(a) Determine algebraically whether the functions below are even, odd or neither. i. r+6 f(x)=- r-r?...
Determine algebraically whether the given function is even, odd, or neither. h(x)= 3x²+4 O Neither Even Odd
1. Cousider the followving periodic function a) Determine whether the following function is odd, even or neither f(x) = sin 2x cos 3a. 2marks] Consider the following periodic function b) ㄫㄨ for -2 < x < 0 for 0< S 2 f(x) = { sin 0 f(x) = f(x + 4). i. Sketch the graph of the function over the interval-6< r <6. 2marks] Find the Fourier Series of f(x). (6marks ii.
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the sine terms, and can be approximate an odd function, so Fo(x) b sinx)+b2 sin(2x)+ b, sin(3x)+. Why is there no b, term in the series F, (x)? 1. 2. Using steps similar to those outlined for even functions, develop a rule for finding the coefficients to approximate any odd function on the interval [-π, π]. 3. If f (x)sin...
Determine algebraically whether the function is odd, even, or neither. g(x) = -5x4 - 3x2 +9 O Odd, g(-x) = -g(x) O Even, g(-x) = g(x) Odd, g(-x) = g(x) O Neither even nor odd Even, g(-x) = -g(x)
1. Determine whether the function f(x) = (x2 - 1) sin 5x is even, odd, or neither. A. Even B. Odd C. Neither 2. a). Find the Fourier sine series of the function f(x) shown below. b). Sketch the extended function f(x) that includes its two periodic extensions. TT/2 TT Formula to use: The sine series is f(x) = 6 sin NIT P where b. - EL " (x) sin " xd
*Fourier Series a) Skatch the graph of f(x) from -2n <x <3x. Hence, determine whether the function is even, odd or neither (3 marks) b) Gihen that b find a, and a,. Hence, write f(x)in a Fourier series (11 marks)
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)
Odd and Even Functions An even function has the property f(x) =f(-x). Consider the function f(x) Now, f (-a)-(-a)"-d f(a) An odd function has the property f(-x)-f(x). Consider the function f(x) Now, f (-a) = (-a)' =-a3 =-f(a) Declarative & Procedural Knowledge Comment on the meaning of the definitions of even and odd functions in term of transformations. (i) (ii) Show that functions of the formx) are even. bx2 +c Show, that f(x) = asin xis odd and g(x) =...
3. Consider the function defined by f(x) = 1, 0 < r< a, | 0, a< x < T, where 0a < T (a) Sketch the odd and even periodic extension of f (x) on the interval -3n < x < 3« for aT/2 (b) Find the half-range Fourier sine series expansion of f(x) for arbitrary a. (e) To what value does the half-range Fourier sine series expansion converge at r a? [8 marks 3. Consider the function defined by...
Section 1.3 3. For each of the following, determine algebraically whether function is even, odd, or neither: A) f(x) = 5x - 7 Answer B)f(x) = 8x - 7x + 10 Answer c) f(x) - 6x2 + 1 Answer