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
19
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
30
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
52
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