Given that:
MECH2407: Multivariable Calculus and Partial Differential Equations 4. (a) Given two periodic functions as below in...
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)
(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...
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...
Consider a periodic function f(x) given as -7, f(x) = { - < x < 0, 0 < x <, TT – I, f(x) = f(x + 27). i) Sketch the graph of f(x) in the interval –37 < x < 37. Then, deter- mine whether f(x) is even, odd or neither. (3 marks) ii) Hence, find the Fourier series of f(x). (12 marks)
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)...
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...
a) Given a non-linear partial market model: 50 - 2 Q = 4P – 4, where P and Q, respectively represent price (RM) and quantity (unit). i) State which is the demand function and which is the supply function. (2 marks) ii) Provide reason for your answer in a). (5 marks) iii) Find the values of the endogenous variables via elimination technique. Show your works. (6 marks) iv) Illustrate your finding in c) with a suitable sketch and labels them...
Use MATLAB to : ("j" is the imaginary number.) The term lo is the fundamental frequency of the periodic signal, 2/T, where T is the period. Frequencies nlo, where n is an integer, are the harmonics. This infinite sum is an exact representation of the original function. If we use a finite sum, where n goes from -N to N, we will get an approximation "X-(t)". In this problem we will calculate and plot the Fourier series representation of a...
#1,2,4,5,7 Chapter 4 Functions of Two Variables Applied Calculus 272 4.3 Exercises For problems 1 through 6, find f/, f, and f, for the function given. Confirm that ух 3. f(t,y)-Sxy 4. f(x, y)-e" 5.(x,y)-In(axy+2x-6y) y4-5 7. Find the critical points of f(x,y)-ys_x3 + 15x2-12y+12 and use the Second Derivative Test to classify them. If the test fails, say "the test fails. 3 Chapter 4 Functions of Two Variables Applied Calculus 272 4.3 Exercises For problems 1 through 6, find...