MATH2001/2901/7000 School of Mathematics and Physics

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(1) (3 marks) Find the least squares approximation of cosh(x) over the interval [?1, 1] by a
polynomial of degree at most 2. Note that you should make use of the inner product

(2) Let V be an inner product space, and let S = {e1, e2, . . . , en} be an orthonormal subset
of V.

(b) (3 marks) Prove that the inequality in part (a) is an equality if and only if v ?span(S).
Show all working.

(3) (3 marks) Consider the matrix

Find an orthogonal matrix P and a diagonal matrix D satisfying A = P DP ?1
. Show
all working.

(4) (3 marks) Consider the quadratic form

Determine an orthogonal change of variables that removes any cross terms in Q(x, y, z).
Show all working.

(5) (3 marks) Evaluate the double integral

(6) (3 marks) Find the volume of the region between the surface z = y + 3x + 6 and the
region D in the x-y plane indicated in the diagram below.

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