MA 508 Spring 2014
Review Exercises TEST 1: Sections 17.2, 17.3, 17.4, 17.6
1.
(a) Is f even? Odd? Neither?
1) f (x) = sin x ln(1 + x2 )
2) f (x) = sin(sin x)
(b) Is f periodic? If it is, find its period.
1) f (x) = sin x + cos 2x
2) f (x) = cos |x|
2. Prove that f is both even and odd if and only if it is identically zero.
3. Is f even? Odd? Neither?
1) f (x) = xe−2x
2) f (x) =
2
x
x+3
4. Is f periodic? If it is, find its period.
1) f (x) = sin x + (sin x)2
2) f (x) = sin |x|
5. Let f (x) be even and let g(x) be odd. Is F (x) = g(−f (−x)) odd?
6. Is f (x) = | cos(πx/4)| even? Odd? Neither? Periodic?
Work out the Fourier series of f , given over the period. At which value of x, if any, does the
series fail to converge to f (x)? To what values does it converge at those points?
7. f (x) = 1 on −5 ≤ x < −2, and f (x) = x on −2 ≤ x < −1.
8. f (x) = 5 + 3 cos x on (−π, π)
9. f (x) = x2 on (−π, π)
10. f (x) = 5 − x on (−1, 1)
11. f (x) = 1 − 2 sin2 x, 0 < x ≤ π.
12. f (x) = 1 on −1 ≤ x < 1, and f (x) = 0 on 1 ≤ x < 2 and −2 ≤ x < −1
13. f (x) = | cos x| on −
π
π
≤x<
2
2
14. For the given function sketch two graphs corresponding HRC, QRS.
f (x) = x − 1 on 0 ≤ x < 1.
15. For the given function find HRS
f (x) = sin x on 0 ≤ x < π/2.
16. For the given function derive the QRC expansion
f (x) = cos x on 0 ≤ x < π/2
Sketch four graphs corresponding HRC, HRS, QRC, QRS.
17. For the given function derive the HRC expansion
f (x) = 2 + x on 0 ≤ x < π
Sketch four graphs corresponding HRC, HRS, QRC, QRS.
18. Work out the complex exponential form of f , given over the period. At which value of x, if
any, does the series fail to converge to f (x)? To what values does it converge at those points?
f (x) = x,
−2π ≤ x < 2π
19. Work out the complex exponential form of f , given over the period. At which value of x, if
any, does the series fail to converge to f (x)? To what values does it converge at those points?
f (x) = sin x,
−π ≤ x < π
20. Work out the complex exponential form of f , given over the period. At which value of x, if
any, does the series fail to converge to f (x)? To what values does it converge at those points?
f (x) = cos2 x,
−π ≤ x < π
21. Work out the complex exponential form of the Fourier series of f (x), given over the period.
Sketch the graph of f (x). At which value of x, if any, does the series fail to converge to f (x)?
To what values does it converge at those points?
f (x) = π/2 − x,
−π/2 ≤ x < π/2.
22. Are these functions piecewise continuous on [−π, π]?
a) f (x) =
1
1−x
1
x
23. Are these functions piecewise continuous on [−π, π]?
b) f (x) = sin
a) f (x) =
1
1 − ex
2
x − 10
24. Are these functions piecewise continuous on [−π, π]?
b) f (x) = sin
a) f (x) =
ex
14 + x
1
sin x
25. Is f (x) = sin(1/(x − 3.2)) piecewise continuous on [0, π]?
b) f (x) =
26. The function f (x) = x, 0 < x < 1, f (x) = −1, 1 ≤ x ≤ 2 given over the period. Sketch the
graph of fe (x).
27. Write the Parseval equality for f (x) given over the period.
f (x) = | sin x|,
−π/2 ≤ x < π/2
28. Write the Parseval equality for f (x) given over the period.
f (x) = |x|,
−π ≤ x < π
29. Compute ||E3 ||, where f (x) given over the period.
f (x) = 3x − 1,
−2 ≤ x < 2
Write the Bessel inequality for f (x) and n = 3.
30. Compute ||E5 ||, where f (x) given over the period.
f (x) = x2 ,
−2π ≤ x < 2π
Write the Bessel inequality for f (x) and n = 5.
31. Show that if f (x) given over period [−π, π] then
2
||En || =
2
where ||En || =
Rπ
−π
Z π
−π
f (x)2 dx − π(2a20 +
2
(f (x) − Sn (x)) dx and Sn (x) = a0 +
n
X
k=1
Pn
k=0 (ak
32. Let n 6= m. Prove that the integral
Z l
−l
is equal to 0 (orthogonality property).
cos
(a2k + b2k )),
nπx
mπx
sin
dx
l
l
cos(kx) + bk sin(kx)).