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Unit FP3: Further Pure Mathematics 3

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  1. FP3.1 - Hyperbolic functions

    1. 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.

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

  2. FP3.2 - Further coordinate systems

    1. 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.

    2. 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.

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

    4. Simple loci problems.

  3. FP3.3 - Differentiation

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

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

  4. FP3.4 - Integration

    1. Integration of hyperbolic functions and expressions involving them.

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

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

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

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

    6. 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.

  5. FP3.5 - Vectors

    1. 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.

    2. 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.

    3. 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.

  6. FP3.6 - Further matrix algebra

    1. FP3.6.1Linear transformations of column

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

    2. FP3.6.2Combination of transformations

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

    3. FP3.6.3Transpose of a matrix

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

    4. FP3.6.4Evaluation of 3 × 3 determinants

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

    5. FP3.6.5Inverse of 3 × 3 matrices

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

    6. FP3.6.6Inverse transformations and combinations

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

    7. FP3.6.7Eigenvalues and eigenvectors

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

    8. FP3.6.8Reduction of symmetric matrices

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