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Unit P3: Pure Mathematics 3

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6 topics · 18 learning objectives

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  1. P3.1 - Algebra and functions

    1. P3.1.1Simplification of rational Denominators of rational

      Simplification of rational Denominators of rational expressions will be linear or expressions including factorising quadratic, and cancelling, and algebraic 1 ax +b x3 +1 division. e.g.,,. ax + b px2 + qx + r x2 −1.

    2. P3.1.2Definition of a function

      Definition of a function.; Domain The concept of a function as a one-one or many-one and range of functions. mapping from (or a subset of) to.; The notation Composition of functions.; Inverse f: x and f(x) will be used. functions and their graphs. ℝ ℝ ℝ Students should know that fg will mean ‘do g first, then f ’.; Students should know that if f −1 exists, then f −1f(x) = ff −1(x) = x.

    3. P3.1.3modulus function

      The modulus function.; Students should be able to sketch the graphs of y = | ax + b | and the graphs of y = | f(x) | and y = f(| x |), given the graph of y = f(x).; For example, sketch the graph with equation y = | 2x − 1 | and use the graph to solve the equation | 2x − 1 | = x + 5 or the inequality | 2x − 1 | > x + 5.

    4. P3.1.4Combined transformations of functions

      Combine transformations y = af(x), y = f(x) + a, y = f(x + a) and y = f(ax), and sketch the resulting graphs; transformations of the form y = f(ax + b) are not required.

  2. P3.2 - Trigonometry

    1. P3.2.1Secant, cosecant and cotangent

      Knowledge of secant, cosecant and Angles measured in both degrees and radians. cotangent and of arcsin, arccos and arctan.; Their relationships to sine, cosine and tangent.; Understanding of their graphs and appropriate restricted domains.

    2. P3.2.2Further trigonometric identities

      Knowledge and use of sec2 θ = 1 + tan2 θ and cosec2 θ = 1 + cot2 θ.

    3. P3.2.3Knowledge

      Knowledge and use of double angle To include application to half angles.; Knowledge of the formulae; use of formulae for t (tan 1 θ) formulae will not be required.; Students should be sin (A ± B), cos (A ± B) and able to solve equations such as a cos θ + b sin θ = c in a tan (A ± B) and of expressions for given interval, and to prove identities such as a cos θ + b sin θ in the equivalent cos x cos 2x + sin x sin 2x ≡ cos x. forms of r cos (θ ± a) or r sin (θ ± a).

  3. P3.3 - Exponentials and logarithms

    1. P3.3.1function ex and its graph

      The function ex and its graph.; To include the graph of y = eax + b + c.

    2. P3.3.2function ln x and its graph

      The function ln x and its graph; ln x Solution of equations of the form eax + b = p and as the inverse function of ex. ln (ax + b) = q is expected.

    3. P3.3.3Use logarithmic graphs to estimate

      Use logarithmic graphs to estimate Plot log y against log x and obtain a straight line where the parameters in relationships of the intercept is log a and the gradient is n. form y = axn Plot log y against x and obtain a straight line where the and y = kbx. intercept is log k and the gradient is log b.

  4. P3.4 - Differentiation

    1. P3.4.1Differentiating exp, log and trig functions

      Differentiation of ekx, ln kx, sin kx, cos kx, tan kx and their sums and differences.

    2. P3.4.2Differentiation

      Differentiation using the product Differentiation of cosec x, cot x and sec x are required. rule, the quotient rule and the chain Skill will be expected in the differentiation of functions rule. generated from standard functions using products, quotients and composition, such as e3x 2x4 sin x,, cos x2 and tan2 2x. x.

    3. P3.4.3Reciprocal derivative relationship

      Use dy/dx = 1/(dx/dy), including finding dy/dx when x is given as a function of y.

    4. P3.4.4exponential

      Understand and use exponential Students should be familiar with terms such as ‘initial’, growth and decay. ‘meaning when’ t = 0.; Students may need to explore the behaviour for large values of t or to consider whether the range of values predicted is appropriate.; Consideration of a second improved model may be required. d Knowledge and use of the result (ax) = ax ln a is dx expected.

  5. P3.5 - Integration

    1. P3.5.11

      1 To include integration of standard functions such as sin 3x, Integration of ekx,, sin kx, cos kx xn e5x, and their sums and differences. 2x.

    2. P3.5.2Integration by recognition

      Integration by recognition of For example, to include integration of tan x, sec2 2x. known derivatives to include Students are expected to be able to use trigonometric integrals of the form identities to integrate, for example, sin2 x, tan2 x, cos2 3x. ∫ f′(x) dx = ln (f(x)) + c and f(x) ∫ f′(x)[f(x)]n dx = [ f(x) ]n+1 + c n +1.

  6. P3.6 - Numerical methods

    1. P3.6.1Locating roots using sign changes

      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.

    2. P3.6.2Iteration and approximate equation solving

      Approximate solution of equations Solution of equations by use of iterative procedures, for using simple iterative methods, which leads will be given. including recurrence relations of the form x = f(x) n + 1 n.