Free Printable M6 - Repaso Parcial 1

1.

2.- Las funciones pares son las que cumplen con la siguiente condición:

a)

$f\left(x\right)\ =\ f\left(x\right)$

b)

$f\left(x\right)\ =\ f\left(-x\right)$

c)

$f\left(-x\right)\ =\ -f\left(x\right)$

d)

$f\left(-x\right)\ =\ f\left(x\right)$

2.

3.- En matemáticas la expresión siguiente se utiliza para representar a

a)

Una serie de Legendre

b)

Una serie de Fourier

c)

Una serie de Morgan

d)

Una serie de Crammer

3.

4.- Las funciones impares son aquellas que cumplen con la siguiente condición:

a)

$f\left(-x\right)\ =\ -f\left(x\right)$

b)

$f\left(x\right)\ =\ f\left(x\right)$

c)

$f\left(-x\right)\ =\ f\left(x\right)$

d)

$-f\left(x\right)\ =\ f\left(x\right)$

4.

5.- La serie de Fourier de una función definida en el intervalo (-p, p) o en el medio intervalo es:

a)

b)

c)

d)

5.

8.- Son algunas de las áreas de aplicación de las series de Fourier

a)

Analisis de potencial y voltages

b)

Música, Cálculo de presión y volúmenes

c)

Ingeniería, análisis vibratorio, acustica, óptica

d)

Análisis de presión y temperatura

6.

The value of sin $n\pi$ is

a)

1

b)

0

c)

-1

d)

n

7.

The value of cos $n\pi=$ ------ and Cos 0=-------

a)

0 and 0

b)

$\left(-1\right)^n$ and 0

c)

$\left(-1\right)^n$ and 1

d)

1 and 1

8.

If f(x) = sinh(x) defined in -π<x<π then a₀and a_n values are

a)

a₀ = 0 , a_n= 1

b)

a₀ = 1 , a_n= 1

c)

a₀ = 0 , a_n= 0

d)

a₀ = 1 , a_n= 1

9.

Which of the following functions cannot be expressed as a Fourier series ?

a)

tanx

b)

x

c)

xsinx

d)

cosx

10.

If\ b_n\ =\frac{4}{n\pi}\left\{1-\left(-1\right)^n\right\}\ ,\ What\ is\ the\ value\ of\ b_3\ ?

a)

$\frac{4}{n\pi}$

b)

$\frac{8}{n\pi}$

c)

$\frac{-8}{n\pi}$

d)

0

11.

A periodic function is defined as:

a)

f(x+P) = f(x)

b)

f(x) = -f(x)

c)

f(x+nP) = f(x)

d)

f(x) = f(-x)

12.

The particular conditions that a function f(x) must fulfill in order that it may be expanded as a Fourier series is:

a)

Gibbs phenomenon

b)

Fourier–Mellin theorem

c)

Dirichlet conditions

d)

Convolution theorem

13.

A periodic function is defined as:

a)

f(x+P) = f(x)

b)

f(x) = -f(x)

c)

f(x+nP) = f(x)

d)

f(x) = f(-x)

14.

At a point of finite discontinuity, x₀, the Fourier series converges to:

a)

b)

c)

d)

15.

One of the Dirichlet's conditions for convergence is that a function may have finite number of maxima or minima in any one period. Say true or false.

a)

True

b)

False

16.

The graph of y =|sin x| is periodic, with a period of . Say True or False.

a)

True

b)

False

17.

f(x) expanded in a fourier series should satisfy certain condition known as ……….

a)

periodic

b)

Dirichlet's

c)

prerequisite

d)

continuous

18.

Find the $a_0$ of Fourier Series of f(x)= $x^2,\ \ \ 0<x<2\pi$

a)

$\frac{4\pi^2}{3}$

b)

$\frac{2\pi^2}{3}$

c)

$-\frac{4\pi^2}{3}$

d)

$\frac{4\pi^{ }}{3}$

19.

Find the $b_n$ of Fourier Series of f(x)= $e^x,\ \ \ -\pi<x<\pi$

a)

$-\frac{2n\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

b)

$\frac{2n\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

c)

$-\frac{2\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

d)

$-\frac{2n\cos nx}{\left(1+n^2\right)}.\sinh\pi$

20.

For function $f\left(x\right)=\left|\sin\ x\right|,\$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$ , the $a_n$ is equal to

a)

$\frac{4}{\pi}\int_0^{\frac{\pi}{2}}\sin\ \left(x\ \right)\cos\ \left(2nx\right)\ dx$

b)

0

c)

$\frac{2}{\pi}\int_0^{\frac{\pi}{2}}\sin\ \left(x\ \right)\cos\ \left(2nx\right)\ dx$

d)

I am not sure, please help me!

21.

What function am I?

a)

Even

b)

Odd

c)

Neither even nor odd

22.

What will be the value of

a_0

in the Fourier Series of the above function in

\left(-4,\ 4\right)

?

a)

1/4

b)

4

c)

-4

d)

0

23.

Find the $b_n$ of Fourier Series of f(x)= $x^2,\ \ \ -\pi<x<\pi$

a)

Zero

b)

$\frac{3\pi}{2}$

c)

$-2\pi$

d)

$-1$

24.

Find the $a_0$ of Fourier Series of f(x)= $4-x^2,\ \ \ 0<x<2$

a)

$\frac{8}{3}$

b)

$\frac{7}{3}$

c)

$\frac{4}{3}$

d)

$\frac{5}{3}$

25.

Find the F.S. of $f\left(x\right)\ =\ -\cos x\ \ \ \ in\ \ -\pi\le x\le\pi$

a)

$-\frac{2n\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

b)

$\frac{2n\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

c)

$\frac{2n\left(1+\cos n\pi\right)}{\left(n^2-1\right)}\ \ if\ \ n\ne1$

d)

$\frac{2\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

M6 - Repaso Parcial 1 - 2025

Total questions: 25

Worksheet time: 45mins

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