Mathematics Complex number Functions: sinusoids

Sin function, cosine function Differentiation Integration

1 2

Quadratic equation Quadratic equations:

Solution:

Example:

02 cbxax

aacbbx

242

0132 2 xx

21or1

2213

222433 2

x

a=2b=-3c=1

3

Quadratic equation Real solutions if What about this case?

Example: 012 x

2 42

b b acxa

042 acb

042 acb

1j jx 4

Complex Number – rectangular form

Complex number:

Rectangular form:

jbaz

Real

Imaginary, j (a,b)

a

b

Re z = a, Im z = b

Real part Imaginary part

5

Calculations in rectangular form z = a + jb w = c + jd Addition/subtraction

z+w = (a+c) + j(b+d) z-w = (a-c) + j(b-d)

Multiplication zw = (a+jb)(c+jd) = ac + jad + jbc + j*j*bd

= ac + jad + jbc - bd= (ac-bd) + j (ad+bc)

6

Complex numbers

A = 1 + j3 B = 2 – j4

A+B = AB =

7

Calculations in rectangular form z = a + jb w = c + jd Division

2 2

a jb c jdz a jbw c jd c jd c jd

ac bd j bc adzw c d

8

Exercise 1

Solution:

2 ?2

z jw j

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Complex Number – polar form Rectangular form:

Polar form:

jbaz

Real

Imaginary, j (a,b)

a=cos

b=sin

abz

baz

ezzz j

1

22

tan

sincos jzez j

10

Example z=-1-j

zz

342j

z e

11

Radian angle Radian

Standard unit of angular measure Equal to the length of the arc of a unit circle 2/360

12

Exercise 2 Express the following numbers in polar

form:

31

31

31

3

2

1

jz

jz

jz

13

Exercise 3 Express the following numbers in polar

form:

53

2

1

zjz

jz

14

Exercise 4 Given:

z1=2+3j z2=-2-3j

Calculate (z1)2/ z2

15

Exercise 5 Simplify 2

93 221

jj ee

answer

29

323

1 jjj

eee

16

Conjugate

Complex number: Conjugate:

jbaz

Real

Imaginary, j (a,b)

a

b

jejbaz *

b

17

Euler’s Relation

sincossincosje

jej

j

jee

ee

jj

jj

2sin

2cos

Functions and Graphs

x: input For each input exactly one output x=3, y= 11 x: independent variable f(x): function output / dependent

variable

18

2 2f x x

Different kinds of functions Constant function

Linear functions

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Different kinds of functions Quadratic function

Polynomial functions

20

Different kinds of functions Exponential function

21

Different kinds of functions Trigonometric function sin:

cosine:

22

23

Sinusoids

Basis of all signals A sinusoid is a signal that has its

magnitude changes in time according to a sine function sin()

sin

(in degree)24

Sinusoids

Radian Standard unit of angular measure Equal to the length of the arc of a unit

circle 2/360

25

Sinusoids

Radian: 2/360

26

27

Change of Amplitude

sin(x)

5 sin(x)

sin(x)

A sin (x)

How large the sine wave is

Change of Amplitude

28

29

Change of Frequencysin(x)

sin(2x)

Sin(3x)

sin(bx)How fast the sine wave is changing

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Time shift and phase shift

x(t) is the received signal If this signal is received t0 seconds

later x(t+ t0)

Exercise 6 Sketch: y=sin(2x) Sketch: y=sin(2x+1)

31

Exercise 7 Sketch: y=sin(2x+pi/6) Sketch: y=3sin(x+pi/4)

32

Differentiation

Related to “find” velocity of an object at a particular instant

Object moves along the x-axis and its displacement s at time is:

Velocity at t=2?

33

Review of Calculus -Differentiation Differentiation is the mathematical process to

evaluate the derivative of a function (signal) Derivative refers to the instantaneous rate of

change of a function (signal) For a function f(x) to be continuous at a point,

the function must exist at the point, and a small change in x produces only a small change in f(x)

34

y

x

y = f(x)

Small change in x, i.e. x will cause a small change in y, i.e. y, since y = f(x)

x

y

Derivative and the slope of a curve

35

y

x

P (x1, y1)

Q (x2, y2) The slope of the line through P and Q is

The slope at any point of a curve (say point P) is the limiting value of the slope of the PQ as Q approaches P, i.e., when x is very small

2 1

2 1

y ymx x

The derivative

Given y=f(x), the derivative of y with respect to x is given by

The derivative of a curve y=f(x) at a point (e.g., x1) is the slope of the curve at that point 36

dy df x f xdx dx

0

limh

f x h f xh

Derivative of a constant

y=f(x)=c, c is a constant

37

dy d cdx dx

0lim 0h

c ch

Derivative of a straight line

y=f(x)=3x

38

3dy d xdx dx

0

3 3lim 3h

x h xh

Derivative of t^2

39

2ds t d tdt dt

2 2

0 0lim lim 2 2h h

t h tt h t

h

Exercise 8 of finding the derivative

40

3y f x x

Slope at x=2 is 12

Basic Differentiation Formulas

41

f(x)

Constant, c 0

df xdx

nx 1nnx

xe xe

xa lnxa a

ln x 1x

Differentiation Differentiation is the mathematical

process to obtain the derivative of a function

Given y=f(x)=3-2x, differentiate y with respect to x (i.e., derivative of y with respect to x)

42

3 2 3 2dy d d dx xdx dx dx dx

0 2 2d xdx

Exercise 9

Given Differentiate y with respect to x:

Given Differentiate y with respect to x

43

2 2y f x x

2

3 22xy f x

x

Derivative of commonly used functions

44

Derivative of commonly used functions

45

Derivative of commonly used functions

46

Exercise 10

47

2

3 22xy f x

x

2

3 22

dy d xdx dx x

Finding the maximum/minimum One important applications of

differentiation is to find the maximum/minimum point(s) of a curve

48

Find the maximum/minimum

Step 1: Find

Step 2: Set

49

dydx

3 2 25 10 100 3 10 10dy d x x x x xdx dx

0dydx

23 10 10 0

10 100 4 3 104.1387or 0.8054

2 3

x x

x

Exercise 11

Find the derivatives of the following functions:

50

2

2

1.

2. 2 3 2

3. 3 2

y x

y x x

y x

Exercise 12 A rectangular box without lid is to be

made from a square cardboard of sides 18cm by cutting equal squares from each corner and then folding up the sides.

Find the length of the side of the square that must be cut off if the volume of the box is to be maximized.

51

Review of Calculus -Integration

Integration is a mathematical operation that allows the evaluation of the total sum of a function within a certain evaluation window

Example application It is known that, on average, the Internet

traffic y in normal weekdays is given by y=f(x), where x is the time of a day (00:00 to 23:59)

For planning the networking system, need to know the total traffic during the office hour from 9:00 am to 5:00pm

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Sum of a function

If the traffic is constant in a day, the total traffic can be obtained as,

53

17 9y c

Traffic is not constant???

Sum of a function

54

9 : 00 1

10 : 00 1

... 17 : 00 1

y f hr

f hr

f hr

55

Anti-differentiation

Integration: reverse operation of differentiation

By differentiating a function y=f(x), we get the change of y, dy for a small change in x

By integrating dy/dx, we get back the original function y

56

Example y=x, dy/dx=1

57 58

The indefinite integral

If

F(x) is known as the indefinite integral of f(x)

The constant c is needed since the derivative of a constant is zero

The result of an anti-derivative is not unique ( c can be any value)

59

d F x f x f x dx F x cdx

Example

60

38x dx F x c 38d F x xdx

38x

1n nd x nxdx

4 32 8d x xdx

3 48 2x dx x c

Find the indefinite integral of

Solution:

Exercise 13

61

39 xe dx F x c 39 xd F x edx

39 xeFind the indefinite integral of

Solution:

A table of Integrals

62

function Integral

1

1

nx cn

1x

ln x c

nx

xexe c

sin x cos x c

cos x sin x c

Exercise 14

63

The definite integral

Given

The total sum of the function from x=a to x=b:

64

f x dx F x c

x b x a

x b x a

F x c F x c

F x F x

The definite integral

The definite integral of a function f(x) is defined as,

a: lower limit b: upper limit The definite integral: area under the

curve of y=f(x) from x =a to x=b (summation) 65

b

x b x aa

f x dx F x F x

Properties of definite integral

Linearity

Inequality If m≤f(x)≤M, then

66

b b b

a a a

f x g x dx f x dx g x dx

b

a

m b a f x dx M b a

Properties of definite integral

Additivity of integration on interval If a≤c ≤b, then

Reserve limit of definite integral

67

b c b

a a c

f x dx f x dx f x dx

b a

a b

f x dx f x dx 68

Application of Integration

b

a

dxxfAarea

69

Example Sketch the function Shade in the area defined by the integral Compute the integral

f(x) = x + 1

0

22

1

21

dxxfA

dxxfA

70

Example

0

22

1

2

21

21 2

12241

21

2

dxxfA

xxdxxfA

02

0

2

2

xx

71

Exercise 15

Compute the integral Shade in the area defined by the integral

x

dtxF1

2Solution

x>1

Exercise 16 Find the area of the region that is

bounded by the line x=1, the x-axis and the curve

72

Area between graphs of function

73

Example Find out the area lies below y=x+2,

above that of y=x^2, and between x=-1 and x=2

74

Exercise 17 Find the area of the combined region

bounded by the x-axis and the curve

75