Download Mapping chart P1-3 to C1-4 - Pearson Schools and FE Colleges

EDEXCEL GCE
Edexcel Advanced Subsidiary GCE in Mathematics
Further Mathematics
Pure Mathematics
For teaching from September 2004
First award in January 2005
Edexcel Advanced GCE in
Mathematics
Further Mathematics
Pure Mathematics
For teaching from September 2004
First award in June 2005
Mapping from C1–C4 to P1–P3 with Heinemann textbook references
1
Specification content
Unit C1 – Core Mathematics 1
The examination
The examination will consist of one 1½ hour paper. It will contain about ten questions of varying length. The mark allocations per question will be stated on the
paper. All questions should be attempted.
Formulae which candidates are expected to know are given in the appendix to this unit and these will not appear in the booklet, Mathematical Formulae including
Statistical Formulae and Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage.
For this unit, candidates may not have access to any calculating aids, including log tables and slide rules.
Preamble
Construction and presentation of rigorous mathematical arguments through appropriate use of precise statements and logical deduction, involving correct use of
symbols and appropriate connecting language is required. Candidates are expected to exhibit correct understanding and use of mathematical language and
grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’, and notation such as
, ,  and .
NOTES
HEINEMANN TEXTBOOK REFERENCE
Laws of indices for all rational exponents.
The equivalence of am/n and nam should be known.
P1:
Use and manipulation of surds.
Candidates should
denominators.
NEW C1 SPECIFICATION
1. Algebra and functions
be
able
to
rationalise P1:
1.1 Laws of indices for rational exponents
1.2 Using and manipulating surds
Manipulating surds
Rationalising the denominator
2
NEW C1 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
Quadratic functions and their graphs.
P1:
1.8 Quadratic functions and their graphs
The discriminant of a quadratic function.
P1:
1.8 Quadratic functions and their graphs
Completing the square. Solution of quadratic Solution of quadratic equations by factorisation, use P1:
equations.
of the formula and completing the square.
1.9
Completing the square of a quadratic
function
Completing the square for ax2 + bx + c
1.10 Solving quadratic equations
Solving quadratic equations by
factorisation
Solving quadratic equations by
completing the square
Solving quadratic equations by using
the formula
How to check whether a quadratic can
be factorised
Simultaneous equations: analytical solution by For example, where one equation is linear and one P1:
substitution.
equation is quadratic
1.11 Solving two simultaneous equations,
one linear and one quadratic
Solution of linear and quadratic inequalities.
1.12 Solving linear and quadratic
inequalities in one variable
For example, ax + b > cx + d,
px2 + qx + r  0, px2 + qx + r < ax + b.
P1:
Algebraic manipulation of polynomials, including Candidates should be able to use brackets. P1:
expanding brackets and collecting like terms, Factorisation of polynomials of degree n, n  3,
factorisation.
eg x3 + 4x2 + 3x. The notation f(x) may be used. (Use
of the factor theorem is not required.)
1.3
Processing polynomials
Adding and subtracting polynomials
Multiplying one polynomial by another
1.4
Factorising polynomials
3
NEW C1 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
Graphs of functions; sketching curves defined by Functions to include simple cubic functions and the P2:
simple equations.
reciprocal function y = k/x with x  0.
2.2
Graphing a function
2.6
Sketching simple graphs of the form
y = xn
Even and odd functions
P2:
2.8
Solving equations graphically
Knowledge of the effect of simple transformations Candidates should be able to apply one of these P2:
on the graph of y = f(x) as represented by y = af(x), transformations to any of the above functions
y = f(x) + a, y = f(x + a), y = f(ax).
[quadratics, cubics, reciprocal] and sketch the
resulting graph.
2.7
Transforming graphs
The transformation y = af(x)
The transformation y = f(x) + a
The transformation y = f(x + a)
The transformation y = f(ax)
Geometrical interpretation of algebraic solution of
equations. Use of intersection points of graphs of
functions to solve equations.
Knowledge of the term asymptote is expected.
Given the graph of any function y = f(x) candidates
should be able to sketch the graph resulting from one
of these transformations.
4
NEW C1 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
2. Coordinate geometry in the (x, y) plane
Equation of a straight line, including the forms
y – y1 = m(x – x1) and ax  by  c  0 .
To include
(i) the equation of a line through two given points,
(ii) the equation of a line parallel (or perpendicular) to
a given line through a given point. For example, the
line perpendicular to the line 3x + 4y = 18 through the
point (2, 3) has equation y – 3 = 43 (x – 2).
P1:
3.1
The equation of a straight line in the
form y – y1 = m(x – x1)
General method for finding the
equation of a straight line from one
point and the gradient
3.2
The equation of a straight line in the
form
y  y1
x  x1
=
y 2  y1
x2  x1
General method for finding the
equation of a straight line from two
points on the line
Conditions for two straight lines to be parallel or
perpendicular to each other.
3.3
The general equation of a straight
line
P1:
3.4
Parallel and perpendicular lines
P1:
4.1
Sequences including those given by a
general formula for the nth term
Representing a series graphically
P2:
3.1
Sequences and recurrence relations
P1:
4.2
Arithmetic series
The sum of an arithmetic series
The sum of any arithmetic series
3. Sequences and series
Sequences, including those given by a formula for
the nth term and those generated by a simple
relation of the form xn+1 = f(xn).
Arithmetic series, including the formula for the
sum of the first n natural numbers.
The general term and the sum to n terms of the series
are required. The proof of the sum formula should be
known.
5
NEW C1 SPECIFICATION
HEINEMANN TEXTBOOK REFERENCE
NOTES
Understanding of  notation will be expected.
P1:
4.3
The sum of the first n natural
numbers
4.6
Further examples on arithmetic [and
geometric] series
5.1
The gradient of a graph
Gradient of a straight line graph
Gradient of a curve
5.2
Using geometry to approximate a
gradient
A numerical approach to rates of
change
5.3
A general approach to rates of
change
second order derivatives.
5.6
Second derivatives
Differentiation of xn, and related sums and E.g., for n  1, the ability to differentiate expressions P1:
2
differences.
x + 5x  3
such as (2x + 5)(x  1) and
is expected.
1/ 2
3x
5.4
A general formula for
Applications of differentiation
tangents and normals.
5.7
Turning points
5.8
Tangents and normals to curves
5.9
Using differentiation to solve
practical problems
4. Differentiation
dy
The derivative of f(x) as the gradient of the tangent For example, knowledge that
is the rate of
dx
to the graph of y = f (x) at a point; the gradient of
the tangent as a limit; interpretation as a rate of change of y with respect to x. Knowledge of the chain
rule is not required.
change;
P1:
The notation f(x) may be used.
to
gradients, Use of differentiation to find equations of tangents P1:
and normals at specific points on a curve.
dy
when
dx
y = xn
6
NEW C1 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
P2:
6.3
Applications of differentiation
Candidates should know that a constant of integration P1:
is required.
6.1
Integration as the inverse of
differentiation
6.3
The solution of the differential
5. Integration
Indefinite integration as the reverse of
differentiation.
equation
Integration of xn.
For example, the ability to integrate expressions such P1:
( x  2) 2
1
as 12 x 2  3x 2 ,
is expected.
1
x2
6.2
dy
= f(x)
dx
Integration of xn where n  ℚ, n  1
Given f(x) and a point on the curve, candidates
should be able to find an equation of the curve in the
form y = f(x).
7
Appendix C1 – Formulae
This appendix lists formulae that candidates are expected to remember and that may not be included in formulae booklets.
Quadratic equations
 b  b  4 ac
2a
2
ax 2  bx  c  0 has roots
Differentiation
function
derivative
xn
nx n  1
Integration
function
integral
xn
1
x n  1 + c , n  1
n 1
8
Unit C2 – Core Mathematics 2
The examination will consist of one 1½ hour paper. It will contain about nine questions of varying length. The mark allocations per question which will be stated
on the paper. All questions should be attempted.
Formulae which candidates are expected to know are given in the appendix to this unit and these will not appear in the booklet, Mathematical Formulae including
Statistical Formulae and Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage.
Candidates are expected to have available a calculator with at least the following keys: +, , , , π, x2, x,
1 y
, x , ln x, ex, x!, sine, cosine and tangent and their
x
inverses in degrees and decimals of a degree, and in radians; memory. Calculators with a facility for symbolic algebra, differentiation and/or integration are not
permitted.
Prerequisites
A knowledge of the specification for C1, its preamble and its associated formulae, is assumed and may be tested.
NEW C2 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
1. Algebra and functions
Simple algebraic division; use of the Factor Only division by (x + a) or (x – a) will be required. P1:
Theorem and the Remainder Theorem.
Candidates should know that if f(x) = 0 when x = a, P1:
then (x – a) is a factor of f(x).
P3:
Candidates may be required to factorise
cubic expressions such as x3 + 3x2 – 4 and
6x3 + 11x2 – x – 6.
1.6
Algebraic division
1.7
The factor theorem
1.2
The remainder theorem and the
factor theorem
Candidates should be familiar with the terms
‘quotient’ and ‘remainder’ and be able to determine
the remainder when the polynomial f(x) is divided by
(ax + b).
9
NEW C2 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
2. Coordinate geometry in the (x, y) plane
Coordinate geometry of the circle using the Candidates should be able to find the radius and the P1:
equation of a circle in the form
coordinates of the centre of the circle given the
(x – a)2 + (y – b)2 = r2 and including use of the equation of the circle, and vice versa.
following circle properties:
P3:
(i) the angle in a semicircle is a right angle;
(ii) the perpendicular from the centre to a chord
bisects the chord;
(iii) the perpendicularity of radius and tangent.
8.13 Simple geometrical properties of a
circle
3.1
The circle
The equations and properties of
tangents to a circle
3.2
Sketching curves given by cartesian
equations
Sketching the graph of y =
1
f ( x)
3. Sequences and series
The sum of a finite geometric series; the sum to The general term and the sum to n terms are P1:
infinity of a convergent geometric series, including required.
P1:
the use of r  1 .
The proof of the sum formula should be known.
Binomial expansion of (1  x)n for positive integer Expansion of (a + bx)n may be required.
P2:
4.4
Geometric series
The sum of a geometric series
4.5
The sum to infinity of a convergent
geometric series
4.6
Further examples on [arithmetic
and] geometric series
3.2
Binomial series for a positive,
integral index
Binomial expressions
Pascal’s triangular array
The expansion of (a + b)n, where n is a
positive integer using Pascal’s triangle
3.3
Expanding (a + b)n
n
n. The notations n! and   .
r
 
10
NEW C2 SPECIFICATION
HEINEMANN TEXTBOOK REFERENCE
NOTES
4. Trigonometry
The sine and cosine rules, and the area of a triangle
in the form 12 ab sin C.
Radian measure, including use for arc length and Use of the formulae s = rθ and A =
area of sector.
1
2
r2θ for a circle.
P1:
8.11 The sine rule and the cosine rule
The sine rule
The cosine rule
The area of a triangle
P1:
2.1
Radian measure
2.2
Length of an arc of a circle
2.3
Area of a sector of a circle
2.4
The three basic trigonometric
functions for any angle
Some special results
2.5
Drawing the graphs of sin x, cos x
and tan x
Graphing sin x
Graphing cos x
Graphing tan x
Graphing sin nx, cos nx and tan nx
Graphing –sin x, –cos x and –tan x
Graphing sin (–x), cos (–x) and tan (–x)
Graphing n sin x, n cos x and in tan x
Graphing sin (x + n), cos (x + n) and
tan (x + n)
2.6
The identities tan  
Sine, cosine and tangent functions. Their graphs, Knowledge of graphs of curves with equations such P1:
symmetries and periodicity.
as y = 3 sin x, y = sin(x + π/6), y = sin 2x is expected.
Knowledge and use of tan  =
sin2 + cos2  = 1.
sin 
, and
cos 
P1:
cos 2  + sin2   1
sin 
and
cos
11
NEW C2 SPECIFICATION
HEINEMANN TEXTBOOK REFERENCE
NOTES
Solution of simple trigonometric equations in a Candidates should be able to solve equations such as
given interval.
sin (x  π/2) = 34 for 0 < x < 2π,
cos (x + 30) =
1
2
P1:
2.7
Solving simple equations in a given
interval
P2:
5.4
Equations of the form ax = b
for 180 < x < 180,
tan 2x = 1 for 90 < x < 270,
6 cos2 x + sin x  5 = 0 for 0  x < 360,
 
sin2  x   = 12 for –  x < .
6

5. Exponentials and logarithms
y  ax and its graph.
Laws of logarithms.
To include
log a xy = log a x + log a y,
x
log a = log a x  log a y,
y
loga xk = k log a x,
1
log a   log a x ,
x
log a a = 1.
P2:
5.3
Laws of logarithms
The solution of equations of the form ax = b.
Candidates may use the change of base formula.
P2:
5.4
Equations of the form ax = b
Applications of differentiation to maxima and The notation f(x) may be used for the second order P1:
minima and stationary points, increasing and derivative.
decreasing functions.
To include applications to curve sketching. Maxima P1:
and minima problems may be set in the context of a
practical problem.
5.8
Tangents and normals to curves
5.9
Using differentiation to solve
practical problems
6. Differentiation
12
NEW C2 SPECIFICATION
HEINEMANN TEXTBOOK REFERENCE
NOTES
P2:
6.3
Applications of differentiation
P1:
6.4
Boundary conditions
6.5
Definite integrals
6.6
Finding the area of a region bounded
by lines and a curve
7.4
Numerical integration: the
trapezium rule
The trapezium rule
7. Integration
Evaluation of definite integrals.
Interpretation of the definite integral as the area
under a curve.
Candidates will be expected to be able to evaluate P1:
the area of a region bounded by a curve and given
straight lines.
E.g. find the finite area bounded by the curve
y = 6x – x2 and the line y = 2x.
 x dy will not be required.
Approximation of area under a curve using the
trapezium rule.
E.g. evaluate

1
0
(2 x  1) dx using the values of P2:
(2 x  1) at x = 0, 0.25, 0.5, 0.75 and 1.
13
Appendix C2 – Formulae
This appendix lists formulae that candidates are expected to remember and that may not be included in formulae booklets.
Laws of logarithms
log a x  log a y  log a ( xy)
 x
log a x  log a y  log a  
 y
k log a x  log a x k
Trigonometry
In the triangle ABC
a
b
c


sin A sin B sin C
area =
1
2
ab sin C
Area
area under a curve 

b
a
y dx ( y  0)
14
Unit C3 – Core Mathematics 3
The examination
The examination will consist of one 1½ hour paper. It will contain about seven questions of varying length. The mark allocations per question which will be stated
on the paper. All questions should be attempted.
Formulae which candidates are expected to know are given in the appendix to this unit and these will not appear in the booklet, Mathematical Formulae including
Statistical Formulae and Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage.
Candidates are expected to have available a calculator with at least the following keys: +, , , , π, x2, x,
1 y
, x , ln x, ex, x!, sine, cosine and tangent and their
x
inverses in degrees and decimals of a degree, and in radians; memory. Calculators with a facility for symbolic algebra, differentiation and/or integration are not
permitted.
Prerequisites
A knowledge of the specification for C1and C2, their preambles, prerequisites and associated formulae, is assumed and may be tested.
Preamble
Methods of proof, including proof by contradiction and disproof by counter-example, are required. At least one question on the paper will require the use of proof.
15
NEW C3 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
1. Algebra and functions
Simplification of rational expressions including Denominators of rational expressions will be linear or P2:
factorising and cancelling, and algebraic division.
x3  1
ax + b
1
quadratic, eg
,
,
.
ax + b px2 + qx + r x 2  1
1.1
Simplifying rational algebraic
expressions
Cancelling common factors
Multiplying fractions
Dividing fractions
Adding fractions
Subtracting fractions
P2:
1.2
Equations containing algebraic
fractions
P1:
1.6
Algebraic division
Definition of a function. Domain and range of The concept of a function as a one-one or many-one P2:
functions. Composition of functions. Inverse mapping from ℝ (or a subset of ℝ) to ℝ. The
functions and their graphs.
notation f : x ↦ . . . and f(x) will be used.
2.1
Mappings and functions
Introducing the notation of functions
2.3
Composite functions
2.4
Inverse functions
2.5
Modulus functions
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.
The modulus function.
Candidates should be able to sketch the graphs of y = P2:
ax + b and the graphs of y = f(x) and
y = f(x), given the graph of y = f(x).
The modulus function x, x  ℝ
The modulus function f(x)
The modulus function f(x), x  ℝ
16
NEW C3 SPECIFICATION
HEINEMANN TEXTBOOK REFERENCE
NOTES
Candidates should be 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 +

4
).
The graph of y = f(ax + b) will not be required.
2. Trigonometry
Knowledge of secant, cosecant and cotangent and of Angles measured in both degrees and radians.
arcsin, arccos and arctan. Their relationships to sine,
cosine and tangent. Understanding of their graphs
and appropriate restricted domains.
4.1
Secant, cosecant and cotangent
Drawing the graphs of sec x, cosec x
and cot x
4.2
Inverse trigonometric functions
P2:
4.3
Identities
Knowledge and use of double angle formulae; use To include application to half-angles. Knowledge of P2:
of formulae for sin(A  B), cos (A  B) and tan (A  the t (tan 12  ) formulae will not be required.
B) and of expressions for a cos  + b sin  in the
equivalent forms of r cos (  a) or r sin (  a).
Candidates should be able to solve equations such as
a cos  + b sin  = c in a given interval, and to prove
simple identities such as cos x cos 2x + sin x sin 2x 
cos x.
4.4
The addition formulae sin (A  B),
cos (A  B), tan (A  B) and simple
applications
4.5
Double angle formulae
Half angle formulae
4.6
Sum and product formulae
4.7
Solving trigonometric equations
4.8
Alternative forms of a cos  + b sin 
Knowledge and use of sec2  = 1 + tan2  and cosec2
 = 1 + cot2 .
P2:
17
NOTES
HEINEMANN TEXTBOOK REFERENCE
The function ex and its graph.
To include the graph of y = eax + b + c.
P2:
5.1
Exponential functions
The function ln x and its graph; ln x as the inverse
function of ex.
Solution of equations of the form eax + b = p and ln (ax
+ b) = q is expected.
P2:
5.2
The natural logarithmic function
P2:
6.1
The differentiation of ex
6.2
The differentiation of ln x, x > 0
P3:
2.7
Differentiating trigonometric
functions
Differentiation using the product rule, the quotient Differentiation of cosec x, cot x and sec x are P3:
rule and the chain rule.
required. Skill will be expected in the differentiation
of functions generated from standard forms using
products, quotients and composition, such as 2x4 sin
e3 x
x,
, cos x2 and tan2 2x.
x
2.1
Composite functions
2.2
Differentiating composite functions
using the chain rule
2.3
Differentiating products
2.4
Differentiating quotients
2.5
Connected rates of change
2.6
Further examples using
differentiation
2.8
Differentiating relations given
implicitly
NEW C3 SPECIFICATION
3. Exponentials and logarithms
4. Differentiation
Differentiation of ex, ln x , sin x, cos x, tan x and
their sums and differences.
The use of
dy
1

.
dx  dx 
 
 dy 
E.g. finding
dy
for x = sin 3y.
dx
P3:
18
NEW C3 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
5. Numerical methods
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.
P2:
8.1
Locating the roots of f(x) = 0
Approximate solution of equations using simple Solution of equations by use of iterative procedures P2:
iterative methods, including recurrence relations of for which leads will be given
the form xn+1 = f(xn).
8.2
Simple iterative methods
Starting points for iteration
Convergence
How to check the degree of accuracy of
a root
19
Appendix C3 – Formulae
This appendix lists formulae that candidates are expected to remember and that may not be included in formulae booklets.
Trigonometry
cos 2 A  sin 2 A  1
sec 2 A  1  tan 2 A
2
2
cosec A  1  cot A
sin 2 A  2 sin A cos A
cos 2 A  cos2 A  sin 2 A
tan 2 A 
2 tan A
1  tan2 A
Differentiation
function
derivative
sin kx
cos kx
ekx
k cos kx
–k sin kx
kekx
1
x
f  ( x) + g ( x)
f  ( x) g( x)  f ( x) g ( x)
f  ( g ( x )) g  ( x )
ln x
f (x) + g (x)
f ( x) g ( x)
f (g (x))
20
Unit C4 – Core Mathematics 4
The examination
The examination will consist of one 1½ hour paper. It will contain about seven questions of varying length. The mark allocations per question which will be stated
on the paper. All questions should be attempted.
Formulae which candidates are expected to know are given in the appendix to this unit and these will not appear in the booklet, Mathematical Formulae including
Statistical Formulae and Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage.
Candidates are expected to have available a calculator with at least the following keys: +, , , , π, x2, x,
1 y
, x , ln x, ex, x!, sine, cosine and tangent and their
x
inverses in degrees and decimals of a degree, and in radians; memory. Calculators with a facility for symbolic algebra, differentiation and/or integration are not
permitted.
Prerequisites
A knowledge of the specification for C1, C2 and C3 and their preambles, prerequisites and associated formulae, is assumed and may be tested.
NEW C4 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
1. Algebra and functions
Rational functions. Partial fractions (denominators Partial fractions to include denominators such as (ax
not more complicated than repeated linear terms).
+ 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.
P3:
1.1
Partial fractions
Linear factors in the denominator
Quadratic factor in the denominator
Repeated factor in the denominator
Improper algebraic fractions
Quadratic factors in the denominator such as
(x2 + a), a > 0, are not required.
21
NEW C4 SPECIFICATION
NOTES
HEINEMANN TEXTBOOK REFERENCE
2. Coordinate geometry in the (x, y) plane
Parametric equations of curves and conversion Candidates should be able to find the area under a P3:
between Cartesian and parametric forms.
curve given its parametric equations. Candidates
will not be expected to sketch a curve from its
parametric equations.
3.3
Sketching curves given by
parametric equations
Parametric equations that easily
transform to a cartesian equation
Curves that cannot easily be sketched
from their cartesian equation or whose
cartesian equation is difficult to obtain
Curves with trigonometric parametric
equations
P3:
1.3
The binomial series (1 + x)n, n  ℚ,
x < 1
defined The finding of equations of tangents and normals to P3:
curves given parametrically or implicitly is required.
2.8
Differentiating relations given
implicitly
2.9
Differentiating functions given
parametrically
4.7
Exponential growth and decay
Forming and solving simple differential
equations
3. Sequences and series
Binomial series for any rational n.
b
, candidates should be able to obtain the
a
expansion of (ax + b)n, and the expansion of rational
functions by decomposition into partial fractions.
For x 
4. Differentiation
Differentiation of simple
implicitly or parametrically.
Exponential growth and decay.
functions
Knowledge and use of the result
d x
(a ) = ax ln a
dx
is expected.
P3:
22
HEINEMANN TEXTBOOK REFERENCE
NEW C4 SPECIFICATION
NOTES
Formation of simple differential equations.
Questions involving connected rates of change may P3:
be set.
4.7
Exponential growth and decay
Forming and solving simple differential
equations
To include integration of standard functions such as P2:
1
sin 3x, sec2 2x, tan x, e5x,
.
2x
7.1
The integration of ex
7.2
The integration of
4.1
Integrating standard functions
  y 2 d x is required but not   x 2 d y . Candidates P2:
should be able to find a volume of revolution, given
parametric equations.
7.3
Using integration to find areas and
volumes
The area under a curve
Volumes of revolution
P3:
4.6
Further examples on areas of regions
and volumes of solids of revolution
Simple cases of integration by substitution and Except in the simplest of cases the substitution will P3:
integration by parts. These methods as the reverse be given. The integral ln x d x is required. More than
processes of the chain and product rules
one application of integration by parts may be
respectively.
required, for example x 2e x dx .
4.3
Integrating using substitutions
4.4
Integrating by parts
4.5
A systematic approach to integration
5. Integration
Integration of ex,
1
, sinx, cosx.
x
Candidates should recognise integrals of the form
P3:
 f ( x)
dx = ln f(x) + c.

 f( x)
1
x
Candidates are expected to be able to use
trigonometric identities to integrate, for example,
sin2 x, tan2 x, cos2 3x.
Evaluation of volume of revolution.


23
HEINEMANN TEXTBOOK REFERENCE
NEW C4 SPECIFICATION
NOTES
Simple cases of integration using partial fractions.
Integration of rational expressions such as those P3:
arising from partial fractions,
2
3
e.g.
,
.
3x  5 ( x  1) 2
4.2
Integrating using identities
Note that the integration of other rational expressions,
x
2
such as 2
and
is also required (see
x 5
(2 x  1) 4
above paragraphs).
Analytical solution of simple first order differential
equations with separable variables.
General and particular solutions will be required.
P3:
4.7
Exponential growth and decay
Forming and solving simple differential
equations
Numerical integration of functions.
Application of the trapezium rule to functions
covered in C3 and C4. Use of increasing number of
trapezia to improve accuracy and estimate error will
be required. Questions will not require more than
three iterations.
P2:
7.4
Numerical integration: the
trapezium rule
The trapezium rule
P3:
5.1
Some definitions
Directed line segment
Displacement vector
Simpson’s Rule is not required.
6. Vectors
Vectors in two and three dimensions.
24
HEINEMANN TEXTBOOK REFERENCE
NEW C4 SPECIFICATION
NOTES
Magnitude of a vector.
Candidates should be able to find a unit vector in the P3:
direction of a, and be familiar with a.
Algebraic operations of vector addition and
multiplication by scalars, and their geometrical
interpretations.
Position vectors. The distance between two points.


5.1
Some definitions
Modulus of a vector
Equality of vectors
The zero vector
Negative vector
5.2
More definitions and operations on
vectors
Unit vector
P3:
5.2
More definitions and operations on
vectors
Scalar multiplication of a vector
Parallel vectors
Adding vectors
The commutative law
The associative law
Subtracting vectors
Non-parallel vectors
P3:
5.3
Position vectors
Position vector of the mid-point of a
line
5.4
Cartesian components of a vector in
two dimensions
5.5
Cartesian coordinates in three
dimensions
Finding the distance between two
points

OB  OA  AB  b a .
The distance d between two points (x1 , y1 , z1) and (x2
, y2 , z2) is given by
d 2 = (x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2 .
25
NEW C4 SPECIFICATION
Vector equations of lines.
NOTES
To include the forms r = a + tb and
r = c + t(d – c).
HEINEMANN TEXTBOOK REFERENCE
P3:
5.6
Cartesian components of a vector in
three dimensions
5.8
The vector equation of a straight line
5.9
Deriving cartesian equations of a
straight line from a vector equation
5.7
The scalar product of two vectors
The scalar product in terms of cartesian
coordinates
The projection of a vector on to another
vector
Intersection ,or otherwise, of two lines.
The scalar product. Its use for calculating the angle Candidates should know that for

between two lines.
OA = a = a1i + a2j + a3k and

OB = b = b1i + b2j + b3k then
P3:
a . b = a1b1 + a2b2 + a3b3 and
a.b
cos AOB =
.
ab
Candidates should know that if a . b = 0, and that
a and b are non-zero vectors, then a and b are
perpendicular.
26
Appendix C4 – Formulae
This appendix lists formulae that candidates are expected to remember and that may not be included in formulae booklets.
Integration
function
integral
cos kx
1
sin kx + c
k
1
 cos kx + c
k
1 kx
e +c
k
sin kx
ekx
1
x
f  ( x)  g ( x)
f  (g ( x)) g ( x)
ln x + c, x  0
f ( x) + g ( x) + c
f ( g ( x )) + c
Vectors
 x   a
   
 y    b   xa  yb  zc
 z  c 
   
27