Question

[16 marks] Provide concise answers and brief justification and reasoning (a) Determine the period of the function f(t) = n+|cos(t)(3+)sin(3tt) (b) Consider the 4-periodic function g(t) that is defined as g(t)=-(x-1)2+1 for te (0,2. Does an approximation of this function by a Fourier series converge faster for the odd or the even extension. (c) Consider the function h(t) cos2(t)+sin2 (t). Are the coefficients by of the Fourier series of this function equal to 0 for all n? (d) Consider the function f(x,y) =-x +xyexp(-x) + sin x. The function g(x,y) is the Taylor series of f(x,y) around the point (0,0) which includes all terms up to first order. What is g(-n/6, n/3)? (e) Let r(t) be the smooth trajectory of a particle in R3 for t e [1,5]. The speed of the particle increases linearly from r(1)= 0 at t 1 to r ( 5)1= 20. What is the length of the trajectory r(t) between f 1 and t 5? (f Consider a vector field F(x,y) on R2. AB is the straight line path from a point a to a point b and BA the straight line path from point b to a. C is a circular path that connects b to a. The value of the line integral ofF along AB is 1, that is, fAR F(r)dr 1. If possible, determine the values of fR F(r)dr and fe F(r)dr without assuming that F is conservative. (g) Consider three linearly dependent vectors v, v2 and v3 in R3, all of length 1. The angle between vi and v2 is m/4 and the angle between v2 and v3 is t/3. Are the two vectors w and w2, defined by wi = v1 X V2 and w2 v2 X V3, linearly independent? The symbol x denotes the cross product. (h) Is the set (x e R| |x|> 3) of all real numbers x E R with absolute value greater than 3 a vector space?

Answer 1

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