YEAR 13 - Mathematics Pure (C3) Term 1 plan

Transcription

1 Week Topic YEAR 13 - Mathematics Pure (C3) Term 1 plan Algebra and functions Simplification of rational expressions including factorising and cancelling. Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs. 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). 3 Exponentials and Logarithms The function e x and its graph. (Link with Differentiation section Differentiation of y=a x ) n x as the inverse function of e x The function ln x and its graph 4-5 Trigonometry Knowledge of secant, cosecant and cotangent and of arcsin, arccos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains Knowledge and use of 1 + tan² = sec² 1 + cot² = cosec² Knowledge and use of formulae for sin (A B), cos (A B) and tan (A B); Knowledge and use of double angle formulae Knowledge and use of the expressions for acos +bsin in the equivalent forms of rcos( ± ) or rsin( ± )

2 6-7 Differentiation Differentiation of e x, sinx,cosx and their sums and differences. Differentiation using the product rule, the quotient rule, the chain rule. Differentiation of tanx The use of dy dx = 1/ dy dx Differentation of lnx 8-9 Numerical Methods Location of the 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 solutions of equations using simple iterative methods, including recurrence relations of the form x f ( x ) n 1 n. Week Topic YEAR 13 - Mathematics Pure (C4) Term 2 plan Partial Fractions Rational functions. Partial fractions (denominators not more complicated than repeated linear terms). Identifying partial fractions Finding partial fractions Improper partial fractions The mixed bag Coordinate Geometry in the (x,y) plane Cartesian and parametric equations of curves and conversion between the two forms. Cartesian or parametric Points on parametrics Parametric pictures

3 15 The binominal Expansion Binomial series for any rational n. For x, b a, students should be able to obtain the expansion of (ax 1 b)n, and the expansion of rational functions by decomposition into partial fractions Differentiation What students need to learn: Differentiation of ex, ln x, sin x, cos x, tan x and their sums and differences. Differentiation using the product rule, the quotient rule and the chain rule. Differentiation of cosec x, cot x and sec x are required. Skill will be expected in the differentiation of functions generated from standard forms using products, quotients and composition, such as 2x4 sin x, e3x x, cos x2 and tan2 2x. The use of dy dx 5 1 ( dx dy ). E.g. finding dy dx for x 5 sin 3y. Differentiation of simple functions defined implicitly or parametrically. The finding of equations of tangents and normals to curves given parametrically or implicitly is required. Exponential growth and decay. Knowledge and use of the result d dx (ax ) 5 ax ln a is expected. Formation of simple differential equations. Questions involving connected rates of change may be set. 18 Vectors Vectors in two and three dimensions. Magnitude of a vector. Students should be able to find a unit vector in the direction of a, and be familiar with a. Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations. Position vectors. OB 2 OA 5 AB 5 b 2 a. The distance between two points. The distance d between two points (x1, y1, z1) and (x2, y2, z2) is given by d 2 5 (x1 2 x2)2 1 ( y1 2 y2)2 1 (z1 2 z2)2. Vector equations of lines. To include the forms r 5 a 1 tb and r 5 c 1 t(d 2 c). Intersection, or otherwise, of two lines. The scalar product. Its use for calculating the angle between two lines. Students should know that for OA 5 a 5 a1i 1 a2 j 1 a3k and OB 5 b 5 b1i 1 b2 j 1 b3k then a. b 5 a1b1 1 a2b2 1 a3b3 and cos AOB 5 a. b a b. Students should know that if a. b 5 0, and a and b are non-zero vectors, then a and b are perpendicular.

4 19-20 Integration Integration of ex, 1 x, sin x, cos x. To include integration of standard functions such as sin 3x, sec2 2x, tan x, e5x, 1 2x. Students should recognise integrals of the form f 9(x) f(x) dx 5 ln f(x) 1 c. Students are expected to be able to use trigonometric identities to integrate, for example, sin2 x, tan2 x, cos2 3x. Evaluation of volume of revolution. p y2 dx is required, but not p x2 dy. Students should be able to find a volume of revolution, given parametric equations. Simple cases of integration by substitution and integration by parts. These methods as the reverse processes of the chain and product rules respectively. 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 ex dx. Simple cases of integration using partial fractions. Integration of rational expressions such as those arising from partial fractions, e.g. 2 3x 1 5, 3 (x 2 1)2. Note that the integration of other rational expressions, such as x x2 1 5 and 2 (2x 2 1)4 is also required (see above paragraphs). Analytical solution of simple first order differential equations with separable variables. General and particular solutions will be required. YEAR 13 - Mathematics Pure (C34) Term 3 plan REVISION AND EXAMINATION Revision C3 and C4 Practice C34 Papers

5 Week Topic YEAR 13 - Mathematics Mechanics (M1) Term 1 plan Mathematical Models in Mechanics To be able to define Particle - An object which is small in comparison with other sizes or lengths can be modelled as a particle. Rod - An object with one dimension small in comparison with another (such as a metre ruler or beam) can be modelled as a rod Lamina - An object with one dimension (its thickness) very small in comparison with the other two (its length and width) can be modelled as a lamina. Uniform Body - If an object is uniform then its mass is evenly distributed over its entire volume. Light Object - If the mass of an object is very small in comparison with the masses of other objects, we can model it as being light. Inextensible - If a string does not stretch under a load it is inextensible or inelastic. Smooth - If we want to ignore the effects of friction, we can model a surface as being smooth. Rough Surface - If a surface is not smooth it is said to be rough. We need to consider the friction between the surface and an object moving or tending to move along it. For example, a ski slope might be modelled as a smooth or a rough surface depending on the problem to be solved. Air Resistance - When an object moves through the air it experiences a resistance due to friction. Wind - Unless it is specifically mentioned, you can usually ignore any effects due to the wind in your models. Gravity - The force of attraction between all objects with mass is called gravity. Because the mass of the earth is very large, we can usually assume that all objects are attracted towards the Earth (ignoring any force of attraction between the objects themselves). We usually model the force of the Earth s gravity as uniform, and acting vertically downwards. The acceleration due to gravity is denoted by g and is always assumed to be constant at 9.8ms-2. This value is given on the front of the exam paper.

6 3-5 Kinematics of a Particle Moving in a Straight Line To be able to know The five formulae for solving problems about particles moving in a straight line with constant acceleration are v = u + at s = ½ (u + v)t v2 = u2 + 2as s = ut +½at2 s = vt - ½at2 An object moving vertically in a straight line can be modelled as a particle with a constant downward acceleration of g = 9.8ms-2 The gradient of a speed-time graph illustrating the motion of a particle represents the acceleration of the particle The area under a speed-time graph illustrating the motion of a particle represents the distance moved by the particle Area of trapezium = average of the parallel sides x height = ½ (a + b) x h At constant speed, distance = speed x time

7 6-10 Dynamics of a Particle Moving in a Straight Line To be able to know The unit of force is the Newton (N). It is defined as the force that will cause a mass of 1kg to accelerate at a rate of 1ms-2 F = ma The force due to gravity is called the weight of an object, and it acts vertically downwards. A particle falling freely experiences acceleration of g = 9.8ms-2 W = mg The component of a force of magnitude F acting in a certain direction is F cosine(θ), where Θ is the size of the angle between the force and the direction The maximum or limiting value of the friction FMAX between two surfaces is given by FMAX = μr where μ is the coefficient of friction and R is the normal reaction between the two surfaces If a force P is applied to a block of mass m which is at rest on a rough horizontal surface and P acts at an angle to the horizontal: The normal reaction R is not equal to mg The force tending to pull or push the block along the plane is not equal to P A particle placed on a rough inclined plane will remain at rest if tangent(θ) μ where Θ is the angle the plane makes with the horizontal and μ is the coefficient of friction between the particle and the plane Provided all parts of a connected system are moving in the same straight line you can treat the whole system as a single particle In problems involving particles which are connected by string(s) which pass over pulley(s) you cannot treat the whole system as a single particle. This is because the particles are moving in different directions. The momentum of a body of mass m which is moving with velocity v is mv Momentum = mass x velocity If a constant force F acts for time t then we define the impulse of the force to be Ft Impulse = Force x Time The Impulse-Momentum Principle states that Impulse = Final Momentum Initial Momentum = Change in Momentum I = mv mu The principle of Conservation of Momentum states that Total Momentum Before Impact = Total Momentum After Impact m1u1 + m2u2 = m1v1 + m2v2

8 YEAR 13 - Mathematics Mechanics (M1) Term 2 plan Week Topic Statics of a particle Forces treated as vectors. Resolution of forces. Equilibrium of a paticle under coplanar forces. Weight, normal reaction, tension and thrust, friction Coefficient of friction Only simple cases of the application of the conditions for equilibrium to uncomplicated systems will be required. An understanding of F = R is a situation of equilibrium Moments Moments of a force Simple problems involving coplanar parallel forces acting on a body and conditions for equilibrium in such situations Vectors Magnitude and direction of a vector. Resultant of vectors may also be required. Application of vectors to displacements, velocities, accelerations and forces in a plane. Candidates may be required to resolve a vector into two components or use a vector diagram. Questions may be set involving the unit vectors i and j. Use of changeofdisplacement velocity the case of constant velocity, and of time changeofvelocity acceleration in the case of constant acceleration, will be required. time

9 YEAR 13 - Mathematics Mechanics (M1) Term 3 plan Week Topic Revision and Examination Revision M1 and Practice of Past Papers M1