Maths — Pure Mathematics 3

Underground Mathematics

Cambridge's Underground Mathematics Combining Functions, Geometry of Equations, Circles, Product Rule & Integration by Parts, Chain Rule & Integration by Substitution, Trigonometry: Triangles to Functions, Trigonometry: Compound Angles, Exponentials & Logarithms, Sequences, Thinking about Algebra, Calculus Of Powers, Calculus Trig Log, Polynomials & Rational Functions, Thinking about Functions and Thinking about Numbers collections for Edexcel IAL Maths — 185 resources mapped across 18 specification points, each with suggestions and worked solutions.

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18 specification points

1.1 · Simplification of rational Denominators of rational expressions will be linear or expressions including factorising quadratic, and cancelling, and algebraic 1 ax + b 3 division. e.g. , , x + 1 . ax + b px 2 + qx + r x 2 − 1

5 resources

1.2 · Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs.: • The concept of a function as a one-one or many-one mapping from ℝ (or a subset of ℝ) to ℝ. The notation f : x and f(x) will be used. 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.

19 resources

1.3 · 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.

31 resources

1.4 · Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax). Trigonometry: • Students should be able to sketch the graph of, for example, y = 2f(3x), y = f(−x) + 1, given the graph of y = f(x) or the graph of, for example, y = 3 + sin 2x, y = − cos (x + π⁄4). The graph of y = f(ax + b) will not be required.

9 resources

2.1 · 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. Angles measured in both degrees and radians.

6 resources

2.2 · Knowledge and use of sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.

24 resources

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

38 resources

3.1 · The function eˣ and its graph.: • To include the graph of y = eᵃˣ⁺ᵇ + c

3 resources

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

15 resources

3.3 · 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 = ax n and y = kb x . Plot log y against x and obtain a straight line where the intercept is log k and the gradient is log b .

2 resources

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

13 resources

4.2 · Differentiation using the product rule, the quotient rule and the chain rule.: • Differentiation of cosecx, cotx and secx are required. Skill will be expected in the differentiation of functions generated from standard functions using products, quotients and composition, such as 2x⁴ sin x, e³x⁄x, cos x² and tan² 2x.

13 resources

4.3 · d y 1 For example, finding d y for x = sin 3 y The use of = d x  d x  d x      d y 

15 resources

4.4 · 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. Knowledge and use of the result d ( a x ) = a x ln a is d x expected.

13 resources

5.1 · Integration of eᵏˣ, 1⁄xⁿ, sinkx, cos kx and their sums and differences.: • To include integration of standard functions such as sin3x, 1⁄2x, e⁵ˣ

10 resources

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

7 resources

6.1 · 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.

8 resources

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

4 resources