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MECH2407: Multivariable Calculus and Partial Differential Equations 4. (a) Given two periodic functions as below in part (0) and part (iã) i) f2 1 -1st< (ii) f)-1/2 State the period of the t...
MECH2407: Multivariable Calculus and Partial Differential Equations 4. (a) Given two periodic functions as below in part (0) and part (iã) i) f2 1 -1st< (ii) f)-1/2 State the period of the two periodic functions respectively. Hence, sketch the two given perodic functions for 3 periods. Find the Fourier series for the two given perodic functions over the given interval and expand the series to give the partial sum up to the first three non-zero terms respectively. (16 marks) Use...
4. Consider the periodic function given below: f(x)-x 0 ㄨㄑㄧ (i) State its fundamental period, and sketch the function for 3 periods. (5 marks) i) Find the Fourier series of the given periodic function, and expand the series to give the first three non-zero a and b terms (15 marks) ii) Use the answer obtained in Q4(ii) and the given periodic function, find the sum of the series 4(2n-1 )2 (5 marks)
QUESTION 2 Given two periodic functions, f(t) and g(t) is defined by and f (t) = cos, -<t<t f(t)= f(t +26) g(t) = cos, 0<t<2n g(t) = g(t +21) Sketch the graph of the periodic functions f(t) and g(t) over the interval (-37,37). Sketch in separate graphs. (Please use any online graphing software not hand-drawn). Find the Fourier series of f(t) and g(t). (b) Then, briefly comment what do you observe from the graphs and the Fourier series expansion of...
(a) Determine algebraically whether the functions below are even, odd or neither. i. r+6 f(x)=- r-r? (2 marks) ii. f(x) = 2x sinx (2 marks) (b) A periodic function is defined by: f(x) = 4-x?, -25x52, f(x+4)= f(x) i. Sketch the graph of the function over -10<x<10. (4 marks) ii. Based on result in (i), identify whether the function is even or odd. Give your reason. (2 marks) ii. Calculate the Fourier series expansion of f(x). (12 marks) (c) An...
(Bernoulli Equations) Let p, f : I → R be continous functions defined on an interval I of R. Then for every α є R\ {0, 1), the 1st-order differential equation is called Bernoulli equation. It is a nonlinear ordinary differential equation. (a) Use the literature and describe in brief steps a method to find a solution of equation (1) Hint: See Trench, p.63 (b) Find all solutions to the following two differential equations. Use Mathematica to plot a direction...
The sketch of the following periodic function f (t) given in one period f(t) t2 -1, 0s t s 2 is given as follows f(t) 2 -1 We proceed as follows to find the Fourier series representation of f (t) (Note:Jt2 cos at dt = 2t as at + (a--)sina:Jt2 sin at dt = 2t sin at + sin at. Г t2 sin at dt-tsi. )cos at.) Please scroll to the bottom of page for END of question a) The...
4. Consider the following partial information about a function f(x): S.x2, 0<x<I, (2-x), 1<x<2. Given that the function can be extended and modelled as a Fourier cosine-series: (a) Sketch this extended function in the interval that satisfies: x <4 (b) State the minimum period of this extended function. (C) The general Fourier series is defined as follows: [1 marks] [1 marks] F(x) = 4 + ] Ancos ("E") + ] B, sin("E") [1 marks] State the value of L. (d)...
Fourier Series. Thank you in advance. Q 2(a) [6 Marks] (i) Sketch 3 successive cycles of the periodic function g(t):[4 marks) g() gt3). (ii) Is g(t)an even or odd function of? [2 marks] [14 Marks] (i) Show that the Fourier series coefficients of gl), as defined in part 2(a), may Q 2(b) [11 marks]: be written as follows 0, EVEN (ii) Obtain a value for the constant [3 marks] [5 Marks Q 2(c) What is the power, or mean-squared value,...
Given the periodic function 5 f(1) = { 1 f (+4) 0<i and I<2 2 <r and I<4 otherwise and its graph is displayed below. 6 5 4 y 3 2 1 0 -2 2 4 6 00+ x The function may be approximated by the Fourier series f(t) = 40 + 1 (an cos ( 172 ) + bn sin where L is the half-period of the function. + bn sin ne :)), L Calculate the coefficients of the...
Please help! 1. (Bernoulli Equations) Let p, f : I → R be continous functions defined on an interval I of R. Then for every a є R \ {0, 1), the 1st-order differential equation is called Bernoulli equation. It is a nonlinear ordinary differential equation. (a) Use the literature and describe in brief steps a method to find a solution of equation (1). Hint: See Trench, p.63 (b) Find all solutions to the following two differential equations. Use Mathematica...