Week 9 Fourier series

Question 1

what is a period function?

Answer 1

A periodic function, f(t), repeats at regular intervals T. Where the interval between repetitions,
T, is called the period.

Question 2

When is a period function called even?

Answer 2

• symmetric about the vertical axis

Question 3

When is a period function called odd?

Answer 3

Question 4

The equation for a fourier series calculated for a periodic function f(t) =

(state each definition for the terms)

Answer 4

Question 5

What are the conditions for the fourier series to exist?
(Another way to phrase question?)

Answer 5

for the Fourier series to converge to the function of interest, f(t)

Question 6

How does a series converge?

Answer 6

• A series is a sequence of sums.
• So for a series to converge, these sums have to get closer and closer to a specific limit as we add more and more terms up to infinity

Question 7

what’s the period of these periodic functions?

Answer 7

Question 8

what does A and C represent?

Answer 8

Question 9

Find the period of this function:

Answer 9

Question 10

period of sin and cos?

Answer 10

2 pi

Question 11

general form to solve/check period of a function

Answer 11

Question 12

even or odd function?

Answer 12

even

Question 13

even or odd function?

Answer 13

even

Question 14

even or odd function?

Answer 14

even

Question 15

Define coshx and sinhx

Answer 15

1/2 (ex + e-x) = coshx

1/2 (ex-e-x) = sinhx

these are the decomposition of odd/even components of the function f(x)=ex
where:
fe(x) = coshx
fo(x) = sinhx

Question 16

define tanhx and give its graph

Answer 16

Question 17

is f(x) = x +x2 even or odd?

Answer 17

neither

Question 18

periods for sin(kx) and cos(kx)

Answer 18

period= 2π/ k

Question 19

what are the properties for even and odd functions?

Answer 19

even × even = even
even × odd = odd
odd × odd = even

even + even = even
odd + odd = odd

derivative of even function= odd

derivative of odd function is even

Question 20

Sketch graph of y = e-x

Answer 20

−∞ < x < ∞ and the range is y > 0

Question 21

sketch y = ln x.

Answer 21

domain is x > 0 and the range is −∞ < y < ∞

Question 22

what do these transformations do?
1) y = f (x) + c

2) y = f (x + c)

3. y = cf (x)
4. y = f (cx)

5) -f(x) and f(-x)

Answer 22

1) y+c in y-axis

2) The graph of y = f (x) is shifted to the left if c > 0 or to the right if c < 0. (x-c) coordinates

3) sub in y*a

4) sub in x/c

5) -f(x)= negatise all y values= -y (reflection over y-axis)
f(-x) = negatise all x values= -x (reflection over x-axis)

Question 23

what kind of functions are:
1) x2 , x4 , x6
2) x3 , x5 , x7
3) cosx, sinx, tanx

Answer 23

1) even
2) odd
3) cosx = even (so cos(-x)= cosx)
sinx= odd (sin(=x) = -sinx)
tanx= odd (tan(=x) = -tanx)

Question 24

even or odd function?

Answer 24

odd

Question 25

even or odd function?

Answer 25

odd

Question 26

even or odd function?

Answer 26

odd

Question 27

Calculate the fourier series for this:

Answer 27

Question 28

Integration by parts?

Answer 28

Question 29

when does a fourier series converge?

Answer 29

• the coefficient in front of the sin decreases
• the frequency in the trig bracket increases

Question 30

Answer 30

Question 31

?

Answer 31

Question 32

equation for the derivative of the fourier series

Answer 32

Question 33

does this term converge?

Answer 33

yes, as n increases, each term in the series gets smaller, so it converges

Question 34

equation for the integration of the Fourier series

Answer 34

Question 35

Answer 35