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A pulsed electric voltage that contains a dc component of 3V is defined by

Question : A pulsed electric voltage that contains a dc component of 3V is defined by : 2151785

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

One period of the given function is defined below. Determine whether the function is even, odd, or neither.

1) f(x) = {0 -3 ≤ x < 3 1 3 ≤ x < 5)

A) Even

B) Neither

C) Odd

2) f(x) = { -3x -3 ≤ x < 0 3x 0 ≤ x < 3}

A) Odd

B) Neither

C) Even

3) f(x) = -x sin3x cos5x -5 ≤ x < 5

A) Odd

B) Even

C) Neither

4) f(x) = x cos2x -2 ≤ x < 2

A) Even

B) Neither

C) Odd

5) f(x) = x3 -1 ≤ x < 1

A) Even

B) Neither

C) Odd

6) f(x) = {x2 + 2x -2 ≤ x < 0 2x - x2 0 ≤ x < 2}

A) Even

B) Odd

C) Neither

7) f(x) = { -1 -3 ≤ x < -1 0 -1 ≤ x < 1 1 1 ≤ x < 3}

A) Odd

B) Neither

C) Even

8) f(x) = { -1 -3 ≤ x < 0 0 0 ≤ x < 1 1 1 ≤ x < 3}

A) Neither

B) Odd

C) Even

9) f(x) = {x + 4 -2 ≤ x < 0 x 0 ≤ x < 2}

A) Even

B) Odd

C) Neither

10) f(x) = {1 -3 ≤ x < -1 -1 -1 ≤ x < 2 1 2 ≤ x < 3}

A) Odd

B) Neither

C) Even

Determine whether the Fourier series of the given function will include only sine terms, only cosine terms, or both sine terms and cosine terms.

11) f(x) = {((5 -π ≤ x < -π/2, π/2 ≤ x < π)(6 -π/2 ≤ x < π/2))}

A) only sine terms

B) only cosine terms

C) both sine and cosine terms

12) f(x) = {((3 -π ≤ x < 0)(7 0 ≤ x < π))

A) only sine terms

B) both sine and cosine terms

C) only cosine terms

13) f(x) = {((0 -2 ≤ x < 0)(3 0 ≤ x < 2))

A) only cosine terms

B) only sine terms

C) both sine and cosine terms

14) f(x) = 1 - 5x -6 ≤ x < 6

A) only sine terms

B) both sine and cosine terms

C) only cosine terms

15) f(x) = 5 - x2 -8 ≤ x < 8

A) only sine terms

B) both sine and cosine terms

C) only cosine terms

16) f(x) = 4 - 7x + x2 -7 ≤ x < 7

A) only sine terms

B) both sine and cosine terms

C) only cosine terms

17) f(x) = {((4x -5 ≤ x < 0)(-4x 0 ≤ x < 5))}

A) only sine terms

B) only cosine terms

C) both sine and cosine terms

18) f(x) = {((sinx -5 ≤ x < 0)(7 0 ≤ x < 5))

A) only cosine terms

B) only sine terms

C) both sine and cosine terms

19) f(x) = sin(cosx) -π ≤ x < π

A) only sine terms

B) only cosine terms

C) both sine and cosine terms

20) f(x) = -xcos7x -9 ≤ x < 9

A) both sine and cosine terms

B) only cosine terms

C) only sine terms

Find at least three nonzero terms [including a0, at least two cosine terms (if they are not all zero) and at least two sine terms (if they are not all zero)] of the Fourier series for the given function.

21) f(x) = {((5 -π ≤ x < 0)(6 0 ≤ x <π))}

A) f(x) = (9/2) + (2/π)(sinx + (1/3)sin3x + (1/5)sin5x + . . .)

B) f(x) = (9/2) - (2/π)(sinx + (1/3)sin3x + (1/5)sin5x + . . .)

C) f(x) = (11/2) + (2/π)(sinx + (1/3)sin3x + (1/5)sin5x + . . .)

D) f(x) = (11/2) - (2/π)(sinx + (1/3)sin3x + (1/5)sin5x + . . .)

22) f(x) = {((2 -π ≤ x < (π/2))(3 (π/2) ≤ x < π))}

A) f(x) = (13/4) - (9/π)(cosx - (1/3)cos3x + . . .) + (1/π)(sinx - sin2x + . . .)

B) f(x) = (9/4) - (9/π)(cosx - (1/3)cos3x + . . .) + (1/π)(sinx - sin2x + . . .)

C) f(x) = (13/4) - (1/π)(cosx - (1/3)cos3x + . . .) + (1/π)(sinx - sin2x + . . .)

D) f(x) = (9/4) - (1/π)(cosx - (1/3)cos3x + . . .) + (1/π)(sinx - sin2x + . . .)

23) f(x) = x + (3/2) -1 ≤ x <1

A) f(x) = (3/2) - 2(sinπx - (1/2)sin2πx + . . .)

B) f(x) = (3/2) + (2/π)(sinπx - (1/2)sin2πx + . . .)

C) f(x) = (3/2) - (2/π)(sinπx - (1/2)sin2πx + . . .)

D) f(x) = (3/2) + 2(sinπx - (1/2)sin2πx + . . .)

24) f(x) = {((x + 6 -3 ≤ x < 0)(x 0 ≤ x < 3))}

A) f(x) = 6 - (6/π)(sin(πx/3) + (1/2)sin(2πx/3) + . . .)

B) f(x) = 3 - (6/π)(sinx + (1/2)sin2x + . . .)

C) f(x) = 6 - (6/π)(sinx + (1/2)sin2x + . . .)

D) f(x) = 3 - (6/π)(sin(πx/3) + (1/2)sin(2πx/3) + . . .)

25) f(x) = {((3 -4 ≤ x < 0)(0 0 ≤ x < 4))}

A) f(x) = (1/2) - (2/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

B) f(x) = (3/2) - (6/π)(sinx + (1/3)sin3x + . . .)

C) f(x) = 3 - (6/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

D) f(x) = (3/2) - (6/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

26) f(x) = {((-10 -4 ≤ x < 0)(-2 0 ≤ x < 4))}

A) f(x) = -6 - (16/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

B) f(x) = -6 + (16/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

C) f(x) = 6 - (16/π)(sin(πx/4) + (1/3)sin(3πx/4) + . . .)

D) f(x) = -6 - (16/π)(sinx + (1/3)sin3x + . . .)

27) f(x) = {((-9 -2 ≤ x < -1)(-7 -1 ≤ x < 1)(-9 1≤ x < 2))}

A) f(x) = -8 + (4/π)(cosx - (1/3)cos3x + . . .)

B) f(x) = -6 + (4/π)(cos(πx/2) - (1/3)cos(3πx/2) + . . .)

C) f(x) = -6 + (4/π)(cosx - (1/3)cos3x + . . .)

D) f(x) = -8 + (4/π)(cos(πx/2) - (1/3)cos(3πx/2) + . . .)

28) f(x) = {((4 -π ≤ x < - (π/2), (π/2) ≤ x < π)(5 - (π/2) ≤ x < (π/2)))}

A) f(x) = (11/2) + (2/π)(cosx - (1/3) 3x + . . .)

B) f(x) = (9/2) + (10/π)(cosx - (1/3) 3x + . . .)

C) f(x) = (11/2) + (10/π)(cosx - (1/3) 3x + . . .)

D) f(x) = (9/2) + (2/π)(cosx - (1/3) 3x + . . .)

29) f(x) = x2 + (2/3) -1 ≤ x < 1

A) f(x) = 1 + (4/π2)(cosπx - (1/4)cos2πx + (1/9)cos3πx - . . .)

B) f(x) = 1 - (4/π2)(cosπx - (1/4)cos2πx + (1/9)cos3πx - . . .)

C) f(x) = - 1 - (4/π2)(cosπx - (1/4)cos2πx + (1/9)cos3πx - . . .)

D) f(x) = - 1 + (4/π2)(cosπx - (1/4)cos2πx + (1/9)cos3πx - . . .)

30) f(x) = {((ex -1 ≤ x < 0)(0 0 ≤ x < 1))}

A) f(x) = (1 - e-1/2) - ((1 + e-12 + 1) cosπx + (1 - e-1/4π2 + 1) cos2πx + . . .) +((π(1 + e-1)/π2 + 1) sinπx + (2π(1 - e-1)/4π2 + 1) sin2πx + . . .)

B) f(x) = (1 - e-1/2) + ((1 + e-12 + 1) cosπx + (1 - e-1/4π2 + 1) cos2πx + . . .) -((π(1 + e-1)/π2 + 1) sinπx + (2π(1 - e-1)/4π2 + 1) sin2πx + . . .)

C) f(x) = (1 - e-1/2) + ((e-12 + 1) cosπx - (e-1/4π2 + 1) cos2πx + . . .) - ((πe-12 + 1) sinπx + (2πe-1/4π2 + 1) sin2πx + . . .)

D) f(x) = (1 - e-1/2) - ((e-12 + 1) cosπx - (e-1/4π2 + 1) cos2πx + . . .) - ((π e-12 + 1) sinπx - (2π e-1/4π2 + 1) sin2πx + . . .)

Solve.

31) A pulsed electric voltage that contains a dc component of 3V is defined by

V(t) = {3 0 ≤ t < 5 4 5 ≤ t < 7 3 7 ≤ t < 12}

where t is time (in μs). Find a0 (if it is not zero), two nonzero cosine terms (if they exist), and two nonzero sine terms (if they exist) of the Fourier series for the time-dependent voltage.

A) 3 - (2/π)((1/2)cos(πt/6) - (√(3)/4)cos(πt/3) + . . .)

B) 3 - (2/π)((1/2)cos(πt/6) - (√(3)/4)cos(πt/6) + . . .)

C) (19/6) - (2/π)((1/2)cos(πt/6) - (√(3)/4)cos(πt/6) + . . .)

D) (19/6) - (2/π)((1/2)cos(πt/6) - (√(3)/4)cos(πt/3) + . . .)

32) A pulsed electric voltage that is offset by a dc component of 1V is defined by

V(t) = {1 0 ≤ t < 14 2 14 ≤ t < 28 1 28 ≤ t < 42}

where t is time (in μs). Find a0 (if it is not zero), two nonzero cosine terms (if they exist), and two nonzero sine terms (if they exist) of the Fourier series for the time-dependent voltage.

A) (4/3) - (2/π)((√(3)/2)cost - (√(3)/4)cos2t + . . .)

B) 1 - (2/π)((√(3)/2)cos(πt/21) - (√(3)/4)cos(2πt/21) + . . .)

C) (4/3) - (2/π)((√(3)/2)cos(πt/21) - (√(3)/4)cos(2πt/21) + . . .)

D) (2/3) - (2/π)((√(3)/2)cos(πt/21) - (√(3)/4)cos(2πt/21) + . . .)

33) A pulsed electric voltage that is offset by a dc component of 2V is defined by

V(t) = {2 0 ≤ t < 9 3 9 ≤ t < 45 2 45 ≤ t < 54}

where t is time (in μs). Find a0 (if it is not zero), two nonzero cosine terms (if they exist), and two nonzero sine terms (if they exist) of the Fourier series for the time-dependent voltage.

A) (7/3) - (√(3)/π)(cos(πt/27) + (1/2)cos(2πt/27) + . . .)

B) (5/3) - (2/π)(cos(πt/27) + (1/2)cos(2πt/27) + . . .)

C) (8/3) - (√(3)/π)(cos(πt/27) + (1/2)cos(2πt/27) + . . .)

D) 2 - (√(3)/π)(cos(πt/27) + (1/2)cos(2πt/27) + . . .)

34) An electron beam is made to perform a repeated unidirectional sweep across a phosphor-coated display panel by a time-periodic sweep voltage, V, given as

V(t) = 50t -1 ≤ t < 1

Find a0 (if it is not zero), three nonzero cosine terms (if they exist), and three nonzero sine terms (if they exist) of the Fourier series for the time-dependent voltage.

A) (50/π)(sinπt - (1/2)sin2πt + (1/3)sin3πt - . . .)

B) 100(sinπt - (1/2)sin2πt + (1/3)sin3πt - . . .)

C) (100/π)(sinπt - (1/2)sin2πt + (1/3)sin3πt - . . .)

D) (2/π)(sinπt - (1/2)sin2πt + (1/3)sin3πt - . . .)

35) Expand f(x) = 10 in a half-range sine series for 0 ≤ x < 2.

A) 40(sinπx + (1/4)sin2πx + (1/6)sin3πx + . . .)

B) (40/π)(sin(πx/2) + (1/3)sin(3πx/2) + (1/5)sin(5πx/2) + . . .)

C) 10(sin(πx/2) + (1/3)sin(3πx/2) + (1/5)sin(5πx/2) + . . .)

D) (40/π)(sin(πx/2) + sin(3πx/2) + sin(5πx/2) + . . .)

36) Expand f(x) = 6x in a half-range cosine series for 0 ≤ x < 4.

A) 12 - (96/π2)(cos(πx/4) + cos(3πx/4) + cos(5πx/4) + . . .)

B) 12 - (96/π2)(cos(πx/4) + (1/9)cos(3πx/4) + (1/25)cos(5πx/4) + . . .)

C) 12 - (48/π2)(cos(πx/2) + (1/9)cos(3πx/2) + (1/25)cos(5πx/2) + . . .)

D) 12 - (48/π2)(cosx + (1/9)cos3x + (1/25)cos5x + . . .)

37) Expand f(x) = x2 in a half-range sine series for 0 ≤ x < 8.

A) (128/π3)(π2 - 4)sin(πx/4) - (64/π)sin(πx/2) + (128/27π3)(9π2 - 4)sin(3πx/4) + . . .

B) 128(π2 - 4)sin(πx/8) - 64sin(πx/4) + (128/27)(9π2 - 4)sin(3πx/8) + . . .

C) (128/π3)(π2 - 4)sin(πx/8) - (64/π)sin(πx/4) + (128/27π3)(9π2 - 4)sin(3πx/8) + . . .

D) (128/π3)sin(πx/8) - (64/π)sin(πx/4) + (128/27π3)sin(3πx/8) + . . .

38) Expand f(x) = 6x2 in a half-range cosine series for 0 ≤ x < 4.

A) 32 - (384/π2)(cos(πx/4) - cos(πx/2) + cos(3πx/4) + . . .)

B) 32 - (384/π2)(cos(πx/4) - (1/4)cos(πx/2) + (1/9)cos(3πx/4) + . . .)

C) 32 - (384/π2)(cos(πx/2) - (1/4)cosπx + (1/9)cos(3πx/2) + . . .)

D) 32 - (96/π2)(cos(πx/4) + (1/4)cos(πx/2) + (1/9)cos(3πx/4) + . . .)

Solution
5 (1 Ratings )

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