MT1710: Question Sheet 1 To be handed in: Tuesday 9 October, 2018 (by 1pm, in the Maths office, McCrea Room 118) 1. Sketch graphs to illustrate the following functions, describing the curves and lines you draw: (a) y = [1; [2- x2; 0? x ?1, 1?x<2. ? 0?x?1, (b) y = { 2 - x; 1<x?2, (2-x)2; 2?x ?3. (c) y = |2.x| - |2 - x| , -2?x?3. 2. Find formulae for the functions, in the form y = f(x), corresponding to each of the graphs below: (a) y (b) y 2 ? ? ? -4 -2 x 2 ? -4 -2 2 x N.B. In (b), the interval 0 ? x ? 2 shows the quadrant of a circle of radius 2. 3. (a) The even function f1 is defined on the interval -2 < x < 2. Given that the functional form of f1 is given by Question 1(a) on the interval 0 ? x ? 2, give the functional form for f1 on the interval -2 ? x < 0. The odd function g1 is defined on the interval -2 < x < 2. Given that the functional form of g1 is given by Question 1(a) on the interval 0 ? x ? 2, give the functional form for g1 on the interval -2 ? x < 0. (b) The even function f2 is defined on the interval -3 ? x < 3. Given that the functional form of f2 is given by Question 1(b) on the interval 0 ? x ? 3, give the functional form for f2 on the interval -3 ? x < 0. The odd function g2 is defined on the interval -3 ? x < 3. Given that the functional form of g2 is given by Question 1(b) on the interval 0 < x ? 3, give the functional form for 92 on the interval -3 ? x < 0. [Hint: Sketch the functions f1, etc, to check their functional form!] 1
4. The following functions are periodic with period a. Sketch the curves for -a ? x ? 3a and give the values of y in each case for x = - & and x = 2?. In each case use the properties of periodic functions to justify your answers: [ 1; 0 <x?? , (a) y = 0; a < x < a . (square wave) , (b) y= x, 0<x?a. (saw - tooth) , a V VI 812 (c) y= cos(-) a 2 ? , S-(x +?); ?? ? x ?0, (d) y = 1x -a; 0 ? x ? ? · 5. (Unseen). A function f satisfies the relation 2f (u) cos v = f (u + v) + f (u - v) for all real values of u and v. Show that for real values of x, (i) f(x)+ f(-x) = 2a cos x, =0, (ii) f ( TT - x ) + f (-x) (iii) f(T-x)+f(x) = 2bsinx, where a and b are certain constants.