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Question 1 of 40 Quiz ID: q1

A function f(x) satisfies f(-x) = -f(x) for all x in its domain. What is the value of ∫ from -a to a of f(x) dx, where a > 0?

2∫ from 0 to a of f(x) dx

0

∫ from -a to 0 of f(x) dx

Cannot be determined without knowing f(x)

Question 2 of 40 Quiz ID: q2

Which of the following properties is true for the product of an even function and an odd function?

The product is even.

The product is odd.

The product could be even or odd depending on the functions.

The product is neither even nor odd.

Question 3 of 40 Quiz ID: q3

Two distinct functions f and g are said to be orthogonal over an interval [a, b] if:

f(x) * g(x) = 0 for all x in [a, b]

∫ from a to b of f(x) dx = ∫ from a to b of g(x) dx

∫ from a to b of f(x)g(x) dx = 0

f(x) = -g(x) for all x in [a, b]

Question 4 of 40 Quiz ID: q4

For a 2π-periodic function, the Fourier coefficient a_n is given by:

(1/π) ∫ from -π to π of f(x) sin(nx) dx

(1/π) ∫ from -π to π of f(x) dx

(1/π) ∫ from -π to π of f(x) cos(nx) dx

(1/(2π)) ∫ from -π to π of f(x) cos(nx) dx

Question 5 of 40 Quiz ID: q5

According to Dirichlet's Theorem, at a point of discontinuity x=c, the Fourier series of a function f converges to:

f(c)

f(c+)

f(c-)

(f(c-) + f(c+)) / 2

Question 6 of 40 Quiz ID: q6

For an odd function defined on [-π, π], which Fourier coefficients are necessarily zero?

a_n only

b_n only

a_0 and a_n

a_0 and b_n

Question 7 of 40 Quiz ID: q7

To find the half-range cosine series for a function defined on [0, π], how must the function be extended to the interval [-π, π]?

As an even function

As an odd function

Periodically

It does not need to be extended

Question 8 of 40 Quiz ID: q8

The Fourier series of a function f(x) with period 2l is given by f(x) = a_0/2 + Σ [a_n cos(nπx/l) + b_n sin(nπx/l)]. What is the formula for b_n?

(1/l) ∫ from -l to l of f(x) sin(nπx/l) dx

(1/π) ∫ from -l to l of f(x) sin(nπx/l) dx

(2/l) ∫ from 0 to l of f(x) sin(nπx/l) dx

(1/l) ∫ from 0 to l of f(x) sin(nπx/l) dx

Question 9 of 40 Quiz ID: q9

Parseval's formula for a 2c-periodic function f(x) states that ∫ from -c to c of [f(x)]² dx is equal to:

c (a_0² + Σ (a_n² + b_n²))

c (a_0²/2 + Σ (a_n² + b_n²))

(c/2) (a_0² + Σ (a_n² + b_n²))

(c/2) (a_0²/2 + Σ (a_n² + b_n²))

Question 10 of 40 Quiz ID: q10

In the complex form of the Fourier series for a 2π-periodic function, the coefficient c_n is given by:

(1/(2π)) ∫ from -π to π of f(x) e^{-inx} dx

(1/π) ∫ from -π to π of f(x) e^{-inx} dx

(1/(2π)) ∫ from -π to π of f(x) cos(nx) dx

(1/π) ∫ from -π to π of f(x) e^{inx} dx

Question 11 of 40 Quiz ID: q11

For the function f(x) = x² defined on [-π, π] and made periodic, which statement about its Fourier series is true?

It is a sine series because x² is an odd function.

It is a cosine series because x² is an even function.

It contains both sine and cosine terms.

It is a constant series.

Question 12 of 40 Quiz ID: q12

The Dirichlet conditions guarantee that the Fourier series of a function f:

Converges absolutely everywhere.

Converges to f(x) at every point x.

Converges to f(x) at points of continuity and to the average of limits at jump discontinuities.

Has coefficients that decrease exponentially.

Question 13 of 40 Quiz ID: q13

What is the value of the integral ∫ from c to c+T of sin(2nπt/T) cos(2mπt/T) dt for integers m, n?

T/2

T

0

Cannot be determined without c

Question 14 of 40 Quiz ID: q14

A function is defined piecewise on one period as f(x) = { 1 for 0<x<π; -1 for π<x<2π }. What is its period and what type of function is it?

Period π, Even function

Period 2π, Odd function

Period 2π, Even function

Period 2π, Neither even nor odd

Question 15 of 40 Quiz ID: q15

When finding the Fourier series for a general period 2l, the argument of the trigonometric functions changes from nx to:

nπx / l

2nπx / l

nx / l

nπx / (2l)

Question 16 of 40 Quiz ID: q16

According to Parseval's formula, if the Fourier coefficients of a function are all zero, what must be true about the function on the interval (-c, c)?

The function is constant.

The function is zero at every point.

The integral of the square of the function is zero.

The function is continuous.

Question 17 of 40 Quiz ID: q17

The complex Fourier coefficient c_n for a 2π-periodic function is related to the real coefficients a_n and b_n by:

c_n = a_n + i b_n

c_n = (a_n - i b_n) / 2

c_n = (a_n + i b_n) / 2

c_n = a_n / 2

Question 18 of 40 Quiz ID: q18

For a half-range sine series expansion on [0, c], the extension of the function to [-c, c] must be:

Even

Odd

Periodic

Constant

Question 19 of 40 Quiz ID: q19

What is the fundamental period of the function g(x) = cos(2x) + sin(3x)?

π

2π

6π

2π/3

Question 20 of 40 Quiz ID: q20

The Fourier series representation of a function is primarily useful because it:

Simplifies differentiation.

Represents a complex function as a sum of simpler sinusoidal components.

Always converges faster than a Taylor series.

Is easier to compute than other series.

Question 21 of 40 Quiz ID: q21

If a function f is both even and odd, what must be true?

It must be constant.

It must be zero everywhere.

It must be periodic.

It must be differentiable.

Question 22 of 40 Quiz ID: q22

The Fourier series of a continuous and smooth periodic function will typically have coefficients that decay:

Not at all (remain constant)

Linearly (like 1/n)

Faster than 1/n (e.g., 1/n², exponentially)

Erratically

Question 23 of 40 Quiz ID: q23

For the piecewise function f(x) = { 0 for -π<x<0; x for 0<x<π } (period 2π), which coefficient formula is correct?

a_0 = (1/π) ∫ from -π to π f(x) dx = (1/π) ∫ from 0 to π x dx

b_n = (1/π) ∫ from -π to π f(x) sin(nx) dx = (2/π) ∫ from 0 to π x sin(nx) dx

a_n = (1/π) ∫ from -π to π f(x) cos(nx) dx = 0

All of the above are potential simplifications based on the function's definition.

Question 24 of 40 Quiz ID: q24

The Gibbs phenomenon refers to:

The failure of the Fourier series to converge for some functions.

Overshoots and oscillations of the Fourier series near a jump discontinuity, even as the number of terms increases.

The slow convergence of the Fourier series for functions with corners.

The error in calculating Fourier coefficients numerically.

Question 25 of 40 Quiz ID: q25

The Fourier series expansion of the function f(x) = π - x on (0, 2π) will have:

Only sine terms

Only cosine terms

Both sine and cosine terms

No terms (it's already a Fourier series)

Question 26 of 40 Quiz ID: q26

If you want to analyze the frequency components of a non-sinusoidal voltage waveform in an AC circuit, the most appropriate mathematical tool is:

Taylor series

Laurent series

Fourier series

Geometric series

Question 27 of 40 Quiz ID: q27

For a function defined on [0, L], the half-range cosine expansion and the half-range sine expansion will:

Always be identical.

Always be different.

Converge to the same function on (0, L).

Converge to different functions on (0, L).

Question 28 of 40 Quiz ID: q28

The value of the integral ∫ from -π to π of cos(4x) sin(5x) dx is:

π

π/2

0

1

Question 29 of 40 Quiz ID: q29

A key difference between the complex form of the Fourier series and the real form is that the complex form:

Is only valid for even functions.

Uses a single set of coefficients c_n for both positive and negative frequencies.

Is more difficult to compute.

Cannot represent functions with discontinuities.

Question 30 of 40 Quiz ID: q30

The process of finding the Fourier series of a function is essentially a:

Interpolation problem.

Extrapolation problem.

Projection onto a basis of orthogonal functions.

Numerical integration problem.

Question 31 of 40 Quiz ID: q31

The function f(x) = sin(x) + cos(x) is:

Even

Odd

Neither even nor odd

Both even and odd

Question 32 of 40 Quiz ID: q32

If a Fourier series has only sine terms, what can be concluded about the original function?

It is an even function.

It is an odd function.

It is constant.

It is defined on [0, π].

Question 33 of 40 Quiz ID: q33

The convergence of the Fourier series at a specific point depends on:

The global behavior of the function.

The behavior of the function in an arbitrarily small neighborhood around that point.

The value of the function at all other points.

The period of the function.

Question 34 of 40 Quiz ID: q34

For a function with period T, the fundamental frequency ω₀ is:

T

1/T

2π

2π/T

Question 35 of 40 Quiz ID: q35

The integral ∫ from -π to π of x³ cos(2x) dx for a 2π-periodic extension of f(x)=x³ on (-π, π) will be:

Non-zero

Zero

π⁴/2

2π⁴

Question 36 of 40 Quiz ID: q36

Parseval's identity is significant in signal processing because it:

Allows calculation of the function's derivative from its coefficients.

Relates the total power of a signal to the sum of the powers of its frequency components.

Provides the fastest algorithm for computing Fourier coefficients.

Guarantees uniform convergence of the Fourier series.

Question 37 of 40 Quiz ID: q37

When extending a function defined on [0, L] to create a half-range expansion, the resulting periodic function on the whole real line:

Is always continuous.

Is always smooth.

May have discontinuities or corners, depending on the original function and the type of extension.

Has a period of L.

Question 38 of 40 Quiz ID: q38

The Fourier series representation is particularly advantageous over a power series (Taylor series) for representing:

Analytic functions.

Polynomials.

Discontinuous periodic functions.

Exponential functions.

Question 39 of 40 Quiz ID: q39

The formula for the Fourier coefficient a_0 includes a factor of 1/2 in the series expansion (a_0/2) primarily to:

Make the formula for a_0 consistent with the formula for a_n when n=0.

Ensure the series converges.

Simplify the calculation of the integral.

Make the function even.

Question 40 of 40 Quiz ID: q40

A 'low-pass filter' applied to a Fourier series would:

Attenuate high-frequency components (large n).

Attenuate low-frequency components (small n).

Remove all sine terms.

Remove all cosine terms.

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