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Unit 4: Fourier Series - Practice Quiz

MTH174 — Engineering Mathematics 60 Questions
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1 A Fourier series represents a periodic function as a sum of which type of functions?

Introduction and Euler's formulae Easy
A. Only exponential functions
B. Logarithmic and rational functions
C. Sines and cosines
D. Only polynomial functions

2 For a Fourier series with period , what is the constant term usually written as?

Introduction and Euler's formulae Easy
A.
B.
C.
D.

3 Which formula gives the Fourier cosine coefficient for a function of period ?

Introduction and Euler's formulae Easy
A.
B.
C.
D.

4 Euler's formula connecting complex exponentials with trigonometric functions is

Introduction and Euler's formulae Easy
A.
B.
C.
D.

5 Which set of conditions is commonly associated with the existence of a Fourier expansion?

Conditions for a Fourier expansion Easy
A. Cauchy conditions
B. Dirichlet conditions
C. Bernoulli conditions
D. Newton conditions

6 A function satisfying Dirichlet conditions should have how many finite discontinuities in one period?

Conditions for a Fourier expansion Easy
A. An infinite number only
B. Exactly one discontinuity
C. No finite discontinuities
D. A finite number

7 Under suitable conditions, at a point where is continuous, its Fourier series converges to

Conditions for a Fourier expansion Easy
A.
B.
C.
D.

8 A piecewise smooth function on a finite interval is generally suitable for a Fourier expansion because it has

Conditions for a Fourier expansion Easy
A. Constant derivative everywhere
B. No defined values
C. Finite variation
D. Only odd powers

9 At a jump discontinuity, the Fourier series converges to

Functions having points of discontinuity Easy
A. The average of the two limits
B. The larger limit
C. The right-hand limit
D. The left-hand limit

10 If and at a discontinuity, the Fourier series converges there to

Functions having points of discontinuity Easy
A.
B.
C.
D.

11 A discontinuity where the left and right limits are finite but unequal is called a

Functions having points of discontinuity Easy
A. Jump discontinuity
B. Stationary discontinuity
C. Removable discontinuity
D. Periodic discontinuity

12 At a point of continuity, the left-hand and right-hand limits of a function are

Functions having points of discontinuity Easy
A. Equal to each other
B. Always zero
C. Always opposite
D. Both infinite

13 For a Fourier series defined on , the sine and cosine terms generally have arguments involving

Change of interval Easy
A.
B.
C.
D.

14 The fundamental period associated with the interval is

Change of interval Easy
A.
B.
C.
D.

15 For a function defined on , what value of is used in the general interval ?

Change of interval Easy
A.
B.
C.
D.

16 A function is even if it satisfies

Even and odd functions Easy
A.
B.
C.
D.

17 A function is odd if it satisfies

Even and odd functions Easy
A.
B.
C.
D.

18 Which function is even?

Even and odd functions Easy
A.
B.
C.
D.

19 For an even function integrated over , the integral can be written as

Even and odd functions Easy
A.
B.
C.
D.

20 A half-range Fourier series is formed when a function is defined on

Half range series Easy
A. A single point
B. A half interval
C. An infinite two-dimensional region
D. Only the whole real line

21 For on , what is the value of the sine coefficient in its Fourier series?

Introduction and Euler's formulae Medium
A.
B.
C.
D.

22 If a Fourier series is written as , what is for the constant function on ?

Introduction and Euler's formulae Medium
A.
B.
C.
D.

23 Which Euler identity correctly connects the exponential and trigonometric forms used in Fourier analysis?

Introduction and Euler's formulae Medium
A.
B.
C.
D.

24 Which condition is generally sufficient for a function to possess a Fourier expansion on a finite interval?

Conditions for a Fourier expansion Medium
A. It must have zero average value
B. It must be continuous everywhere
C. It must satisfy Dirichlet-type conditions
D. It must be bounded and periodic only

25 A function is piecewise smooth on and has finitely many finite discontinuities. What can be concluded about its Fourier series?

Conditions for a Fourier expansion Medium
A. It has a Fourier expansion under Dirichlet conditions
B. It converges uniformly at every point
C. It has only cosine terms
D. It cannot be represented by a Fourier series

26 If a periodic function has infinitely many unbounded oscillations in every finite subinterval, which Fourier-series condition is most directly violated?

Conditions for a Fourier expansion Medium
A. The function has a zero constant coefficient
B. The function is defined on a symmetric interval
C. The function has a nonzero period
D. The function has a finite number of extrema

27 At a jump discontinuity , where and , to what value does the Fourier series converge at ?

Functions having points of discontinuity Medium
A.
B.
C.
D.

28 Suppose a periodic function is defined by for and for . What is the Fourier-series value at ?

Functions having points of discontinuity Medium
A.
B.
C.
D.

29 At a point where a periodic function is continuous, what value does its convergent Fourier series approach?

Functions having points of discontinuity Medium
A. The function value at that point
B. The value of the first Fourier term
C. The average over the full period
D. The larger neighboring value

30 For a Fourier series on the interval , which trigonometric basis is appropriate?

Change of interval Medium
A. and
B. and
C. and
D. and

31 A function has period . In a Fourier series written on , what is the argument of the th sine and cosine terms?

Change of interval Medium
A.
B.
C.
D.

32 A function has period . In a Fourier series written on , what is the argument of the th sine and cosine terms?

Change of interval Medium
A.
B.
C.
D.

33 When changing the Fourier-series interval from to , which coefficient formula is correct for ?

Change of interval Medium
A. for every function
B.
C.
D.

34 If is even on , which Fourier coefficients must vanish?

Even and odd functions Medium
A. All coefficients
B. All coefficients
C. Only
D. Only the coefficients with even index

35 If is odd on , which statement about its Fourier series is correct?

Even and odd functions Medium
A. It contains both terms with equal coefficients
B. It contains a nonzero constant term only
C. It contains only sine terms
D. It contains only cosine terms

36 For on , which simplification is valid when calculating its Fourier coefficients?

Even and odd functions Medium
A. Only sine integrals need to be evaluated
B. All integrals over become zero
C. Cosine integrals can be doubled over
D. The function should first be treated as odd

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

Half range series Medium
A.
B.
C.
D.

38 The half-range sine series on corresponds to which type of extension of to ?

Half range series Medium
A. A constant extension
B. A periodic extension without reflection
C. An even extension
D. An odd extension

39 For the half-range cosine series of on , which coefficients are obtained using the convention ?

Half range series Medium
A. and
B. and
C. and
D. and

40 For a half-range cosine series on , the corresponding extension of to is:

Half range series Medium
A. Antiperiodic with period
B. Even with
C. Odd and discontinuous at every integer multiple of
D. Linear with zero value at the origin

41 For the -periodic function on , what is its complex Fourier coefficient for ?

Introduction and Euler's formulae Hard
A.
B.
C.
D.

42 If on and is extended periodically with period , which is the correct cosine coefficient for ?

Introduction and Euler's formulae Hard
A.
B.
C.
D.

43 For a real-valued function with complex Fourier coefficients , which relation is necessary and sufficient for the Fourier series to represent a real-valued function?

Introduction and Euler's formulae Hard
A. for every
B. for every
C. for every
D. for every

44 Which condition set is sufficient for the Fourier series of a periodic function to converge at every point to the midpoint of its one-sided limits?

Conditions for a Fourier expansion Hard
A. The function is differentiable at every interior point
B. The function is continuous only at rational points
C. The function is piecewise smooth with finitely many extrema and discontinuities
D. The function is bounded and periodic

45 A periodic function is continuous everywhere except at finitely many jump discontinuities and has an integrable derivative on each smooth subinterval. What does its Fourier series converge to at a jump point ?

Conditions for a Fourier expansion Hard
A.
B.
C.
D. regardless of the jump

46 Which statement correctly distinguishes a sufficient Dirichlet condition from a necessary condition for Fourier convergence?

Conditions for a Fourier expansion Hard
A. Piecewise smoothness is sufficient but not necessary for all Fourier convergence
B. Piecewise smoothness is necessary for all Fourier convergence
C. Boundedness alone guarantees pointwise Fourier convergence
D. Continuity is sufficient only for finite Fourier sums

47 Let for and for , extended periodically. What is the Fourier series value at ?

Functions having points of discontinuity Hard
A.
B.
C.
D.

48 Suppose a periodic function has a jump of magnitude at . What feature appears in the large- behavior of its Fourier coefficients?

Functions having points of discontinuity Hard
A. They remain exactly constant
B. They generally decay like
C. They decay exponentially for every jump
D. They generally decay like

49 A periodic function is assigned the value at an isolated jump point, while its one-sided limits are and . What is the Fourier series sum at that point?

Functions having points of discontinuity Hard
A.
B.
C.
D.

50 For a piecewise smooth periodic function, the Fourier series converges to the original function at a point precisely when which condition holds?

Functions having points of discontinuity Hard
A. and this common limit equals
B. is larger than both one-sided limits
C.
D.

51 For a Fourier series on , which basis functions correspond to the correct fundamental angular frequency?

Change of interval Hard
A. and
B. and
C. and
D. and

52 A function is represented on by a Fourier series with period . Which coefficient formula is correct for the cosine terms?

Change of interval Hard
A.
B.
C.
D.

53 Under the substitution , a Fourier series on transforms into which standard interval and basis?

Change of interval Hard
A. with and
B. with and
C. with and
D. with and

54 For on , regarded as a function of period , which statement about its Fourier coefficients is correct?

Change of interval Hard
A. Only sine coefficients are nonzero
B. Both sine and cosine coefficients generally occur
C. Only cosine coefficients are nonzero
D. All nonconstant coefficients vanish

55 If is odd on and satisfies the usual integrability conditions, which Fourier coefficients must vanish?

Even and odd functions Hard
A. All , including
B. Only
C. Only coefficients with even index
D. All , but not

56 If is even on , which integral correctly computes its sine coefficient?

Even and odd functions Hard
A.
B.
C.
D.

57 For an even function on , which expression gives its Fourier series in the most reduced form?

Even and odd functions Hard
A.
B.
C.
D.

58 A function satisfies on . Which parity decomposition is correct?

Even and odd functions Hard
A. and
B. and
C. and
D. and

59 The half-range cosine series of on corresponds to which extension across ?

Half range series Hard
A. A zero extension outside
B. An even extension of , namely
C. A periodic extension with period
D. An odd extension of

60 The half-range sine series of on has coefficients . Which coefficient is zero?

Half range series Hard
A.
B.
C. No coefficient is zero
D.
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