Unit FP3: Further Pure Mathematics 3

Definition of the six hyperbolic For example, cosh x = 1 (ex + e−x), functions in terms of exponentials.; Graphs and properties of the 1 2 sech x = =. hyperbolic functions. cosh x ex + e−x Students should be able to derive and use simple identities such as cosh2 x − sinh2 x ≡ 1 and cosh2 x + sinh2 x ≡ cosh 2x and to solve equations such as a cosh x + b sinh x = c.

Understand inverse hyperbolic functions, their graphs, properties and logarithmic forms, including arsinh x = ln(x + √(1 + x²)).

Cartesian and parametric equations Extension of work from FP1. for the ellipse and hyperbola.; Students should be familiar with the equations: x2 y2 + = 1; x = a cos t, y = b sin t. a2 b2 x2 y2 − = 1; x = a sec t, y = b tan t; a2 b2 x = a cosh t, y = b sinh t.

The focus-directrix properties of For example, students should know that, for the ellipse, the ellipse and hyperbola, including b2 = a2(1 − e2), the foci are (ae, 0) and (−ae, 0) and the the eccentricity. equations of the directrices are a a x = + and x = −. e e.

Tangents and normals to these The condition for y = mx + c to be a tangent to these curves curves. is expected to be known.

Simple loci problems.

Differentiation of hyperbolic cosh2x For example, tanh 3x, x sinh2 x,. functions and expressions involving (x +1) them.

Differentiation of inverse functions, For example, arcsin x + x (1 – x2), 1 artanh x2. including trigonometric and 2 hyperbolic functions.

Integration of hyperbolic functions and expressions involving them.

Integration of inverse trigonometric ∫ ∫ For example, arsinh x dx, arctan x dx. and hyperbolic functions.

Integration using hyperbolic and To include the integrals of trigonometric substitutions. 1 1 1 1,,, (a2 + x2) (a2 − x2) (a2 + x2) (x2 − a2).

Use of substitution for integrals In more complicated cases, substitutions will be given. involving quadratic surds.

Derive and use simple reduction formulae for integrals, including powers of sine.

The calculation of arc length and The equation of the curve may be given in cartesian or the area of a surface of revolution. parametric form.; Equations in polar form will not be set.

FP3.5.1Vector product and scalar triple product

The vector product a × b and the The interpretation of | a × b | as an area and a. b × c as a triple scalar product a. b × c. volume.

FP3.5.2Use of vectors in problems

Use of vectors in problems Students may be required to use equivalent cartesian forms involving points, lines and planes. also.; The equation of a line in the form Applications to include (r − a) × b = 0. (i) distance from a point to a plane, (ii) line of intersection of two planes, (iii) shortest distance between two skew lines.

FP3.5.3Equation of a plane

The equation of a plane in the Students may be required to use equivalent cartesian forms forms also. r.n = p, r = a + sb + tc.

Linear transformations of column Extension of work from FP1 to 3 dimensions. vectors in two and three dimensions and their matrix representation.

Combination of transformations.; The transformation represented by AB is the transformation Products of matrices. represented by B followed by the transformation represented by A.

Transpose of a matrix.; Use of the relation (AB)T = BTAT.

Evaluation of 3 × 3 determinants.; Singular and non-singular matrices.

Inverse of 3 × 3 matrices.; Use of the relation (AB)−1 = B−1A−1.

The inverse (when it exists) of a given transformation or combination of transformations.

Eigenvalues and eigenvectors of Normalised vectors may be required. 2 × 2 and 3 × 3 matrices.

Reduction of symmetric matrices to Students should be able to find an orthogonal matrix P such diagonal form. that PTAP is diagonal.