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MATH1021 Notes
MATH1021 NOTES
________________________________________________________________
______________________
MODULE 1: REAL AND COMPLEX NUMBERS
1.1 Number Sets
Symb
ol
Meaning
Example
{…}
Is a set of
A = {a, b, c, d, …}
∈
Is an element of
s
∈ S
⊆
Is a subset of
S
⊆ T
⊂
Is a proper subset of
(Smaller set is strictly a subset of the
larger, cannot be equal)
S
⊂ T
∪
Union
(Elements are members of either one or
both of the sets)
{1, 2}
∩
Intersection
(Elements which are members of both
sets)
{1, 2, 3}
\
Minus/Without
(Elements which are in one set but not in
another)
∪
{3, 4} = {1,
2, 3, 4}
∩ {3, 4, 5} =
{3}
{1, 2, 3}\{3, 4, 5} = {1, 2,
4, 5}
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|
Such that
{x
Symb
ol
Meaning
Example
N
Set of natural numbers
(Whole positive numbers, and zero)
{0, 1, 2, 3,…}
Z
Set of integers
(Whole numbers)
{…, -2, -1, 0, 1, 2,...}
Q
Set of rational numbers
(All numbers of the form n/m where n and
m are integers, and m ≠ 0)
½, -¼
R
Set of real numbers
(Includes all rational and irrational
numbers)
√ 2, π
C
Complex numbers
i
∈
Q
| x > 0}
Absolute Value (Modulus)
 Gives distance on real number line from x to zero is equal to
|x− y |=|y−x|
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MATH1021 Notes
1.2 Intervals
Symbol
Meaning
Example
( or )
Endpoint not included in the set
0 < x < 1 = (0, 1)
[ or ]
Endpoint is included in the set
0 ≤ x ≤ 1 = [0, 1]
(…)
Open interval
[…]
Closed interval
(a, ∞), (-∞,
a)
[a, ∞), (-∞,
a]
Semi-infinite intervals
Note: ∞ and -∞ always take a round
bracket
1.3 Complex Numbers
Imaginary Unit
i 2=−1
i=√ −1
Cartesian Form of a Complex
Number
a+ib
Re (z) = a
Im (z) = b


Purely imaginary number (imaginary number): Any non-zero real multiple
of i
o When b = 0
Complex number: Combination of real and imaginary numbers
o When a = 0
1.4 Arithmetic in Cartesian Form
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MATH1021 Notes
Addition
Add real parts to real parts and imaginary parts to imaginary parts.
Subtraction
Subtract real parts from real parts and imaginary parts from imaginary parts.
Multiplication
Expand the brackets, with i2 able to be simplified to -1, and collect terms into real
and imaginary parts.
Complex Conjugate
 Reflection of z about the real axis
Complex Conjugate
z=a+ib

ź=a−ib
Properties:
o
z+´ w=ź+ ẃ
o
o
zw=ź
´
ẃ
n
n
z´ =ź
If z=a+ib , then z ź= ( a+ib ) (a−ib )=a+b
o
Division
To find z/w, multiply both top and bottom by the complex conjugate of w.
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MATH1021 Notes
1.5 Set of Complex Numbers
Note: The set of complex numbers is not ordered
The Complex Plane
 Horizontal axis  Real axis
 Vertical axis  Imaginary axis

z=a+ib is point ( a , b ) on complex plane
Modulus of a Complex Number
Modulus
z=a+ib
|z|= √ a 2+ b2
|z|= √ z ź

Properties:
o
|zw|=|z||w|
o
o
o
|wz |= ||wz||
|z +w| ≤|z|+|w|
|z−w| ≥| z|−|w|
[triangle inequality]
Note:
|i|=1
Subsets of the Complex Plane
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MATH1021 Notes
Recommended for you
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COMP2123 Finals Notes - Includes all lecture material, and
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49
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Data Structures & Algorithms
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MATH1021 Notes
MODULE 2: POLAR FORMS OF COMPLEX
NUMBERS
2.1 Standard Polar Form
z = a + ib, real part a and imaginary part b
 Distance (modulus), r=|z|= √ a2 +b 2
 Angle  = arg (z) =
tan−1
( ba )
Cartesian Form
z=a+ib
Standard Polar Form
θ+i sinθ
cos ¿
z=r ¿
General Form of Argument: arg (z) =  + 2k
Principal Argument Arg (z): Arg (z) =  [–    ]
Special Angles
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2.2 Polar Exponential Form
Euler’s Formula
e iθ=cosθ + isinθ
Polar Exponential Form
For z=a+ib
z=ℜiθ
2
2
r=|z|= √ a +b
Angle  = arg (z) =
()
tan−1 b
2.3 Arithmetic in Polar Form
Multiplication
zw=r eiθ × se iϕ=(rs) e
Division
i(θ+ϕ)
()
iθ
z re
r e i(θ−ϕ)
= iϕ=
w se
s
Integer Power
n
z n= ( ℜiθ ) =r n einθ
De Moivre’s Theorem
n
(e iθ ) =e inθ
( cosθ + isinθ )n=cos ( n θ ) +isin (n θ) ,
for any n ∈ Z
2.4 Roots of Complex Numbers
Every non-zero complex number has exactly n distinct n-th roots.
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MATH1021 Notes
When finding roots of complex numbers, express it in polar form
z=e iθ
and then add
an integer multiple of 2.
Roots of Complex Numbers
z=e θ+2 kπ , k ∈ Z
1
( θn + 2kn )
z n =e
Note: Some real numbers may have complex roots.
2.5 Roots of Polynomial Equations
A polynomial of degree
has at most
n
n
complex roots.
If the coefficients in a polynomial equation are all real, then all of the non-real
complex roots occur in complex conjugate pairs.
2.6 Sine and Cosine in Terms of Exponentials
cosθ=
eiθ + e−iθ
2
sin θ=
e iθ−e−iθ
2
2.7 Complex Exponential Function
Complex Exponential Function
For z=x +iy
e z =e x+iy =e x e iy= e x( cos y+i sin y)
|e z|=e x , arg (e z) = y
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MATH1021 Notes
MODULE 3: FUNCTIONS
3.1 Functions
Function: If A and B are sets then a function from A to B (written f : A → B ) is a
rule f which assigns to each element x in A exactly one element in B, denoted
x
) (i.e. every element of one set associates to exactly one element of the
f¿
second set).
Given a function f from A to B,
codomain of the function.
The range is { f ( x )∨x ∈ A } .
Vertical Line Test
For every line x=a
one point. Given a
f : A → B , A is the domain and B is the
parallel to the y−axis intersects the graph in exactly
in the fomain of f , f (a) must be unique.
3.2 Combining Functions
Composite Functions
Given two functions f and g , the composite
function g ∘ f is defined by the expression
( g ∘ f )(x ) =g( f ( x )) for all x in the domain of
f such that f ( x) is in the domain of g .
3.3 Injective and Inverse Functions
Injective Functions
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MATH1021 Notes
Injective Functions
A function f is injective (one-to-one) on the
domain A if distinct elements in A are mapped to
distinct elements in B.
f
f
is injective if f ( x 1 )=f ( x 2)
for
x 1=x 2 .
is injective if no horizontal line intersects its
Inverse Functions
Inverse Functions
f : A → B , f −1 : B → A
[with function injective in domain]
The inverse function is a reflection of the function
in the line y=x .
Property of the Inverse Function
f −1( f ( x ) )= xif ∧only if y = f ( x)
Calculating the Inverse Function of f ( x)
 If f is not injective on its natural domain, restrict domain to section
strictly increasing or decreasing
 Formula for f −1 found by rearranging y=f ( x) to make x the
subject,

Swap
x
x=f −1 ( y )
and y to get the inverse in standard notation
3.4 Inverse Trigonometric Functions
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MATH1021 Notes
3.5 Hyperbolic Functions and their Inverse
cosh x
cosh x =
e x + e−x
2
sinh x
sinh x=
e x −e− x
2
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MATH1021 Notes
Identities
cosh2 x− sinh2 x=1
sinh ( x+ y) =sinh x cosh y +sinh y cosh x
cosh (x + y)=cosh x cosh y +sinh x sinh y
sinh x
cosh x
cosh x
coth x=
sinh x
tanh x=
1
cosh x
1
csch x=
sinh x
sech x =
Derivatives
d
( cosh x )=sinh x
dx
d
( sinh x )=cosh x
dx
Inverse
sinh−1 x=ln (⁡x + √ x 2 +1)
cosh−1 (x)=ln (⁡x+ √ x 2−1)
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MODULE 4: LIMITS AND CONTINUITY
4.1 Informal Definition of Limit
Definition of a Limit
In order for a limit to exist, the function must
approach the same value from the left and
right.
lim f ( x) =L
4.2 One-Sided Limit
Left-Hand Limit
L is the limit of f ( x) as x approaches a from the
left if we can make the values of f (x) as close to L as
we like by taking x sufficiently close to a such that
x is less than a .
−¿
x → a f ( x ) =L
lim ¿
¿
Right-Hand Limit
K is the limit of f ( x) as x approaches a from the left if we can make the
values of f (x) as close to K as we like by taking x sufficiently close to a
such that x is greater than a .
x → a+¿ f ( x ) =K
lim ¿
¿
4.3 Basic Limit Laws
The Sum Law
lim ( f ( x )+ g ( x) ) =lim f ( x )+ lim g (x )
x→ a
x→a
x →a
The Difference Law
lim ( f ( x )−g(x ) )=lim f ( x )−lim g ( x )
x→ a
x →a
x→ a
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MATH1021 Notes
The Product Law
(
) (
lim ( f ( x ) × g (x) ) = lim f ( x ) × lim g( x )
x→ a
x→a
x →a
)
The Quotient Law
lim
x→ a
lim f (x )
f ( x ) x→ a
=
g(x ) lim g (x)
The Power Law
n
[
]
lim [ f ( x ) ] = lim f ( x)
x→ a
x →a
n
The Root Law
√
lim √ f ( x)= n lim f ( x)
n
x→ a
x→a
4.4 Limits at Infinity (Horizontal Asymptotes)
Limits at infinity describes when x becomes arbitrarily large and positive (
x → ∞ ) and arbitrarily large and negative ( x→− ∞ ).
Horizontal Asymptotes
lim f ( x )= L
x→∞
or
lim f ( x )= L
x →−∞
Limits at Infinity of Rational Functions
When calculating
lim
x → ±∞
of
x
p(x)
q( x )
, divide the top and bottom by the largest power
appearing in the denominator.
4.5 Infinite Limits (Vertical Asymptotes)
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MATH1021 Notes
Infinite limit describes when function values become arbitrarily large in
magnitude as x approaches fixed point a or as x → ± ∞ .
Infinite Limits
Write
lim f ( x ) =∞
x→ a
if
can be made to
f ( x)
exceed any positive number choosing
sufficiently close to a .
Write
lim f ( x )=∞
x→∞
if
f ( x)
can be made to
exceed any positive number choosing
sufficiently large and positive.
Write
lim f ( x )=∞
x →−∞
if
f ( x)
x
can be made to
lim f (x ) and
Note: The limit laws require that
x
lim f (x)
x→ c
x →± ∞
exist (i.e. are finite).
The limit laws do not apply to infinite limits.
Vertical Asymptotes
is a vertical asymptote of the
curve y=f ( x) if:
The line
x=a
lim f ( x) =∞
lim f ( x) =−∞
x→ a
x→ a
x → a−¿ f ( x ) =∞
lim ¿
x → a−¿ f ( x ) =−∞
lim ¿
¿
¿
+¿
+¿
x → a f ( x ) =∞
x → a f ( x ) =−∞
4.6 The Squeeze Law
Squeeze Law
For
g( x)≤ f (x )≤ h( x ) for all x near a , and
lim g ( x )=L . Then lim f ( x ) exists and,
x→ a
x→ a
lim f ( x ) =L
x→ a
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MATH1021 Notes
The function
f ( x)
h ( x ) and g ( x ) and so
lim f ( x) =lim g ( x )= lim h( x ) .
is squeezed between
x→ a
x→ a
x →a
4.7 Continuous and Discontinuous Functions
Continuous Functions
A real valued function f

f (a) is defined

lim f ( x ) =L exists

lim f ( x ) =f (a)
is continuous at the point
if:
a
x→ a
x→ a
One-Sided Continuity
if
x → a+¿ f ( x ) =f ( a)
.
lim ¿
A function is continuous from the right at a point from
a
A function is continuous from the left at a point
x → a−¿ f ( x ) =f (a)
.
lim ¿
¿
a
if
¿
Continuity on a Closed Interval
A function is continuous on the closed interval [a, b] if it is continuous at every
point inside the interval and is continuous from the right at a and continuous
from the left at b.
Limits of Continuous Functions
To calculate
x=a
lim f ( x) of function
x→ a
to obtain
f (x) that is continuous at
a , substitute
f (a) .
The Composition Law
If
If
f (x)
is continuous and
f (x)
and
g( x)
lim g ( x )= L ,
x→ a
(
)
lim f ( g ( x )) =f lim g ( x ) =f ( L) .
x→ a
x →a
are both continuous then f ( g ( x ))
is also continuous.
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MATH1021 Notes
Discontinuous Functions
Type
Description
Jump
Discontinuiti
es
Occurs when the value of a
limit can be more than one
value
Diagram
Infinite
Discontinuiti
es
Occurs when
Removable
Discontinuiti
es
Occurs when a limit exists and
is finite, meaning there is the
possibility to define a new
function (which is continuous)
and eliminate the discontinuity
lim f ( x )
x→ a
does
not exist
Functions
that are
Discontinuo
us
Everywhere
MODULE 5: DIFFERENTIATION
5.1 The Derivative at a Point

Derivative of function f at point x=a: Slope of the tangent line to the graph
y=f(x) at point (a, f(a))
5.2 The Derivative as a Function
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Derivative of a Function
f
at a point
a
f ( a+h) −f (a)
h
'
df
f (a ) =
dx x=a
f ' ( a) =lim
h→0


|
If
f : I → R is a real valued function of one variable and its derivative
f ' (x) is defined for all x in the open interval I , f is
differentiable on I
If a function f is differentiable at a point a , then f is also
continuous at a
Basic Table of Derivatives
f ( x)
f ' ( x)
sin x
cos x
cos x
−sin x
tan x
sec x
e
x
ln x
x
a
2
e
x
1
(x> 0)
x
a x a−1 (a constant )
5.3 Basic Rules of Differentiation
The Constant Multiple Rule
The Sum/Difference Rule
( kf ) ' =k f ' , k constant
(f ± g )' =f ' ± g '
The Product Rule
The Quotient Rule
(fg )' =f ' g+fg'
( gf ) = f g−fg'
g
'
'
2
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MATH1021 Notes
The Power Rule
'
( x a) =a x a−1 , a constant
5.4 The Chain Rule
Composite Function
Given two functions f
g , the composite
f ∘ g ¿=f (g( x )) for
x in the domain of g such that g( x) is in
the domain of f
function is
all
f ∘g
and
is defined (
Chain Rule
If
y=f (u) where
u=g ( x) ,
dy dy du
= ×
dx du dx
5.5 Implicit Differentiation
Implicit Differentiation
Consists of differentiating both sides of the relation
x
and then solving the equation for
dy
dx
F ( x , y ) =0 with respect to
(using chain rule).
Logarithmic Differentiation
Derivatives of complex functions can be simplified by taking the natural
logarithm of both sides of the equation and using implicit differentiation.
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MATH1021 Notes
Log Laws:

loga m+ log a n=loga mn

loga m− loga n= loga


loga mn=n loga m
loga 1=0

loga a=1

a log x =x
ln x
loga x=
ln a

( mn )
a
5.6 The Mean Value Theorem
Mean Value Theorem
If f is differentiable on some interval I
containing the points a and b. Then there exists
a number c in the open interval (a, b) such that
f ' (c )=
f (b )−f (a)
b− a
It guarantees there is at least one point c Î (a, b) for which the tangent line to
f at x=c has the same slope as the slope of the secant line joining (a, f (a))
and (b, f (b))
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MATH1021 Notes
MODULE 6: APPLICATIONS OF
DIFFERENTIATION
6.1 Functions of One Variable
Extrema
 Local extrema:
o
f has a local maximum at a point

a
if
f (a)≥ f ( x)
when
x
is near the point a
o
f has a local minimum at a point a if f (a)≤ f (x) when x
is near the point a
Global extrema:
o
f has a global maximum at a point a if f (a)≥ f (x) for all
x in its domain
o
f has a local minimum at a point a if f (a)≤ f (x) for all x
in its domain
The Extreme Value Theorem
If
f is continuous on a closed interval
[a , b] , then f attains a global
maximum value f (c) and a global
minimum value f (d) , where c and
d are some real numbers on [a b]
Critical Points
Critical Points
A critical point of
where either
f (x)
f ' (c )=0
is a number
or where
c
f ' (c)
does not exist.
If f is differentiable at the point c
has a local maximum or minimum at
and
c
Calculation of Global Extrema
1. Find all values f at the critical points on [a , b]
2. Calculate the values of f at the endpoints of the interval
3. Compare all the numbers obtained in the first two steps – The largest of all
values is the global maximum value and the smallest is the global
minimum
6.2 Increasing and Decreasing Functions
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f '(x)


represents the slope of the curve at
If
f ' ( x) > 0 , then f
If f ' ( x) < 0 , then f
y=f ( x) .
is increasing
is decreasing
First Derivative Test



If f ' changes from positive to negative, f has a local maximum
If f ' changes from negative to positive, f has a local minimum
If f ' does not change sign, then f has no local maximum or
minimum
6.3 Concavity and Points of Inflexion
Concavity


f ' ' ( x ) >0
Concave downward when f ' ' ( x ) <0
Concave upward when
Point of Inflexion

When there is a point of inflexion (change in concavity),
f ' ' ( x ) =0 or
f ' ' (0) is undefined
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Second Derivative Test
If f and f ' are differentiable functions and



f ' ( c )=0 ,
If f ( c )> 0 , there is a local maximum at x=c
''
If f ( c )< 0 , there is a local minimum at x=c
If f ' ' ( c )=0 , cannot draw any conclusions
''
6.4 Curve Sketching
1. Domain – Find the set of values
2. Intercepts
lim f ( x) and
x→∞
3. Horizontal asymptote – Calculate
4. Vertical asymptotes – A value
for which f ( x)
x
x=a
is defined
lim f (x)
x →−∞
x → a+¿ f ( x ) =± ∞
such that
lim ¿
or
¿
−¿
x → a f ( x ) =± ∞
lim ¿
¿
5. Critical points -
f ' ( c )=0∨f ' ( c ) does not exist
6. Intervals of increase or decrease – Increasing when
when
f ' ( x ) > 0 , decreasing
'
f <0
7. Concavity and points of inflexion –
8. Sketch the curve
f '' ( x ) =0
6.5 L’Hopital’s Rule
L’Hopital’s Rule
Can use when
lim
x→ c
0
0
or
±∞
±∞
form.
f (x)
f ' (x )
=lim
g(x ) x → c g ' (x)
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MODULE 7: TAYLOR POLYNOMIALS
7.1 An Approximation for e x
7.2 Taylor Polynomials About
x=0
Maclaurin Series (Taylor Series About x=0)
'
Pn ( x ) =f ( 0 ) + f ( 0 ) x +
n
f ' '( 0) 2
f (k ) (0) k
x +…=∑
x
2!
k!
k=0
7.3 Taylor Polynomials About
x=a
Taylor Polynomial About
x=a
( )
'
Pn ( x −a)=f ( a )+f ( a ) (x−a)+
n
f ' ' ( a)
f k (a)
2
( x−a ) + …=∑
( x−a)k
2!
k!
k=0
7.4 Remainder Term
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The Remainder Term
Rn ( x ) =f ( x ) −Pn ( x )
f ( x )=lim Pn ( x ) + lim R n ( x)
n→∞
If
n→ ∞
lim R n ( x ) =0 , then the Taylor series of f
n →∞
converges to
Rn ( x ) =
f at that point.
( n +1 )
(c ) n+1
f
x for 0<c < x
( n+1 ) !
| |
Error : |Rn ( x ) |≤
xn +1
(n+1 ) !
Taylor’s Formula
f ( x )=Pn ( x ) + R n (x )
f ( ) ( 0) n f (n+1 )( c) n +1
f ' '(0) 2
x
x +…+
x +
(n+ 1)!
2!
n!
n
f ( x )=f ( 0) + f ' ( 0) x+
Letting n=0 ,
f ' (c )=
f ( x )−f (0)
for 0< c< x ( this isthe MVT )
x
Proof
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MATH1021 Notes
MODULE 8: TAYLOR SERIES
8.1 Infinite Series
∞
An infinite series is
a0 + a1+…+a n+ …=∑ an
n=0
Partial Sum of an Infinite Series
∞
The k-th partial sum,
S k , of the series
is the sum of all the terms
to
a0
an
∑
n=0
ak .
k
S k =a0 + a 0+ …+ a k =∑ a n
n=0
The Sum of an Infinite Series
∞
The sum, the infinite series
∑ an
is the limit
n=0
∞
as
k → ∞ of the partial sums
S k =∑ an ,
n =0
provided the limit exists,
∞
k
n=0
n=0
Sk =lim ∑ an = L
∑ an=lim
k →∞
k→∞
If the sequence fails to converge to a limit the
Geometric Series
∞
1
∑ r n= 1−r
n=0
For
|r|<1 .
8.2 Taylor Series
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MATH1021 Notes
Maclaurin Series (Taylor Series About x=0)
f ( ) (0) k
f ' ' (0) 2
x
x + …=∑
k!
2!
k=0
∞
f (0)+ f ' (0) x +
k
Examples
−1
¿
¿
4
x2 x
cos x=1− + −…+¿
2! 4!
−1
¿
¿
5
x3 x
sin x = x − + −…+ ¿
3! 5 !
3
xn
x2 x
x
e =1+x + + +…+
n!
2! 3 !
1
=1+ x + x 2 +…+ x n
1−x
xn
n!
3
x2 x
ln(1+ x )=x− + +…+¿
2! 3 !
−1¿ n+1
2n
4
x 2 x +…+ x
+
2n!
2! 4!
x3 x5
x 2 n+ 1
sinh x = x + + + …+
(2 n+1) !
3 ! 5!
cosh x =1+
8.3 Euler’s Formula
∞
The infinite series of complex numbers
∞
∞
cn =∑ an +i ∑ b n
∑
n=0
n=0
n=0
.
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MATH1021 Notes
The Complex Exponential Function
Rn ( x ) =f ( x ) −Pn ( x )
f ( x )=lim Pn ( x ) + lim R n ( x)
n→∞
n→ ∞
lim R n ( x ) =0 , then the Taylor series of f
n →∞
If
converges to
f at that point.
zn
z2
=1+z+ +…
2!
n=0 n!
∞
e z =∑
Proving Euler’s Formula
Let z=iθ
3
iθ
e =1+ iθ+
(
iθ
e = 1−
2
(iθ ) (iθ )
+
+…
2!
3!
)(
2
3
θ θ4
θ θ5
+ −… + i θ− + −…
2! 4!
3! 5!
)
iθ
e =cos θ+i sin θ
8.4 The Binomial Series
Binomial Series
The Taylor series for
(1+x ) p , also known as the ‘binomial
series’ is,
1+ px +
∞
¿ 1+∑
n=1
p ( p−1 ) 2 p ( p−1) ( p−2) 3
x+
x +…
2!
3!
p ( p−1 )( p−2 )… (p −n +1) n
x
n!
8.5 Inverse Tan Function
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MATH1021 Notes
A Series for
Note that
−1
tan x
1
1
d
−1
tan−1 x=
dx=tan x +C .
→∫
2
dx
1+x
1+ x2
−1 ¿n x n for|x|≤1
1
=1−x + x 2−…+ ¿
1+ x
1
=1−x 2 + x 4−… for |x|≤1
1+x 2
x3 x5
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MATH1021 Notes
MODULE 9: THE RIEMANN INTEGRAL
9.1 Riemann Sums
To calculate the region bounded by y =f ( x ) and the x−axis between two
points x=a and x=b ,
1. Start with continuous f ( x) defined on closed interval [a , b]
2. Fix an integer N ≥1 and divide the interval into N subintervals of equal
length,
[ x 0 , x 1 ] , [ x 1 , x 2 ] , … ,[ x N −1 , x N ] , where x 0=a∧x N =b
∆ x=
b−a
N
3. The partition points are constructed,
x i=a+∆ x ×i for i =0,1, 2 , … , N
4. Take the minimum value
mi
of
f (x) on each subinterval
M i of f (x) on each subinterval
5. Repeat, taking the maximum value
Properties

Taking height

Taking height

mi gives a lower estimate for area
M i gives an upper estimate for area
When the function increases, the minimum value mi occurs on the left
point of the subinterval
maximum value
so that

Mi
x
[¿ ¿i −1, x i ]
¿
so that
mi=f (x i−1 ) and the
occurs on the right of the subinterval
x
[¿ ¿i−1, x i ]
¿
M i=f ( xi )
When the function decreases, the minimum value
point of the subinterval
x
[¿ ¿i −1, x i ]
¿
so that
mi
occurs on the right
mi=f (x i) and the
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MATH1021 Notes
maximum value
so that
Mi
occurs on the left of the subinterval
x
[¿ ¿i−1, x i ]
¿
M i=f (xi −1 )
Lower and Upper Riemann Sums
Upper Riemann Sum
The area of the i-th rectangle is m i × ∆ x
total area of the smaller rectangles.
and LN
is the
N
LN =( m1 × ∆ x ) × ( m2 × ∆ x ) +…+( mN × ∆ x ) =∑ mi × ∆ x
i=1
Lower Riemann Sum
The area of the i-th rectangle is M i × ∆ x and
total area of the larger rectangles.
UN
is the
N
U N = ( M 1 ×∆ x ) × ( M 2 ×∆ x ) +…+ ( M N × ∆ x ) =∑ M i × ∆ x
i=1
9.2 The Riemann Integral
The Riemann Integral
For f (x) is a continuous function defined on
interval [a , b] ,
b
LN ≤∫ f ( x ) dx ≤ U N
a
As the size of the subintervals is decreased to zero,
the upper and lower Riemann sums approach a
common value .
b
lim L N = lim U N =∫ f ( x ) dx
N →∞
N→∞
a
Non-Positive Functions
 The Riemann sum is equal to the sum of the areas of all the rectangles
above the axis, minus the sum of the areas of the rectangles below the
axis
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MATH1021 Notes
9.3 Calculating Riemann Sums
Calculating Riemann Sums
N
( f ( c 1 ) × ∆ x) × (f ( c 2) ×∆ x) +…+(f ( c n) × ∆ x )=∑ f ( c i ) × ∆ x
i=1
mi ∆ x ≤ f (ci )≤ M i ∆ x
N
LN ≤ ∑ f ( ci )× ∆ x ≤ U N
i=1
9.4 Properties of the Riemann Integral
Properties
1. If m and M are the minimum and maximum values of
[a , b] , then
f
on the interval
b
m×(b −a ) ≤∫ f ( x ) dx ≤ M ×(b−a)
a
2. If
c
is a constant, then
b
b
S N ( cf )= lim c S N ( f )=c lim S N ( f ) =c∫ f ( x ) dx
∫cf ( x ) dx=Nlim
→∞
N→∞
N →∞
a
a
3. For functions
f
and
g defined on the interval
b
b
b
a
a
a
[a , b] ,
∫(f ( x ) +g ( x ))dx=∫ f ( x ) dx+∫ g ( x ) dx
4. If f
and
is defined on the interval
[a , c] , and
b
is a point between a
b , then
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MATH1021 Notes
c
b
c
a
a
b
∫ f ( x ) dx=∫ f ( x ) dx +∫ f ( x ) dx
Reversing the Direction of Integration
When a> b , ∆ x< 0 . Geometrically, it means that in the Riemann sum areas of
rectangles above the axis now count as negative, and areas below the axis count
as positive.
b
a
a
b
∫ f ( x ) dx=−∫ f ( x ) dx
Finding Number of Subintervals to Produce < Given Error
Calculating Riemann Sums < Error
U N −L N <error
U N −L N = ( f ( max )−f ( min ))
a
a
k=b
k=b
( max −min )
<error
N
LN =∑ k; U N =∑ k
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Riemann – Upper and Lower Sums
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MATH1021 Notes
MODULE 10: FUNDAMENTAL THEOREM OF
CALCULUS
10.1 Integrals as Functions
Integral
Let f be a continuous function defined on
interval [a , b] . For any point x in the
interval, we have the definite integral of f
over the smaller interval [a , x ] .
x
F ( x )=∫ f ( t ) dt
a
Properties of Integrals
1.
x
a
x
a1
a1
a
∫ f ( t) dt=∫ f ( t) dt+∫ f ( t ) dt=C +F(x )
f from a 1 ¿ a
2. If f is positive at a point
that point (and vice versa)
where
x , then
F
C is the definite integral of
is an increasing function at
10.2 The Fundamental Theorem of Calculus I
The Fundamental Theorem of Calculus I
Let f (x) be a continuous function defined on
interval [a , b]
be defined by
of the real line and let
F( x )
x
F ( x )=∫ f ( t ) dt
a
Then
F( x )
is a differentiable function of
and
x
F' ( X ) =f (x )
Proof
Importance
 Every continuous function has an antiderivative
 Confirms that differentiation and integration are inverse processes
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
MATH1021 Notes
Antiderivative called the indefinite integral of f – Can always add an
arbitrary constant without changing that the derivative is f (x)
∫ cos x dx =sin x + C
E.g.
10.3 The Fundamental Theorem of Calculus II
The Fundamental Theorem of Calculus II
Let F( x ) be a continuous function defined on
interval [a , b] of the real line. Suppose F is
defined at each point x of the interval, and
that the resulting function F' ( x ) is
continuous. Then,
b
∫ F ' ( x) dx = F ( b)−F (a)
a
Importance
 The change
F ( b )−F ( a )
of a function
F
over an interval denoted,
b
∫ F ' ( x) dx= [ F ( x)] a =F ( b ) −F ( a)
b
a

F' ( x )=f (x)
b
b
a
a
on the interval
[a , b] and,
∫ f (x)dx=∫ F ' ( x) dx = F (b) −F (a)
Leibniz Integral Rule
Leibniz Integral Rule
d
dx
(∫ )
b (x)
f ( t ) dt =f [ b ( x )] ∙
a (x)
d
d
b ( x )−f [a ( x ) ]∙ a(x )
dx
dx
Improper Integrals
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MATH1021 Notes
10.5 The Natural Logarithm and Exponential Functions
The Natural Logarithm
The Natural Logarithm
x
ln x=∫
1
For all
ln x
1
dt
t
0< x <∞ .
is the unique antiderivative to
1
, taking the value at x=1 .
x
Properties:

ln 1=0


ln x> 0 if x> 1
ln x< 0 if 0< x < 1
1
d
ln|x|= for all x ≠ 0
dx
x
d (ax )
a 1
g' ( x )= 1
= =
ax x
ax
dx
ln ( ax)= ln ( x)+C∧ ln (a)= ln(1 )+ C → ln(ax )=ln (a )+ ln(x)
⁡

ln (an )=ln ( a ∙ a ∙ a ∙∙ ∙ a )=ln a+ln a+ …+ln a=n ln a

−n
n
−n
ln (an )+ ln ( a )=ln( a × a ) =ln1=0



( )(
)
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MATH1021 Notes
−n

a
(¿)=−n ln a
ln ¿
Logarithmic Integration
Exponential Function
ln x is an increasing function of
other,
x .
exp x and
ln x
are inverse to each
ln ( exp ( x ) )=x for all x ∈ R
exp ( ln ( x ) )=x for all x ∈(0 , ∞)
Properties:

ax=exp ( ln (ax ) )=exp ( ln ( a )+ln ( x ) ) for all a , x ∈(0 , ∞ )
exp ( r) exp ( s ) =exp(r
⁡ +s)

x=ln( exp ( x ) )
d
exp ( x ) =exp(⁡x)
dx
The General Exponential Function
General Exponential Function
Recall
ln ( a n )=n ln (a)
for
a> 0 .
Applying the exponential function gives
a
x ln¿
a =exp( ¿)
x
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MATH1021 Notes
The Number e
e is the unique value of a with the
property that ln (a)=1 , or a=exp(1) .
e
x ln ¿
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MODULE 11: INTEGRATION TECHNIQUES
11.1 Basic Rules of Integration
Integration (Principle of Linearity)
b
b
a
a
∫ kf ( x ) dx=k ∫ f ( x ) dx
b
b
b
a
a
a
∫ ( f ( x) ± g ( x )) dx=∫ f ( x ) dx ±∫ g ( x ) dx
11.2 Integration by Substitution
Integration by Substitution
Integration by Substitution
d
'
F [ u ( x )] =F [u ( x ) ] u' ( x)
dx
∫ F '[ u ( x ) ]u' ( x ) dx = F [u ( x ) ]+C
∫ f [u ( x )] u' ( x ) dx=∫ f ( u ) du
Integration by Substitution (Definite
Integrals)
b
u( b)
a
u( a)
∫ f [u ( x )] u' ( x ) dx= ∫ f (u ) du
Inverse Trigonometric Identities
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MATH1021 Notes
11.3 Integration by Parts
Integration by Parts
From,
( u v ' ) =u ' v+uv '
∫( u v ' ) dx=uv −∫ (u' v ) dx
Integration by Parts (Definite
Integral)
b
b
∫u dv= [uv ]a−∫ v du
b
a
a
11.4 Partial Fractions
Partial Fractions
 Method for integrating rational functions [in form P(x)/Q(x) where P(x) and
Q(x) are polynomials, and the degree of P(x)< degree of Q(x))
Partial Fractions
A
B
px + q
, a ≠ b=
+
(x−a)(x−b)
( x −a ) (x−b)
Where A ( x−b) + B (x−a )= px+ q
Form of the Rational Function
Form of the Partial Fraction
px + q
,a≠b
( x−a)(x−b)
A
B
+
(x−a) ( x−b)
px + q
(x−a ) 2
A
B
+
( x−a) ( x−b ) 2
p x 2+ qx + r
( x−a)( x−b )( x−c )
B
C
A
+
+
( x−a) (x−b) (x−c )
p x 2 +qx + r
(x−a ) 2 (x−b)
A
B + C
+
(x−a) ( x−a )2 ( x−b)
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MATH1021 Notes
2
p x + qx + r
2
( x−a)( x +bx +c )
A
Bx+C
+
(x−a) (x 2+ bx+ c)
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MATH1021 Notes
MODULE 12: APPLICATIONS OF
INTEGRATION
12.1 Further Integration Techniques
Binomial Theorem
Reduction Formulas
 Obtained when the integral involves an integer parameter n
Example:
Example:
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12.2 Length of a Curve
The length of a curve was calculated by subdividing it into small sections called
the arc length differential, and then adding up all the small-sections to construct
a Riemann sum. Taking the limit as the number of sections tend to infinity gives
a definite integral that gives the total length of the curve.
Consider a function f(x) that is continuous on an interval [a,b].
Arc length differential is,
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12.3 Area Between Two Curves
Shrinking the length of the intervals towards zero,
12.4 Solids of Revolution
The Disc Method
For a solid of revolution, we can get the formula for volume more directly by
finding the relation between a small change in x and the corresponding small
change in volume. Between points x and x + x the volume of the solid of
rotation is approximately that of a disk radius f(x) and thickness x.
Where V is the contribution to the total volume coming from the interval
between x + x.
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Volume: Disc Method
b
2
V =∫ πf ( x ) dx
a
The Shell Method
Rotating a thin rectangle about the y-axis generates a cylindrical shell of
approximate height f(x) and thickness x.
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Volume: Shell Method
b
V =2 π ∫ xf ( x) dx
a
Summary
Shell Method
Disc Method
Representative rectangle is parallel to the
axis of revolution
Representative rectangle is perpendicular
to the axis of revolution
b
V =2 π ∫ xf ( x) dx
a
b
2
V =∫ πf ( x ) dx
a
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MATH1021 Notes
Disc Method
Shell Method
X-axis
Rotation
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Y-axis
Rotation
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Table of Standard Integrals
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MATH1021 Notes
Questions
Volume
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Complex Numbers
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Riemann Integrals
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Taylor’s Formula
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Fundamental Theorem of Calculus
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