Unit P3: Pure Mathematics 3

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.

Use simplification of rational denominators of rational to connect the rule to the data and decision in the question.

This matters because simplification of rational denominators of rational determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply simplification of rational denominators of rational to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Simplification of rational Denominators of rational is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use definition of a function to connect the rule to the data and decision in the question.

This matters because definition of a function determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply definition of a function to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Definition of a function is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use modulus function to connect the rule to the data and decision in the question.

This matters because modulus function determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply modulus function to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: modulus function is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use combined transformations of functions to connect the rule to the data and decision in the question.

This matters because combined transformations of functions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply combined transformations of functions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Combined transformations of functions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use secant, cosecant and cotangent to connect the rule to the data and decision in the question.

This matters because secant, cosecant and cotangent determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply secant, cosecant and cotangent to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Secant, cosecant and cotangent is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use further trigonometric identities to connect the rule to the data and decision in the question.

This matters because further trigonometric identities determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply further trigonometric identities to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Further trigonometric identities is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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).

Use knowledge to connect the rule to the data and decision in the question.

This matters because knowledge determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply knowledge to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Knowledge is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use function ex and its graph to connect the rule to the data and decision in the question.

This matters because function ex and its graph determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply function ex and its graph to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: function ex and its graph is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use function ln x and its graph to connect the rule to the data and decision in the question.

This matters because function ln x and its graph determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply function ln x and its graph to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: function ln x and its graph is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use use logarithmic graphs to estimate to connect the rule to the data and decision in the question.

This matters because use logarithmic graphs to estimate determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply use logarithmic graphs to estimate to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Use logarithmic graphs to estimate is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use differentiating exp, log and trig functions to connect the rule to the data and decision in the question.

This matters because differentiating exp, log and trig functions determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply differentiating exp, log and trig functions to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Differentiating exp, log and trig functions is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use differentiation to connect the rule to the data and decision in the question.

This matters because differentiation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply differentiation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Differentiation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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

Use reciprocal derivative relationship to connect the rule to the data and decision in the question.

This matters because reciprocal derivative relationship determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply reciprocal derivative relationship to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Reciprocal derivative relationship is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use exponential to connect the rule to the data and decision in the question.

This matters because exponential determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply exponential to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: exponential is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use 1 to connect the rule to the data and decision in the question.

This matters because 1 determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply 1 to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: 1 is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use integration by recognition to connect the rule to the data and decision in the question.

This matters because integration by recognition determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply integration by recognition to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Integration by recognition is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use locating roots using sign changes to connect the rule to the data and decision in the question.

This matters because locating roots using sign changes determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply locating roots using sign changes to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Locating roots using sign changes is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

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.

Use iteration and approximate equation solving to connect the rule to the data and decision in the question.

This matters because iteration and approximate equation solving determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply iteration and approximate equation solving to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.