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- 7. Let (a) (b) p(x) = -1-x_ _for-1 < x < 0 +1-x for 0 < x < 1. Find the full Fourier series of p(x) in the interval (-1, 1). Find the first three nonzero terms explicitly. (c) (d) Does it converge pointwise? (e) Does it converge uniformly to p(x) in the interval (-1, 1)? Does it converge in the mean square sense? ORLet f(e) = { , 1 ISI< 2n Find the Fourier series of f (x) over the interval [0,27]Consider an oscillator satisfying the initial valueproblem u''+w2u=0, u(0)=u0, u'(0)=v0 (i) (a)let x1=u, x2=u', and transformequation (i) into the form: x'=Ax, x(0)=x0 (ii) (b)By using the series (23) on page 417 which is (exp(At)=I + Σ∞n=1 (Antn/n!) ), show that expAt=I cos wt +A (sinwt)/w (iii) (c)Find the solution of the initial value problem (ii)
- Determine the nth partial sum of the Fourier Series of: + x, - T < x < 0 f(x) = х, 2 0Q3) a) Evaluate fx-²1₂(3x) dx. b) Expand the following function in a Fourier Bessel series using Bessel functions of the same order as in the indicated boundary condition: ƒ(x)=x², 0Exercise 3. (i) Show that g(x) = cos(kx) 3k k=1 defines a continuous function g : [0, ] → R. (ii) Compute 플 6.³ g(x) dx. You can leave your answer in the form of a power series.у R[0, 1] be the set of all Riemann defined on [0,1] and Let Integrable functions Let & be defned by d (hy) = So I fa - gwoldx for all points PigeR[0, 1]. Explain why is (R50, (I) / not a metric spaceI need help with these: a) Find the Fourier series of f(x) = |x| where −L < x < L. b) What is the Fourier series of the function f of period 2L defined byf(x) = -1 if −L < x< 0,f(x) = 1 if 0 < x < L,What does the series converge to when x = 0?2. Consider the function f : [0, 1] → R which is defined by f(x) = 1, 2x, 0It Let f(t) be defined as f(t) = t 7L -The percentage of female high school graduates who enrolled in college is given in the table below. Year 1970 1980 1990 2000 2010 College Enrollment Percentages for Females 25.5 30.3 38.3 45.6 50.5 Let a represent time (in years) since 1970, and let y represent the corresponding percentage of female high school graduates who enrolled in college. Find the cubic regression model y = F(x) = ax³ + bx² + cx + d, = where a is rounded to 6 decimal places, b is rounded to 4 decimal places, c is rounded to 3 decimal places, and d is rounded to 1 decimal place.2Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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