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Courses How well does this course prepare me for college math?

  1. Apr 16, 2010 #1
    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 modelled 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 modelled 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 modelled 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.
     
  2. jcsd
  3. Apr 16, 2010 #2
    I'd say you're in pretty good shape. Depends where you're trying to start though. Looks like you can probably skip calc 1. Probably still need to take Calc 2 and Calc 3 although you will have seen a lot of that already. Looks like you missed surface integrals, path integrals, integral theorems etc. just from a quick look at you're list. Also Taylor series. Also spherical and cylindrical coordinates. Definitely need to take ODE's still. Looks like you covered a fair amount of linear algebra but if you don't take it you will still be missing out on important things like diagonalization and spectral decomposition. So basically I'd say you can skip calc 1. You will be extremely well prepared for calc 2,3 ODE's and linear algebra but I can't advise you skip any of them.
     
  4. Apr 17, 2010 #3
    Wow, thanks. Makes me feel good that I'll be well prepared or would have seen a lot of the stuff in these classes. How does this course compare to the AP Calculus or other advanced high school math classes taught in the US?

    Also, how do the Mechanics and Statistics portions of this help me? (It always confused me as to why Mechanics needs to be in a course called "Further Mathematics")

    For majoring in Math, what other classes will I normally need apart from these 5? I know it varies from school to school but generally what would I need?
     
  5. Apr 17, 2010 #4
    I'd say that course is much better preparation than AP calculus. There are two AP Calc. classes, AB and BC. AB covers essentially the same material as Calc. I does. Integration, differentiation, applications of both, along with the underpinnings of calculus like limits, continuity, etc. BC is more like calculus I with the majority of calculus II. Your course seems to have covered everything AB would have, along with a good portion of the additional content in BC, with linear algebra and diff. eqs. thrown in.

    So like Phyisab**** said, you seem to be very well prepared for undergrad classes.
     
  6. Apr 17, 2010 #5

    mrb

    User Avatar

    Good grief. When I started my math major I knew, more or less, elementary algebra and I had some vague recollections of plane geometry and trig. Your preparation is fine, trust me.

    The web pages of the math departments at the schools you're looking at will very likely list the required classes for a math major or minor.
     
  7. Apr 17, 2010 #6
    You guys' posts make me happy that I took this class. It was certainly difficult at first because I hadn't completed any of the pre-requisites but after consistent practice and getting a good grasp of the basics in the standard A-Level Math, it actually became somewhat easy. Still need to practice some Mechanics though.

    Now can someone answer my question about the Stats and Mechanics portions?

    Also, I have very little knowledge of Physics but also have a high interest in it. What routes can I take in college which will allow me to pursue both Math and Physics?
     
  8. Apr 17, 2010 #7
    A-level further Math covers more than AP Calculus BC. I actually recommand him to take some stat courses to "complete" his high school maths
     
  9. Apr 17, 2010 #8
    The coverage of linear algebra is surprisingly good. If you are confident in your understanding of the linear algebra listed, I'd recommend skipping any elementary treatments of this subject. The subject is easy, you've already been exposed to its core ideas, and you can pick up the rest of what you need as necessary. I'd recommend instead taking a more rigorous or abstract treatment of linear algebra, the sort you might take after a semester of abstract algebra. This will do you much more good than taking the typical elementary treatment.

    Unfortunately it seems your treatment of elementary calculus was somewhat lacking. Did you cover the various convergence tests for infinite series (ratio test, root test, integral test, etc.)? Did you cover the elementary integration techniques (integration by parts, partial fractions, trigonometric substitution, etc.)? If you did do these things, then you could probably skip calculus II. If not, it's probably in your best interest to take it.

    My recommendations, during your first year of college, are to do something like this:

    First semester: calculus II, ODEs, abstract algebra, university physics I, and then one or two other classes you like.

    Second semester: multivariable calculus, abstract linear algebra, a first course in analysis, university physics II, and then one or two other classes you like.

    This will give you a very strong preparation for the next three years of study, significantly stronger than most undergraduates in the country will have. It will be easy for you to add on a full math major.

    Something to consider is moving one of these courses to the summer prior to officially starting college. Taking just one summer class isn't that bad. There's even something sort of enjoyable about it: because you have only one class, you can easily devote any time needed to get an A without worrying about other classes. A good choice for this would be calculus II or ODEs.

    You could, if you wanted, do both in the summer and get a very awesome start on your college education. This would allow you to add in some extra stuff your first year of college, perhaps other physics courses, more gen. ed. electives, computer science courses, etc. I really would recommend it if you are willing to put forth the effort it would require.

    My recommendations maybe seem ambitious, and they probably are, but I've realized after a year or two of college that most people--myself included--do far less than they are capable of simply because they don't actually know what they're capable of. Amazing and unexpected things often happen when you don't hold yourself back.
     
  10. Apr 17, 2010 #9

    mrb

    User Avatar

    This sounds great if it is possible. The thing is, at the school I graduated from, it would never work. You have to finish Calc 2 before you can take ODEs. You have to finish ODEs before you can take abstract algebra (why? I don't know). You have to take Calc 3 before you can take analysis. Etc. PLUS you must take about 1.5 years worth of general ed requirements, and unless you want to be in a bunch of worthless classes with freshmen when you're a senior and should have moved on to better things, you should get them out of the way early. What this all comes down to is that it's not really feasible to take more than 1 or 2 math and science classes per semester for your first couple years. If this sounds like it's a system specifically designed to hold students back, well, that's how it felt going through it too.

    But are most places different from this? Would the above schedule be feasible at most places?
     
  11. Apr 18, 2010 #10
    Well, it would be feasible at my school, where the math department just ignores all prerequisites. But you're definitely right: it will depend a lot on the particular college.

    As for general education requirements, I guess that's going to vary a lot as well from school to school. Where I'm at presently, if you come in with a few courses that count from high school (say three or four), you can easily just take maybe two classes a semester to fulfill these requirements and be done by the end of your sophomore year. That's assuming you don't do any such classes over the summer, like some people do. Plus you can take up to four of these (no more than one at once) pass/fail, which lets you focus on your major classes and also lets you take classes far from your comfort zone without worrying about grades.

    It's a fantastic system, but I admit that most schools are not like this.

    So yeah, that schedule would be feasible at my school (and I know a person who's basically doing that exact schedule, except I think he took took ODEs in high school and instead of an introduction to analysis he's taking a graduate course in complex analysis).
     
  12. Jul 31, 2010 #11
    I didn't look back at this thread after my last reply. Just did cause I came back wanting to bump this back. To answer your question, I have not covered the convergence tests but have done the integration techniques you mentioned (in the standard A-Level Math class). So should I skip Calculus II and start directly with Calc III? Or should I still take II? Also the schedule you recommended seems awesome for my interests/pursuing my major but it certainly seems rigorous and being a freshman who is leaving his home country to attend college, I would also need time to adjust/socialize which I think such a schedule will not allow. Do you still recommend it? What I'm thinking of right now is 2 Math classes (Calc II/III and Differential Equations), a basic Physics class, a basic Philosophy class and Intermediate Microeconomics (took A-Level Economics in HS which covers a ton of the basic ****). What do you think?

    Also, now that classes will start (I'm gonna be attending UVA) in about three weeks, more opinions on whether I should take Calc II or III, and other general classes I should take being a probable math major, will be appreciated.
     
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