  

  →
  OB	=b = b1i+b2j+b3k

  
  then
  a.b=a₁b₁+a₂b₂+a₃b₃
• and cos AOB = a.b/(IaI IbI)
• Candidates should know that if a.b = 0, and that a and b are non-zero vectors, then a and b are perpendicular.