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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Which formula gives $a_n$ for a Fourier series on $(-L,L)$?

$\frac{1}{L}\int_{-L}^{L}f(x)\cos\left(\frac{n\pi x}{L}\right)\,dx$

$\frac{1}{L}\int_{-L}^{L}f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx$

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

$\frac{1}{L}\int_{-L}^{L}f(x)\cos\left(\frac{nx}{L}\right)\,dx$

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

1 $Which expression represents the Fourier series of a function with period ?$

introduction and Euler's formulae Easy

A.

B.

C.

D.

2 $According to Euler's formulae, what is for a function defined on ?$

introduction and Euler's formulae Easy

A.

B.

C.

D.

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

introduction and Euler's formulae Easy

A.

B.

C.

D.

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

introduction and Euler's formulae Easy

A.

B.

C.

D.

5 $Which statement is a standard sufficient condition for a Fourier expansion?$

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

A.

The function is constant over every period.

B.

The function is piecewise continuous over one period.

C.

The function is polynomial over all real numbers.

D.

The function is differentiable at every point.

6 $At a finite jump discontinuity, to what value does the Fourier series converge?$

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

A.

B.

C.

D.

7 $At a point where is continuous, the Fourier series normally converges to which value?$

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

A.

B.

C.

D.

8 $Which behavior is allowed by the usual Dirichlet conditions over one period?$

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

A.

Unbounded oscillations near every point

B.

A finite number of finite discontinuities

C.

An infinite discontinuity at every point

D.

No defined values anywhere in the period

9 $For a function defined on, which angles appear in its Fourier series?$

change of interval Easy

A.

B.

C.

D.

10 $What is the period associated with a Fourier series defined using the interval ?$

change of interval Easy

A.

B.

C.

D.

11 $Which formula gives for a Fourier series on ?$

change of interval Easy

A.

B.

C.

D.

12 $Which substitution maps onto ?$

change of interval Easy

A.

B.

C.

D.

13 $Which equation defines an even function?$

even and odd functions Easy

A.

B.

C.

D.

14 $Which equation defines an odd function?$

even and odd functions Easy

A.

B.

C.

D.

15 $If is even on, which Fourier coefficients are zero?$

even and odd functions Easy

A.

All cosine coefficients

B.

All sine coefficients

C.

Only the constant coefficient

D.

Only the first sine coefficient

16 $If is odd on, which coefficients are zero?$

even and odd functions Easy

A.

Only and

B.

All and all

C.

and all

D.

and all

17 $Which extension of a function on produces a half-range cosine series?$

half range series Easy

A.

A constant extension

B.

A linear extension

C.

An odd extension

D.

An even extension

18 $Which extension of a function on produces a half-range sine series?$

half range series Easy

A.

An odd extension

B.

An even extension

C.

A quadratic extension

D.

A periodic constant extension

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

half range series Easy

A.

B.

C.

D.

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

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 $If on, which Fourier coefficients are nonzero?$

introduction and Euler's formulae Medium

A.

and

B.

and

C.

and

D.

and

23 $Euler's formula gives . Which expression therefore equals ?$

introduction and Euler's formulae Medium

A.

B.

C.

D.

24 $Applying Parseval's identity to the Fourier series of on leads to which value of ?$

introduction and Euler's formulae Medium

A.

B.

C.

D.

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

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

A.

B.

C.

D.

26 $For on, extended periodically, to what value does its Fourier series converge at ?$

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

A.

B.

C.

D.

27 $Which function on fails the usual Dirichlet conditions because it is unbounded and not absolutely integrable near ?$

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

A.

B.

C.

D.

28 $If the value of a piecewise smooth function is changed at only one point, what happens to its Fourier coefficients?$

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

A.

Only the sine coefficients change

B.

Every coefficient changes equally

C.

Only changes

D.

They remain unchanged

29 $What trigonometric basis is appropriate for a Fourier expansion on an interval of length, treated as one period?$

change of interval Medium

A.

and

B.

and

C.

and

D.

and

30 $Which substitution maps the interval onto ?$

change of interval Medium

A.

B.

C.

D.

31 $A function has period . Which angular frequency appears in its fundamental Fourier terms?$

change of interval Medium

A.

B.

C.

D.

32 $For a Fourier series on, which formula gives the cosine coefficient ?$

change of interval Medium

A.

B.

C.

D.

33 $For on, which parts contribute to the cosine and sine coefficients?$

even and odd functions Medium

A.

contributes to, and contributes to

B.

Both terms contribute only to

C.

Both terms contribute only to

D.

contributes to, and contributes to

34 $What is the even part of ?$

even and odd functions Medium

A.

B.

C.

D.

35 $If is odd on, which expression correctly simplifies its sine coefficient?$

even and odd functions Medium

A.

B.

C.

D.

36 $For on, which statement about its Fourier coefficients is correct?$

even and odd functions Medium

A.

All cosine coefficients are zero

B.

All sine coefficients are zero

C.

Only even cosine coefficients are zero

D.

Only odd sine coefficients are zero

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

half range series Medium

A.

B.

C.

D.

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

half range series Medium

A.

B.

C.

D.

39 $Which extension of a function defined on is used to construct its half-range cosine series?$

half range series Medium

A.

An odd extension to

B.

A zero extension to

C.

A constant extension to

D.

An even extension to

40 $For the half-range cosine series of on, what is the constant term ?$

half range series Medium

A.

B.

C.

D.

41 $For the -periodic extension of on, which set of Euler coefficients is correct?$

introduction and Euler's formulae Hard

A.

,,

B.

,,

C.

,,

D.

,,

42 $Let for a real-valued function . Which relations correctly recover the trigonometric Fourier coefficients?$

introduction and Euler's formulae Hard

A.

and

B.

and

C.

and

D.

and

43 $For on, periodically extended, what are its Euler coefficients for ?$

introduction and Euler's formulae Hard

A.

,

B.

,

C.

,

D.

,

44 $Applying Parseval's identity to the Fourier series of on gives which result?$

introduction and Euler's formulae Hard

A.

B.

C.

D.

45 $A periodic function has finite one-sided limits and at a jump discontinuity. If the Dirichlet conditions hold, to what value does its Fourier series converge at ?$

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

A.

B.

C.

D.

46 $Suppose satisfies the Dirichlet conditions on . A new function differs from only at three isolated points. Which statement is correct?$

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

A.

and have identical Fourier coefficients

B.

and necessarily have different Fourier series

C.

and differ only in their constant coefficients

D.

and have different sine coefficients only

47 $Which condition set is a standard sufficient form of the Dirichlet conditions for Fourier convergence over one period?$

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

A.

The function is differentiable except at one removable discontinuity

B.

The function is bounded with infinitely many oscillations at each point

C.

The function is continuous with an absolutely convergent Fourier series

D.

The function is integrable with finitely many finite jumps and extrema

48 $Define for and for, then extend it with period . What is the Fourier-series sum at ?$

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

A.

B.

C.

D.

49 $Near a rising jump of magnitude, what happens to the Gibbs overshoot as the number of Fourier terms tends to infinity?$

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

A.

Its width tends to zero, and its height approaches exactly

B.

Its width remains fixed, but its height decreases to zero

C.

Its width and height both decrease to zero at equal rates

D.

Its width tends to zero, but its height approaches about

50 $A function defined on is to be expanded with period using a change of interval based at . Which harmonic basis is appropriate?$

change of interval Hard

A.

and

B.

and

C.

and

D.

and

51 $For on, extended periodically with period, what is the coefficient of for ?$

change of interval Hard

A.

B.

C.

D.

52 $Set and, mapping to . How does the coefficient of in the Fourier series of appear in the series for ?$

change of interval Hard

A.

It multiplies with the same value

B.

It multiplies after division by

C.

It multiplies after multiplication by

D.

It multiplies with the same value

53 $A function on satisfies for . In a Fourier expansion centered at, which coefficients must vanish?$

even and odd functions Hard

A.

Only coefficients corresponding to even values of

B.

Only coefficients corresponding to odd values of

C.

All coefficients of

D.

All coefficients of

54 $For on, which description of its Fourier coefficients is correct?$

even and odd functions Hard

A.

,,, and for

B.

,, and for

C.

,,, and for

D.

,,, and for

55 $Suppose on . What can be concluded about the cosine coefficients of ?$

even and odd functions Hard

A.

They equal the cosine coefficients of

B.

They all vanish because the remaining part is odd

C.

They are twice the cosine coefficients of

D.

They cannot be determined without the odd part

56 $Which Fourier series represents the even -periodic function ?$

even and odd functions Hard

A.

B.

C.

D.

57 $What are the coefficients in the half-range sine series of on ?$

half range series Hard

A.

B.

C.

D.

58 $For the half-range cosine series of on, which coefficients are correct?$

half range series Hard

A.

,

B.

,

C.

,

D.

,

59 $For on, what are the half-range sine coefficients?$

half range series Hard

A.

for every

B.

for odd, and for even

C.

for odd, and for even

D.

for odd, and for even

60 $The constant function on is represented by a half-range sine series. Which statement correctly gives its coefficients and endpoint sum at ?$

half range series Hard

A.

for odd and for even; the sum at is

B.

for odd and for even; the sum at is

C.

for odd and for even; the sum at is

D.

for every; the sum at is

Unit 6: Introduction to Fourier series - Practice Quiz

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