Answer Happy

PROBLEM SET 11.1 1-5 PERIOD, FUNDAMENTAL PERIOD The fundamental period is the smallest positive period. Find it for 1. cos x. sin x, cos 2x, sin 2x, cos #rx, sin #x, cos 2mx, sin 2mx 2mx 2mx 2#nx 2. cos nx, sin nx, cos Cos 27x sin k 3. If f(x) and g(x) have period p. show that h(x) = af(x) + bg(x) (a, b, constant) has the period p. Thus all functions of period p form a vector space. 4. Change of scale. If f(x) has period p. show that flax), a 0, and f(x/b), b 0, are periodic functions of x of periods p/a and bp, respectively. Give examples. 5. Show that f= const is periodic with any period but has no fundamental period. 6-10 GRAPHS OF 2T-PERIODIC FUNCTIONS Sketch or graph f(x) which for-<x< is given as follows. 6. f(x)= x 7. f(x) = Isin xl. f(x) sin [x] 8. f(x) = f(x) = |e²| x if -#<x<0 9. f(x) = 17-1 if 0<x<= -cos²x if #<x<0 10. f(x)= -{ cos²x if 0<x<# 11. Calculus review. Review integration techniques for integrals as they are likely to arise from the Euler- formulas, for instance, definite integrals of x cos mx, x² sin xx, e -2x cos nx, etc. 12-21 FOURIER SERIES Find the Fourier series of the given function f(x), which is assumed to have the period 27. Show the details of your work. Sketch or graph the partial sums up to that including cos 5x and sin 5x. 12. f(x) in Prob. 61 13. f(x) in Prob. 9 14. f(x)=x² (-XH) 15. f(x)=x² (0<x<2m) 16. 17. 18. 19. 20. Sk 20 21. 22. CAS EXPERIMENT. Graphing. Write apy graphing partial sums of the following series C from the graph what fix) the series may ap Confirm or disprove your guess by using the formulas, (a) 2(sin x + sin 3r+sin 5 204 sin 2x + 1 in 4+1 + 2 1 cos - cos 3r+ (+4cos dode than t 23. Discontinuities. Verify the last statement in T 24. CAS EXPERIMENT. Orthogonality, 2 for the discontinuities of fis in Proh 2 graph the integral of the product cos mrana various integer m and n of your choice as a function of a and conclude orthogonal