- #1
ahsanxr
- 350
- 6
I'm currently enrolled in A-Level Further Mathematics (the most advanced math course available to us) and have completed pretty much everything, having more difficulty in the technical and complicated algebra involved rather than understanding the topics. I just wanted to know how knowledge of this course prepares me for college level math. I'll most probably be attending either the University of Virginia or Syracuse University this fall, where I will be at least minoring in math. (Note that along with this course I've also taken a normal A-Level Math class which is pretty basic and a pre-requisite for this class)
Here is the content which it covers:
Pure Mathematics:
1. Polynomials and rational functions
− recall and use the relations between the roots
and coefficients of polynomial equations, for
equations of degree 2, 3, 4 only;
− use a given simple substitution to obtain an
equation whose roots are related in a simple
way to those of the original equation;
− sketch graphs of simple rational functions,
including the determination of oblique
asymptotes, in cases where the degree of the
numerator and the denominator are at most 2
(detailed plotting of curves will not be required,
but sketches will generally be expected to show
significant features, such as turning points,
asymptotes and intersections with the axes).
2. Polar coordinates
− understand the relations between cartesian and
polar coordinates (using the convention
r [ 0), and convert equations of curves from
cartesian to polar form and vice versa;
− sketch simple polar curves, for 0 Y θ < 2π or
−π < θ Y π or a subset of either of these intervals
(detailed plotting of curves will not be required, but
sketches will generally be expected to show
significant features, such as symmetry, the form of
the curve at the pole and least/greatest values of
r);
− recall the formula ∫ β
α
2 r
2
1
dθ for the area of a sector,
and use this formula in simple cases.
3. Summation of series
− use the standard results forΣr , Σ 2
r , Σ 3
r to
find related sums;
− use the method of differences to obtain the sum of
a finite series, e.g. by expressing the general term
in partial fractions;
− recognise, by direct consideration of a sum to n
terms, when a series is convergent, and find the
sum to infinity in such cases.
4. Mathematical induction
− use the method of mathematical induction to
establish a given result (questions set may involve
divisibility tests and inequalities, for example);
− recognise situations where conjecture based on a
limited trial followed by inductive proof is a useful
strategy, and carry this out in simple cases e.g.
find the nth derivative of xex.
5. Differentiation and integration
− obtain an expression for (d2x/dx2) in cases where the
relation between y and x is defined implicitly or
parametrically;
− derive and use reduction formulae for the
evaluation of definite integrals in simple cases;
− use integration to find
mean values and centroids of two- and threedimensional
figures (where equations are
expressed in cartesian coordinates, including the
use of a parameter), using strips, discs or shells as
appropriate,
arc lengths (for curves with equations in cartesian
coordinates, including the use of a parameter, or in
polar coordinates),
surface areas of revolution about one of the axes
(for curves with equations in cartesian coordinates,
including the use of a parameter, but not for
curves with equations in polar coordinates).
6. Differential equations
− recall the meaning of the terms ‘complementary
function' and ‘particular integral' in the context of
linear differential equations, and recall that the
general solution is the sum of the complementary
function and a particular integral;
− find the complementary function for a second order
linear differential equation with constant
coefficients;
− recall the form of, and find, a particular integral for
a second order linear differential equation
in the cases where a polynomial or
ebx or a cos px + b sin px is a suitable form, and in
other simple cases find the appropriate
coefficient(s) given a suitable form of particular
integral;
− use a substitution to reduce a given differential
equation to a second order linear equation with
constant coefficients;
− use initial conditions to find a particular solution to a
differential equation, and interpret a solution in terms
of a problem modeled by a differential equation.
7. Complex numbers
understand de Moivre's theorem, for a positive integral
exponent, in terms of the geometrical effect of
multiplication of complex numbers;
− prove de Moivre's theorem for a positive integral
exponent;
− use de Moivre's theorem for positive integral exponent
to express trigonometrical ratios of multiple angles in
terms of powers of trigonometrical ratios of the
fundamental angle;
− use de Moivre's theorem, for a positive or negative
rational exponent
in expressing powers of sin θ and cos θ in terms of
multiple angles,
in the summation of series,
in finding and using the nth roots of unity.
8. Vectors
− use the equation of a plane in any of the forms
ax + by + cz = d or r.n. = p or r = a + λb + μc, and
convert equations of planes from one form to another
as necessary in solving problems;
− recall that the vector product a x b of two vectors can
be expressed either as I a I IbI sin θ n ˆ , where n ˆ is a
unit vector, or in component form as
(a2 b3 – a3 b2) i + (a3 b1 – a1 b3) j + (a1 b2 – a2 b1) k;
− use equations of lines and planes, together with scalar
and vector products where appropriate, to solve
problems concerning distances, angles and
intersections, including
determining whether a line lies in a plane, is parallel to
a plane or intersects a plane, and finding the point of
intersection of a line and a plane when it exists,
finding the perpendicular distance from a point to a
plane,finding the angle between a line and a plane, and the
angle between two planes,
finding an equation for the line of intersection of two
planes,
calculating the shortest distance between two skew
lines,
finding an equation for the common perpendicular to
two skew lines.
9. Matrices and linear spaces
− recall and use the axioms of a linear (vector) space
(restricted to spaces of finite dimension over the field
of real numbers only);
− understand the idea of linear independence, and
determine whether a given set of vectors is dependent
or independent;
− understand the idea of the subspace spanned by a
given set of vectors;
− recall that a basis for a space is a linearly
independent set of vectors that spans the space, and
determine a basis in simple cases;
− recall that the dimension of a space is the number of
vectors in a basis;
− understand the use of matrices to represent linear
transformations from
n →
m;
− understand the terms ‘column space', ‘row space',
‘range space' and ‘null space', and determine the
dimensions of, and bases for, these spaces in simple
cases;
− determine the rank of a square matrix, and use
(without proof) the relation between the rank, the
dimension of the null space and the order of the
matrix;
− use methods associated with matrices and linear
spaces in the context of the solution of a set of linear
equations;
− evaluate the determinant of a square matrix and find
the inverse of a non-singular matrix
(2 x 2 and 3 x 3 matrices only), and recall that the
columns (or rows) of a square matrix are
independent if and only if the determinant is nonzero;
− understand the terms ‘eigenvalue' and ‘eigenvector',
as applied to square matrices;
− find eigenvalues and eigenvectors of 2 x 2 and 3 x 3
matrices (restricted to cases where the eigenvalues
are real and distinct);
− express a matrix in the form QDQ−1, where D is a
diagonal matrix of eigenvalues and Q is a matrix
whose columns are eigenvectors, and use this
expression, e.g. in calculating powers of matrices.
Mechanics:
1. Momentum and impulse
− recall and use the definition of linear
momentum, and show understanding of its
vector nature (in one dimension only);
− recall Newton's experimental law and the
definition of the coefficient of restitution, the
property 0 Y e Y 1, and the meaning of the
terms ‘perfectly elastic' (e = 1) and ‘inelastic'
(e = 0);
− use conservation of linear momentum and/or
Newton's experimental law to solve problems
that may be modeled as the direct impact of two
smooth spheres or the direct or oblique impact
of a smooth sphere with a fixed surface;
− recall and use the definition of the impulse of a
constant force, and relate the impulse acting on
a particle to the change of momentum of the
particle (in one dimension only).
2. Circular motion
− recall and use the radial and transverse
components of acceleration for a particle
moving in a circle with variable speed;
− solve problems which can be modeled by the
motion of a particle in a vertical circle without
loss of energy (including finding the tension in a
string or a normal contact force, locating points
at which these are zero, and conditions for
complete circular motion).
3. Equilibrium of a rigid body under
coplanar forces
− understand and use the result that the effect of
gravity on a rigid body is equivalent to a single
force acting at the centre of mass of the body,
and identify the centre of mass by
considerations of symmetry in suitable cases;
− calculate the moment of a force about a point in
2 dimensional situations only (understanding of
the vector nature of moments is not required);
− recall that if a rigid body is in equilibrium under
the action of coplanar forces then the vector
sum of the forces is zero and the sum of the
moments of the forces about any point is zero,
and the converse of this;
− use Newton's third law in situations involving the
contact of rigid bodies in equilibrium;
− solve problems involving the equilibrium of rigid
bodies under the action of coplanar forces
(problems set will not involve complicated
trigonometry).
4. Rotation of a rigid body
− understand and use the definition of the moment of
inertia of a system of particles about a fixed axis as
Σ 2
mr , and the additive property of moment of
inertia for a rigid body composed of several parts
(the use of integration to find moments of inertia will
not be required);
− use the parallel and perpendicular axes theorems
(proofs of these theorems will not be required);
− recall and use the equation of angular motion
C = Iθ&& for the motion of a rigid body about a fixed
axis (simple cases only, where the moment C
arises from constant forces such as weights or the
tension in a string wrapped around the
circumference of a flywheel; knowledge of couples
is not included and problems will not involve
consideration or calculation of forces acting at the
axis of rotation);
− recall and use the formula 2
2
1
Ιω for the kinetic
energy of a rigid body rotating about a fixed axis;
− use conservation of energy in solving problems
concerning mechanical systems where rotation of a
rigid body about a fixed axis is involved.
5. Simple harmonic motion
− recall a definition of SHM and understand the
concepts of period and amplitude;
− use standard SHM formulae in the course of
solving problems;
− set up the differential equation of motion in
problems leading to SHM, recall and use
appropriate forms of solution, and identify the
period and amplitude of the motion;
− recognise situations where an exact equation of
motion may be approximated by an SHM equation,
carry out necessary approximations (e.g. small
angle approximations or binomial approximations)
and appreciate the conditions necessary for such
approximations to be useful.
Statistics:
6. Further work on distributions
− use the definition of the distribution function as a
probability to deduce the form of a distribution
function in simple cases, e.g. to find the distribution
function for Y, where Y = X
3 and X has a given
distribution;
− understand conditions under which a geometric
distribution or negative exponential distribution may
be a suitable probability model;
− recall and use the formula for the calculation of
geometric or negative exponential probabilities;
− recall and use the means and variances of a
geometric distribution and negative exponential
distribution.
7. Inference using normal and
t-distributions
− formulate hypotheses and apply a hypothesis test
concerning the population mean using a small
sample drawn from a normal population of
unknown variance, using a t-test;
− calculate a pooled estimate of a population
variance from two samples (calculations based on
either raw or summarised data may be required);
− formulate hypotheses concerning the difference of
population means, and apply, as appropriate,
a 2-sample t-test,
a paired sample t-test,
a test using a normal distribution
(the ability to select the test appropriate to the
circumstances of a problem is expected);
− determine a confidence interval for a population
mean, based on a small sample from a normal
population with unknown variance, using a
t-distribution;
− determine a confidence interval for a difference of
population means, using a t-distribution, or a
normal distribution, as appropriate.
8. χ2–tests (chi-square) − fit a theoretical distribution, as prescribed by a
given hypothesis, to given data (questions will not
involve lengthy calculations);
− use a χ
2-test, with the appropriate number of
degrees of freedom, to carry out the corresponding
goodness of fit analysis (classes should be
combined so that each expected frequency is at
least 5);
− use a χ
2-test, with the appropriate number of
degrees of freedom, for independence in a
contingency table (Yates’ correction is not required,
but classes should be combined so that the
expected frequency in each cell is at least 5).
9. Bivariate data
− understand the concept of least squares,
regression lines and correlation in the context of a
scatter diagram;
− calculate, both from simple raw data and from
summarised data, the equations of regression lines
and the product moment correlation coefficient, and
appreciate the distinction between the regression
line of y on x and that of x on y;
− recall and use the facts that both regression lines
pass through the mean centre ( x
, y ) and that the
product moment correlation coefficient r and the
regression coefficients b1, b2 are related by
r
2 = b1b2;
− select and use, in the context of a problem, the
appropriate regression line to estimate a value, and
understand the uncertainties associated with such
estimations;
− relate, in simple terms, the value of the product
moment correlation coefficient to the appearance of
the scatter diagram, with particular reference to the
interpretation of cases where the value of the
product moment correlation coefficient is close to
+1, −1 or 0;
− carry out a hypothesis test based on the product
moment correlation coefficient.
--------
Thanks for the help.
Here is the content which it covers:
Pure Mathematics:
1. Polynomials and rational functions
− recall and use the relations between the roots
and coefficients of polynomial equations, for
equations of degree 2, 3, 4 only;
− use a given simple substitution to obtain an
equation whose roots are related in a simple
way to those of the original equation;
− sketch graphs of simple rational functions,
including the determination of oblique
asymptotes, in cases where the degree of the
numerator and the denominator are at most 2
(detailed plotting of curves will not be required,
but sketches will generally be expected to show
significant features, such as turning points,
asymptotes and intersections with the axes).
2. Polar coordinates
− understand the relations between cartesian and
polar coordinates (using the convention
r [ 0), and convert equations of curves from
cartesian to polar form and vice versa;
− sketch simple polar curves, for 0 Y θ < 2π or
−π < θ Y π or a subset of either of these intervals
(detailed plotting of curves will not be required, but
sketches will generally be expected to show
significant features, such as symmetry, the form of
the curve at the pole and least/greatest values of
r);
− recall the formula ∫ β
α
2 r
2
1
dθ for the area of a sector,
and use this formula in simple cases.
3. Summation of series
− use the standard results forΣr , Σ 2
r , Σ 3
r to
find related sums;
− use the method of differences to obtain the sum of
a finite series, e.g. by expressing the general term
in partial fractions;
− recognise, by direct consideration of a sum to n
terms, when a series is convergent, and find the
sum to infinity in such cases.
4. Mathematical induction
− use the method of mathematical induction to
establish a given result (questions set may involve
divisibility tests and inequalities, for example);
− recognise situations where conjecture based on a
limited trial followed by inductive proof is a useful
strategy, and carry this out in simple cases e.g.
find the nth derivative of xex.
5. Differentiation and integration
− obtain an expression for (d2x/dx2) in cases where the
relation between y and x is defined implicitly or
parametrically;
− derive and use reduction formulae for the
evaluation of definite integrals in simple cases;
− use integration to find
mean values and centroids of two- and threedimensional
figures (where equations are
expressed in cartesian coordinates, including the
use of a parameter), using strips, discs or shells as
appropriate,
arc lengths (for curves with equations in cartesian
coordinates, including the use of a parameter, or in
polar coordinates),
surface areas of revolution about one of the axes
(for curves with equations in cartesian coordinates,
including the use of a parameter, but not for
curves with equations in polar coordinates).
6. Differential equations
− recall the meaning of the terms ‘complementary
function' and ‘particular integral' in the context of
linear differential equations, and recall that the
general solution is the sum of the complementary
function and a particular integral;
− find the complementary function for a second order
linear differential equation with constant
coefficients;
− recall the form of, and find, a particular integral for
a second order linear differential equation
in the cases where a polynomial or
ebx or a cos px + b sin px is a suitable form, and in
other simple cases find the appropriate
coefficient(s) given a suitable form of particular
integral;
− use a substitution to reduce a given differential
equation to a second order linear equation with
constant coefficients;
− use initial conditions to find a particular solution to a
differential equation, and interpret a solution in terms
of a problem modeled by a differential equation.
7. Complex numbers
understand de Moivre's theorem, for a positive integral
exponent, in terms of the geometrical effect of
multiplication of complex numbers;
− prove de Moivre's theorem for a positive integral
exponent;
− use de Moivre's theorem for positive integral exponent
to express trigonometrical ratios of multiple angles in
terms of powers of trigonometrical ratios of the
fundamental angle;
− use de Moivre's theorem, for a positive or negative
rational exponent
in expressing powers of sin θ and cos θ in terms of
multiple angles,
in the summation of series,
in finding and using the nth roots of unity.
8. Vectors
− use the equation of a plane in any of the forms
ax + by + cz = d or r.n. = p or r = a + λb + μc, and
convert equations of planes from one form to another
as necessary in solving problems;
− recall that the vector product a x b of two vectors can
be expressed either as I a I IbI sin θ n ˆ , where n ˆ is a
unit vector, or in component form as
(a2 b3 – a3 b2) i + (a3 b1 – a1 b3) j + (a1 b2 – a2 b1) k;
− use equations of lines and planes, together with scalar
and vector products where appropriate, to solve
problems concerning distances, angles and
intersections, including
determining whether a line lies in a plane, is parallel to
a plane or intersects a plane, and finding the point of
intersection of a line and a plane when it exists,
finding the perpendicular distance from a point to a
plane,finding the angle between a line and a plane, and the
angle between two planes,
finding an equation for the line of intersection of two
planes,
calculating the shortest distance between two skew
lines,
finding an equation for the common perpendicular to
two skew lines.
9. Matrices and linear spaces
− recall and use the axioms of a linear (vector) space
(restricted to spaces of finite dimension over the field
of real numbers only);
− understand the idea of linear independence, and
determine whether a given set of vectors is dependent
or independent;
− understand the idea of the subspace spanned by a
given set of vectors;
− recall that a basis for a space is a linearly
independent set of vectors that spans the space, and
determine a basis in simple cases;
− recall that the dimension of a space is the number of
vectors in a basis;
− understand the use of matrices to represent linear
transformations from
n →
m;
− understand the terms ‘column space', ‘row space',
‘range space' and ‘null space', and determine the
dimensions of, and bases for, these spaces in simple
cases;
− determine the rank of a square matrix, and use
(without proof) the relation between the rank, the
dimension of the null space and the order of the
matrix;
− use methods associated with matrices and linear
spaces in the context of the solution of a set of linear
equations;
− evaluate the determinant of a square matrix and find
the inverse of a non-singular matrix
(2 x 2 and 3 x 3 matrices only), and recall that the
columns (or rows) of a square matrix are
independent if and only if the determinant is nonzero;
− understand the terms ‘eigenvalue' and ‘eigenvector',
as applied to square matrices;
− find eigenvalues and eigenvectors of 2 x 2 and 3 x 3
matrices (restricted to cases where the eigenvalues
are real and distinct);
− express a matrix in the form QDQ−1, where D is a
diagonal matrix of eigenvalues and Q is a matrix
whose columns are eigenvectors, and use this
expression, e.g. in calculating powers of matrices.
Mechanics:
1. Momentum and impulse
− recall and use the definition of linear
momentum, and show understanding of its
vector nature (in one dimension only);
− recall Newton's experimental law and the
definition of the coefficient of restitution, the
property 0 Y e Y 1, and the meaning of the
terms ‘perfectly elastic' (e = 1) and ‘inelastic'
(e = 0);
− use conservation of linear momentum and/or
Newton's experimental law to solve problems
that may be modeled as the direct impact of two
smooth spheres or the direct or oblique impact
of a smooth sphere with a fixed surface;
− recall and use the definition of the impulse of a
constant force, and relate the impulse acting on
a particle to the change of momentum of the
particle (in one dimension only).
2. Circular motion
− recall and use the radial and transverse
components of acceleration for a particle
moving in a circle with variable speed;
− solve problems which can be modeled by the
motion of a particle in a vertical circle without
loss of energy (including finding the tension in a
string or a normal contact force, locating points
at which these are zero, and conditions for
complete circular motion).
3. Equilibrium of a rigid body under
coplanar forces
− understand and use the result that the effect of
gravity on a rigid body is equivalent to a single
force acting at the centre of mass of the body,
and identify the centre of mass by
considerations of symmetry in suitable cases;
− calculate the moment of a force about a point in
2 dimensional situations only (understanding of
the vector nature of moments is not required);
− recall that if a rigid body is in equilibrium under
the action of coplanar forces then the vector
sum of the forces is zero and the sum of the
moments of the forces about any point is zero,
and the converse of this;
− use Newton's third law in situations involving the
contact of rigid bodies in equilibrium;
− solve problems involving the equilibrium of rigid
bodies under the action of coplanar forces
(problems set will not involve complicated
trigonometry).
4. Rotation of a rigid body
− understand and use the definition of the moment of
inertia of a system of particles about a fixed axis as
Σ 2
mr , and the additive property of moment of
inertia for a rigid body composed of several parts
(the use of integration to find moments of inertia will
not be required);
− use the parallel and perpendicular axes theorems
(proofs of these theorems will not be required);
− recall and use the equation of angular motion
C = Iθ&& for the motion of a rigid body about a fixed
axis (simple cases only, where the moment C
arises from constant forces such as weights or the
tension in a string wrapped around the
circumference of a flywheel; knowledge of couples
is not included and problems will not involve
consideration or calculation of forces acting at the
axis of rotation);
− recall and use the formula 2
2
1
Ιω for the kinetic
energy of a rigid body rotating about a fixed axis;
− use conservation of energy in solving problems
concerning mechanical systems where rotation of a
rigid body about a fixed axis is involved.
5. Simple harmonic motion
− recall a definition of SHM and understand the
concepts of period and amplitude;
− use standard SHM formulae in the course of
solving problems;
− set up the differential equation of motion in
problems leading to SHM, recall and use
appropriate forms of solution, and identify the
period and amplitude of the motion;
− recognise situations where an exact equation of
motion may be approximated by an SHM equation,
carry out necessary approximations (e.g. small
angle approximations or binomial approximations)
and appreciate the conditions necessary for such
approximations to be useful.
Statistics:
6. Further work on distributions
− use the definition of the distribution function as a
probability to deduce the form of a distribution
function in simple cases, e.g. to find the distribution
function for Y, where Y = X
3 and X has a given
distribution;
− understand conditions under which a geometric
distribution or negative exponential distribution may
be a suitable probability model;
− recall and use the formula for the calculation of
geometric or negative exponential probabilities;
− recall and use the means and variances of a
geometric distribution and negative exponential
distribution.
7. Inference using normal and
t-distributions
− formulate hypotheses and apply a hypothesis test
concerning the population mean using a small
sample drawn from a normal population of
unknown variance, using a t-test;
− calculate a pooled estimate of a population
variance from two samples (calculations based on
either raw or summarised data may be required);
− formulate hypotheses concerning the difference of
population means, and apply, as appropriate,
a 2-sample t-test,
a paired sample t-test,
a test using a normal distribution
(the ability to select the test appropriate to the
circumstances of a problem is expected);
− determine a confidence interval for a population
mean, based on a small sample from a normal
population with unknown variance, using a
t-distribution;
− determine a confidence interval for a difference of
population means, using a t-distribution, or a
normal distribution, as appropriate.
8. χ2–tests (chi-square) − fit a theoretical distribution, as prescribed by a
given hypothesis, to given data (questions will not
involve lengthy calculations);
− use a χ
2-test, with the appropriate number of
degrees of freedom, to carry out the corresponding
goodness of fit analysis (classes should be
combined so that each expected frequency is at
least 5);
− use a χ
2-test, with the appropriate number of
degrees of freedom, for independence in a
contingency table (Yates’ correction is not required,
but classes should be combined so that the
expected frequency in each cell is at least 5).
9. Bivariate data
− understand the concept of least squares,
regression lines and correlation in the context of a
scatter diagram;
− calculate, both from simple raw data and from
summarised data, the equations of regression lines
and the product moment correlation coefficient, and
appreciate the distinction between the regression
line of y on x and that of x on y;
− recall and use the facts that both regression lines
pass through the mean centre ( x
, y ) and that the
product moment correlation coefficient r and the
regression coefficients b1, b2 are related by
r
2 = b1b2;
− select and use, in the context of a problem, the
appropriate regression line to estimate a value, and
understand the uncertainties associated with such
estimations;
− relate, in simple terms, the value of the product
moment correlation coefficient to the appearance of
the scatter diagram, with particular reference to the
interpretation of cases where the value of the
product moment correlation coefficient is close to
+1, −1 or 0;
− carry out a hypothesis test based on the product
moment correlation coefficient.
--------
Thanks for the help.