Advanced Calculus
Lecture for first year
by Nguyen Xuan Hung
Index
FAQ
Sets
Concepts
Union of sets
Intersections
Complementary sets
Index
FAQ
Definition of Sets
Definition
Sets are to consist of objects with common properties
Notation
A, B, C, D,E, etc.
Each element belongs to A : x  A
Example
A={0,1, 2, 3, 4, 5, 6, 7}
B={x| -x3-x +2 =0}
Index
FAQ
Definition of Sets
- Set of Natural number
N={1, 2, 3, 4, …}
- Set of Integer number
Z={0, ±1, ± 2, ± 3, ± 4, …}
- Set of Rational number
Q={1/2, 1, 3/2, 4, 5/7,…}
- Set of Real number
Index
Q
∩
Z
∩
N
∩
R={1/2, 1, 1.3333,1.41, …}
R
FAQ
Operators on sets
∩
Union of sets A
B = {x  A or x  B }
Intersection of sets
Index
A ∩ B = {x  A and x  B }
FAQ
Operators on sets
Complement of sets U\A = {x U and x  A }
U
A
Index
FAQ
Functions
Functions and their Graphs
Injectivity and Surjectivity
Index
FAQ
Definition of Functions
Definition
∩
∩
Given sets X
R and Y R. A function f : X  Y
is a rule which assigns an element f(x) of the set Y
for every x in X.
f : X  Y or y = f(x)
Let f : X  Y be a function. The set X is the
domain of definition D(f) of the function f. The
set Y is the target domain R(f) of the function f.
The set f(X) = { f(a) | aX }  Y is the range of
the function f.
Index
FAQ
Graphs of Functions
In calculus we are usually concerned with functions
f:

defined in terms of explicit expressions for f  x  .
The product
2
  x, y  | x, y 
 is called the plane.
It is usually pictured
by drawing the x -axis horizontally and the y -axis vertically. The graph of
a function f :
Index

is the graph of the set
 x,f  x  | x  .
FAQ
Graphs of Functions
Examples
 
Below are the graphs of the functions f  x   sin x 2 ,
g  x   x 4  2x 3  x 2  2x, and h  x   2sin( x ). Which is which?
h x   2
sin( x )
Index
 
f  x   sin x 2
g  x   x 4  2 x 3  x 2  2x
FAQ
Curves and Graphs
Problem
Answer
Index
Which of the following curves in the plane are graphs
of functions?
The first two curves are not graphs of functions since they do
not correspond to a rule which associates a unique y-value to
any given x-value. Graphically this means that there are
vertical lines which intersect the first two curves at more than
1 point.
FAQ
Injective Functions
Definition
A function f :X
A YB is injective or one-to-one if
f  x   f  y   x  y.
A one-to-one function associates at most one point in the set X to any
given point in the set Y.
Problem
Answer
Index
Which of the following graphs are graphs of one-to-one
functions?
None of the above graphs are graphs of one-to-one functions
since they correspond to rules which associate several xvalues to some y-values. This follows since there are
horizontal lines intersecting the graphs at more than 1 point.
FAQ
Surjective Functions
Definition
Definition
A function f : A  B is surjective or onto if
f  A   B, i.e., if y  B : x  A such that f  x   y.
A function f : A  B is bijective if it is both one-to-one
and surjective. For a bijective function f,
y  B : ! x  A such that f  x   y .
The notation "! x  A " means that "there is a unique
element x in the set A " having the specified property.
Observe that the property of being surjective or onto depends on how the
set B in the above is defined. Possibly reducing the set B any mapping
f: A  B can always be made surjective.
Index
FAQ
Composed Functions (1)
Definition
Let f,g: 
be two functions.
The composed function f g is defined by setting
f g  x   f  g  x  .
Observe that the composed function f o g can be defined by the above
formula whenever the range of the function g is contained in the
domain of definition of the function f.
2
w y
y  x 1
Example
x 2  1  f g  x  with y  g  x   x 2  1
and w  f  y   y . The composition is
x-axis
y-axis
w-axis
w  y = x 2  1.
There are infinitely many other ways to represent the above function as a
composed function. This is never unique. The composition used depends on
the computation to be performed.
Index
FAQ
Composed Functions (2)
Observations
Assume that f and g are functions for which the
composed function h = f o g is defined.
1. If both f and g are increasing, then also h is
increasing.
2. If f is increasing and g decreasing, then h is
decreasing.
3. If f is decreasing and g increasing, then h is
decreasing.
4. If both f and g are decreasing, then h is
increasing.
Index
FAQ
Inverse Functions
If a function f: A  B is injective, then one can solve x in terms of y
from the equation y = f(x) provided that y is in the range of f. This
defines the inverse function of the function f.
Let f : A  B be a function. If there is a function g : B  A
Definition
such that f g is the identity on B and g f is the identity on A,
then the function f is called invertible, and the function g is the
inverse function of the function f.
The above condition means in shorthand:
b  B : f  g  b    b and a  A : g  f  a    a.
Notation
Warning
Index
The inverse function g of a function f
is denoted by f 1.
Do not confuse f 1  x  with f  x  
1
1
.
f x
Here the operation
“-1” is applied to the
function f rather than
the values of the
function.
FAQ
Finding Inverse Functions
To find the inverse function of a given function f: A  B one can simply
solve x in terms of y from the equation y = f(x). If solving is possible
and the solution is unique, then the function f has an inverse function,
and the solution defines the inverse function.
Example
x 1
x 2
x 1
solve x in terms of y from the equation y 
.
x 2
To find the inverse function of the function f  x  
x 1
 y  x  2   x  1   y  1 x  1  2y
x 2
1  2y
x
provided that x  2 and y  1.
y 1
y
f
We conclude that the inverse function of the
1  2x
function f is f 1  x  
. The graphs of
x 1
f and of f 1 are symmetric with respect to
f-1
the line y  x.
Index
y=x
FAQ
The Logarithm
Let a > 0. We know that the exponential function ax is increasing if a >
1 and decreasing if a < 1. In both cases the function ax is injective.
Hence the exponential function has an inverse function.
Definition
The inverse function of the exponential function
a x is the logarithmic function with base a.
Notation
loga  x   the value of the logarithmic function with
base a at the point x.
Definition
Notation
Index
The logarithmic function with base e is called
the natural logarithm or simply the logarithm.
loge  x   log  x   ln  x .
FAQ
Properties of the Logarithm
The exponential function has the property a
xy
a a
x
y
a 
x
and
y
 a xy .
These formulae imply the following formulae for the logarithm.
1. loga  xy   loga  x   loga  y 
 
2. y loga  x   loga x y
3. loga w  
Proof
log w 
log  a 
. Here log w   the natural logarithm of w.
The formulae 1 and 2 follow directly from the properties of the
exponential function.
log w 
   log w   x log a   x  log a 
w  a  log w   log a
x
x
On the other hand, w  a  loga w   x. Hence loga w  
x
Index
log w 
log  a 
FAQ
.
The Inverse Function of the Sine
Function
The sine function is not
injective since there are
horizontal lines intersecting the
curve at infinitely many points.
y=sin(x)
Hence one cannot solve x in terms of y uniquely from the equation y=sin(x).
In fact, there are no solutions if y > 1 or y < -1. If -1  y  1, there are
infinitely many solutions. The solution becomes unique, if we require it to be
between -/2 and /2. This is equivalent to restricting the domain of
definition of the sine function to the interval [-/2, /2].
y=arcsin(x)
Definition The arcsin function is the inverse function of
y=sin(x)
  
the function sin :   ,    1,1.
 2 2
Index
FAQ
The Inverse Function of the Cosine
and the Tangent Functions
Definition
The arccos function is the inverse function of
the function cos : 0,    1,1.
The arctan function is the inverse function of
arccos(x)
  
the function tan :   ,   .
 2 2
tan(x)
arctan(x)
cos(x)
Index
FAQ
New Functions from Old
Piecewise Defined Functions
Deformations of Functions
Composed Functions
Inverse Functions
Inverses of Exponential Functions
Inverses of Trigonometric Functions
Index
FAQ
Piecewise Defined Functions (1)
Sometimes it is necessary to define a function by giving
several expressions, for the function, which are valid on
certain specified intervals. Such a function is a piecewise
defined function.
Definition
The absolute value |x| is an example of a piecewise defined function.
We have |x| = x if x0 and |x| = -x otherwise. Computations with the
absolute value have to be done using its definition as a piecewise
defined function.
Problem
Express f  x   1  x  2 as a piecewise defined function.
Solution
We have to strip the absolute values from the
expression by starting with the innermost absolute
values.
Index
FAQ
Piecewise Defined Functions (2)
Problem
Express f  x   1  x  2 as a piecewise defined function.
Solution
Observe first that

 1   x  2
f x  

 1  2  x 
if x  2

3 x
i.e. f  x   
if x  2

 x 1
if x  2
if x  2
Next we observe that
 x  3 if x  3
3x  
3  x if x  3
and
 x  1 if x  1
x 1  
.
1  x if x  1
Combine the above to get
if x  3
x  3
3  x if 2  x  3

f x  
 x  1 if 1  x  2
1  x if x  1
Index
f(x)
FAQ
Simple Deformations (1)
Let f be a given function, and let a be a real number.
The following picture illustrates how the graph of
the function f gets deformed as we replace the
values f(x) by a f(x).
1.5 f(x)
By multiplying the function by a positive constant
a the graph gets stretched in the vertical
direction if a>1 and squeezed if a<1.
f(x)
By multiplying the function by a negative
constant a the graph gets first reflected about
the x-axis and then stretched in the vertical
direction if a<-1 and squeezed if 0>a>-1.
0.5f(x)
Index
FAQ
Simple Deformations (2)
The effect, on the graph, of
multiplying a function with a
constant is either stretching,
squeezing or, if the constant is
negative, then first reflecting
and
then
stretching
or
squeezing.
1.5 f(x)
f(x)+1.7
f(x)
f(x)-1.7
Adding a constant to a
function means a vertical
translation in the graph. The
picture on the right illustrates
this situation.
Index
0.5f(x)
FAQ
Simple Deformations (3)
Let f be a given function, and let b be a real number.
The following problem illustrates how the graph of the function
f gets deformed as we replace the values f(x) by f(x+b).
Problem
The picture on the right shows
the graphs of functions f(x-1),
f(x) and f(x+1). Which is which?
x-1 takes a value x0
Solution
when x= x0 +1.
Similarly x+1 takes a value x0 when
x= x0 -1.
We conclude that the black graph
must be the graph of the function f(x),
and that the other graphs are as
labeled in the picture.
Index
f(x-1)
f(x+1)
f(x)
FAQ