According to Euler's formula, how is $a_0$ calculated for a function defined on $(-\pi,\pi)$?

$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$

$a_0=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx$

$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos x\,dx$

$a_0=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin x\,dx$

Which Euler formula gives the coefficient $a_n$ on the interval $(-\pi,\pi)$?

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx$

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$

$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx$

$a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx$

Which Euler formula gives the coefficient $b_n$ on the interval $(-\pi,\pi)$?

$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx$

$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx$

$b_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx$

$b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$

For a function defined on $(-l,l)$, which formula gives the cosine coefficient $a_n$?

$a_n=\frac{1}{l}\int_{-l}^{l}f(x)\cos\left(\frac{n\pi x}{l}\right)dx$

$a_n=\frac{1}{l}\int_{-l}^{l}f(x)\sin\left(\frac{n\pi x}{l}\right)dx$

$a_n=\frac{1}{\pi}\int_{-l}^{l}f(x)\cos\left(\frac{n\pi x}{l}\right)dx$

$a_n=\frac{1}{2l}\int_{-l}^{l}f(x)\cos\left(\frac{n\pi x}{l}\right)dx$

MTH165 Unit 6: Introduction to Fourier series - MCQ Practice | B. Tech. CSE

1 $What is the standard Fourier series representation of a function with period ?$

Introduction and Euler's formulae Easy

A.

B.

C.

D.

2 $According to Euler's formula, how is calculated for a function defined on ?$

Introduction and Euler's formulae Easy

A.

B.

C.

D.

3 $Which Euler formula gives the coefficient on the interval ?$

Introduction and Euler's formulae Easy

A.

B.

C.

D.

4 $Which Euler formula gives the coefficient on the interval ?$

Introduction and Euler's formulae Easy

A.

B.

C.

D.

5 $Which statement is a Dirichlet condition for a Fourier expansion over one period?$

Conditions for a Fourier expansion and functions having points of discontinuity Easy

A.

The function must have infinitely many maxima

B.

The function must have finitely many extrema

C.

The function must be constant on each interval

D.

The function must be differentiable everywhere

6 $At a point where is continuous, to what value does its Fourier series converge?$

Conditions for a Fourier expansion and functions having points of discontinuity Easy

A.

B.

C.

D.

7 $At a jump discontinuity, to what value does the Fourier series converge?$

Conditions for a Fourier expansion and functions having points of discontinuity Easy

A.

B.

C.

D.

8 $Which type of discontinuity is allowed in the usual Dirichlet conditions for a Fourier series?$

Conditions for a Fourier expansion and functions having points of discontinuity Easy

A.

An essential singularity

B.

An unbounded oscillation

C.

An infinite discontinuity

D.

A finite jump discontinuity

9 $For a Fourier series on, which angle replaces in the standard trigonometric terms?$

Change of interval Easy

A.

B.

C.

D.

10 $What is the period of a Fourier series developed on the interval ?$

Change of interval Easy

A.

B.

C.

D.

11 $Which substitution maps the interval to ?$

Change of interval Easy

A.

B.

C.

D.

12 $For a function defined on, which formula gives the cosine coefficient ?$

Change of interval Easy

A.

B.

C.

D.

13 $Which relation defines an even function?$

Even and odd functions Easy

A.

B.

C.

D.

14 $Which relation defines an odd function?$

Even and odd functions Easy

A.

B.

C.

D.

15 $Which Fourier coefficients are zero when is even on ?$

Even and odd functions Easy

A.

All coefficients

B.

Only the coefficient

C.

Only the coefficient

D.

All coefficients

16 $What form does the Fourier series of an odd function take?$

Even and odd functions Easy

A.

A constant series only

B.

A sine series only

C.

A cosine series only

D.

A polynomial series only

17 $Which extension is used to obtain a half-range cosine series for a function defined on ?$

Half range series Easy

A.

A constant extension

B.

A linear extension

C.

An odd extension

D.

An even extension

18 $Which extension is used to obtain a half-range sine series for a function defined on ?$

Half range series Easy

A.

An odd extension

B.

A linear extension

C.

An even extension

D.

A constant extension

19 $Which formula gives in a half-range cosine series on ?$

Half range series Easy

A.

B.

C.

D.

20 $Which formula gives in a half-range sine series on ?$

Half range series Easy

A.

B.

C.

D.

21 $For the Fourier series of on, what is the sine coefficient ?$

Introduction and Euler's formulae Medium

A.

B.

C.

D.

22 $In the Fourier series for on, what is ?$

Introduction and Euler's formulae Medium

A.

B.

C.

D.

23 $If is written in standard Fourier form on, which coefficient statement is correct?$

Introduction and Euler's formulae Medium

A.

B.

C.

D.

24 $For on, what is the Fourier cosine coefficient ?$

Introduction and Euler's formulae Medium

A.

B.

C.

D.

25 $A periodic function has one-sided limits and at a jump discontinuity. If the Dirichlet conditions hold, to what value does its Fourier series converge at ?$

Conditions for a Fourier expansion and functions having points of discontinuity Medium

A.

B.

C.

D.

26 $Let for, extended periodically with period . To what value does its Fourier series converge at ?$

Conditions for a Fourier expansion and functions having points of discontinuity Medium

A.

B.

C.

D.

27 $Which function on a finite interval satisfies a standard sufficient set of Dirichlet conditions for Fourier expansion?$

Conditions for a Fourier expansion and functions having points of discontinuity Medium

A.

An unbounded function with a non-integrable singularity

B.

A function with infinitely many extrema in every subinterval

C.

A function whose absolute integral is infinite

D.

A bounded piecewise smooth function with finitely many jumps

28 $Define for and for, then extend it periodically. What is the sum of its Fourier series at ?$

Conditions for a Fourier expansion and functions having points of discontinuity Medium

A.

B.

C.

D.

29 $For a function defined on, which trigonometric terms appear in its Fourier series?$

Change of interval Medium

A.

and

B.

and

C.

and

D.

and

30 $A function is expanded over an interval of length and extended periodically. Which is the fundamental cosine term?$

Change of interval Medium

A.

B.

C.

D.

31 $Which substitution maps the interval onto ?$

Change of interval Medium

A.

B.

C.

D.

32 $For a function of period expanded over, which expression gives its cosine coefficient ?$

Change of interval Medium

A.

B.

C.

D.

33 $If is even on, which coefficient formula is correct?$

Even and odd functions Medium

A.

B.

C.

D.

34 $Suppose is odd and is even on . Which statement about their product is correct?$

Even and odd functions Medium

A.

is odd and has zero cosine coefficients

B.

is even and has zero sine coefficients

C.

is odd and has zero sine coefficients

D.

is even and has zero cosine coefficients

35 $For any positive integer, what is ?$

Even and odd functions Medium

A.

B.

C.

D.

36 $A Fourier series on contains only terms of the form and converges to at every continuity point. What symmetry does have at those points?$

Even and odd functions Medium

A.

B.

C.

D.

37 $For the half-range sine series of on, what is the coefficient ?$

Half range series Medium

A.

B.

C.

D.

38 $For the half-range cosine series of on, what is the coefficient ?$

Half range series Medium

A.

B.

C.

D.

39 $The half-range sine series of on is evaluated at . To what value does the series converge?$

Half range series Medium

A.

B.

C.

D.

40 $For the half-range sine series of on, which expression correctly describes ?$

Half range series Medium

A.

for odd and for even

B.

for even and for odd

C.

for odd and for even

D.

for odd and for even

41 $Let be -periodic and defined by for and for . Using the convention, which coefficients are correct?$

Introduction and Euler's formulae Hard

A.

,,

B.

,,

C.

,,

D.

,,

42 $A real-valued -periodic function has complex Fourier coefficient, where . What are its trigonometric coefficients and ?$

Introduction and Euler's formulae Hard

A.

and

B.

and

C.

and

D.

and

43 $Suppose the th harmonic of is . If, what is the th harmonic of ?$

Introduction and Euler's formulae Hard

A.

B.

C.

D.

44 $Define a -periodic function by for and for . Assuming the Dirichlet conditions hold, to what values does its Fourier series converge at and ?$

Conditions for a Fourier expansion and functions having points of discontinuity Hard

A.

at both and

B.

at both and

C.

at and at

D.

at and at

45 $Which function on fails to have finite Fourier coefficients because it is not integrable near ?$

Conditions for a Fourier expansion and functions having points of discontinuity Hard

A.

for

B.

for

C.

for

D.

for

46 $Let for, except that, and extend periodically. Which statement about its Fourier series is correct?$

Conditions for a Fourier expansion and functions having points of discontinuity Hard

A.

Its coefficients equal those of, and its sums at and are and

B.

Its coefficients equal those of, and its sums at and are and

C.

Its sine coefficients change, and its sums at and are and

D.

Its constant coefficient increases, and its sums at and are and

47 $For a piecewise smooth periodic function with a jump of magnitude, which statement best describes the Gibbs phenomenon as the number of Fourier terms tends to infinity?$

Conditions for a Fourier expansion and functions having points of discontinuity Hard

A.

The overshoot approaches about, while the affected neighborhood becomes narrower

B.

The overshoot approaches about, while the affected neighborhood becomes narrower

C.

The overshoot approaches zero uniformly, while the affected neighborhood remains fixed

D.

The overshoot remains exactly, while the affected neighborhood becomes wider

48 $For a Fourier expansion on, which coefficient formula and harmonic basis are correctly matched?$

Change of interval Hard

A.

with basis

B.

with basis

C.

with basis

D.

with basis

49 $Which substitution maps onto and converts the Fourier basis to and ?$

Change of interval Hard

A.

B.

C.

D.

50 $What is the Fourier series of on, extended with period ?$

Change of interval Hard

A.

B.

C.

D.

51 $Using a basis centered at the midpoint, which Fourier series represents on, extended with period ?$

Change of interval Hard

A.

B.

C.

D.

52 $On the symmetric interval, consider . Which Fourier coefficients must vanish?$

Even and odd functions Hard

A.

Every and only the odd must vanish

B.

Every odd-indexed and must vanish

C.

and every must vanish

D.

Every even-indexed and must vanish

53 $A function on satisfies and . What are the coefficients and ?$

Even and odd functions Hard

A.

and

B.

and

C.

and

D.

and

54 $A -periodic function satisfies the half-wave symmetry . Which conclusion follows without assuming that is even or odd?$

Even and odd functions Hard

A.

, and all even-indexed sine and cosine coefficients vanish

B.

may be nonzero, and all sine coefficients vanish

C.

may be nonzero, and all cosine coefficients vanish

D.

, and all odd-indexed sine and cosine coefficients vanish

55 $A -periodic function is even and also satisfies . Which harmonic structure is possible?$

Even and odd functions Hard

A.

Only sine terms with even indices

B.

Only sine terms with odd indices

C.

Only cosine terms with odd indices

D.

Only cosine terms with even indices

56 $For on, what are the coefficients in its half-range sine series ?$

Half range series Hard

A.

B.

C.

D.

57 $For on, which coefficients occur in the half-range cosine series ?$

Half range series Hard

A.

,

B.

,

C.

,

D.

,

58 $Let be continuous on with and . How do its half-range sine and cosine series behave at the endpoints?$

Half range series Hard

A.

Both series sum to at both endpoints because of periodic extension

B.

The sine series sums to and, while the cosine series sums to at both endpoints

C.

Both series sum to and at their corresponding endpoints

D.

The sine series sums to at both endpoints, while the cosine series sums to and

59 $What is the half-range sine expansion of on ?$

Half range series Hard

A.

B.

C.

D.

60 $For on, compare the asymptotic decay of coefficients in its half-range sine and cosine expansions. Which statement is correct?$

Half range series Hard

A.

Both sine and cosine coefficients decay as

B.

Sine coefficients decay as, while nonzero cosine coefficients decay as

C.

Sine coefficients decay as, while nonzero cosine coefficients decay as

D.

Both sine and cosine coefficients decay as

Unit 6: Introduction to Fourier series - Practice Quiz

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