,.
1 Power Series and Limits
2 .
functions of one Variable
. Functions of Two
3 or More Variables
.
4 Directional Derivative and the Gradient Operator
5
.
Taylor Series and Stationary Points
6
. Integration Essentials
.
7 Integration of a function of One Variable
8
. Integration of a function of Two Variables
.
9 Integration of a function of Three Variables
. Power Series & Limits
1
A power series in about the point so has the form
: an(-cd" =
do + a, k x)
-
+ a2( -
x +... +
an(x - x +...
Maclauvin Series : about 0
= 0 f(x) f(0) f'(ok + f "(o(
= +
+... + f "(o)x" +...
2! n !
Taylor Series : about the point (o f(x) f(x0) + +'Pco)(x x0) f "((0)(x (c) +...
= -
+ -
+ f "(x0)(x -
xo)" +...
2! n !
If we stop evaluating the after (x-col we get an approximation for the function f(x)
series Taylor polynomial -
.
between the function and the Taylor polynomial is the error or remainder Rnx f (C) -Cont
**
The difference =
where the unknown pointa lies between x and xo .
(n+ 1) !
We can use the fact that c lies between and so to obtain bounds on the error .
The Ratio Test : a test for convergence the -
on converges if :m bu
series =1 The series diverseis
if > 1 . If L = 1 then another test must be done .
"
where bn ankzeo)" the anypoco" cecolim anti
iflim
= so series will converge = =
n0
an(x-x)" an
The power series converges for all values of such that
C2-Co lim an = R where R is the radius of convergence
n ->0
Anti
Alternating Series Test : let an be a
decreasing sequence with ant0 eg .
n
Then
*
-1 an =
a
,
-an , as is convergent
Eg
.
for what
+1
values of does converge ?
(x -
3)n
need to examine
Lim
limkc-3 k-3 when KC-31-131
n+ 1
= =
converges ,
247 the end limits
o
n> ok-3n
n +1
= 2 -
decreasing series with limit tending too-converges by alternating series test
= harmoniseries -
divesis
converges for 2
, Differentiating a Power Series differentiate a
: we can power series (f(x) =
do + ap + az" +... + and" +... ) to
get the power series approximation for f'bs) (f'() a + 2023 3933 +... nanah" +... ) We can apply
= +
, + .
the ratio test again to check the radius of convergence
-
the series will converge for KCKRf ·
Hence for
convergence we require lim (n lan <I +
Kim n mman
nanech-
the first limit
im nim
can be rearranged to get
Combining these limits in the ratio test
. 1 bcR Re =
Differentiating a
convergent power series keeps it convergent with the
original radius of convergence R .
Isame is true for integrating)
Changing the interval : eg. F =
- +
ux
...
UX
- U = > 13 when u= 1x = 1
,
u= 10 = 3
use coordinates and straight line equation (-1 ,
-1) (1 , 3) = u
= then replace u in the series
-
1) -
k) = 3 -
1x3
-
2
Products of Power Series : the product of two functions is the product of their individual power series . The
expansion of the product of the series gives the new power series .
. functions of 1 variable
2
Definition 1: A function is a that assigns to each element in set A one element in set B
rule . The set
A is called the domain (possible inputs) and the set of all possible values of f(x) is the range (possible
outputs) .
Definition 2 : the graph of the function f is defined to be the Cartesian graph of y
= f(x)
Notation : x + (a b] ,
= a(x> b
x + (a . b) = acx(b
Limit of a function & Continuity : a limit of a function is the value that the function f(x) approaches as
approaches a
given value.
Suppose that f(x) is defined over the open interval (a b) , ,
and f(x) = 1 . So is the value approaches and
L is the value of the function at > =o .
The function f(x) is continuous at c=
if ( + = -
= L
The limit from the left is 2- = lim
x x
f(x) and we can
say f(x) = L or limf(x + h)
n >0
= L
L
-
L+ (then f(xo)
-
= = =
(
The limit from the right is L += lim f(x)
x
# L-* L , then the
x-
function is discontinuous .
limit can have indeterminate form eg f(x)
A .
=
E is not defined at = 0
,
but the limit as -o does exist
This limit can be found using ('Hopitals rule or a
power series for sinc. Thus we can make the
function continuous by modifying the definition : f(x)
=
1 Power Series and Limits
2 .
functions of one Variable
. Functions of Two
3 or More Variables
.
4 Directional Derivative and the Gradient Operator
5
.
Taylor Series and Stationary Points
6
. Integration Essentials
.
7 Integration of a function of One Variable
8
. Integration of a function of Two Variables
.
9 Integration of a function of Three Variables
. Power Series & Limits
1
A power series in about the point so has the form
: an(-cd" =
do + a, k x)
-
+ a2( -
x +... +
an(x - x +...
Maclauvin Series : about 0
= 0 f(x) f(0) f'(ok + f "(o(
= +
+... + f "(o)x" +...
2! n !
Taylor Series : about the point (o f(x) f(x0) + +'Pco)(x x0) f "((0)(x (c) +...
= -
+ -
+ f "(x0)(x -
xo)" +...
2! n !
If we stop evaluating the after (x-col we get an approximation for the function f(x)
series Taylor polynomial -
.
between the function and the Taylor polynomial is the error or remainder Rnx f (C) -Cont
**
The difference =
where the unknown pointa lies between x and xo .
(n+ 1) !
We can use the fact that c lies between and so to obtain bounds on the error .
The Ratio Test : a test for convergence the -
on converges if :m bu
series =1 The series diverseis
if > 1 . If L = 1 then another test must be done .
"
where bn ankzeo)" the anypoco" cecolim anti
iflim
= so series will converge = =
n0
an(x-x)" an
The power series converges for all values of such that
C2-Co lim an = R where R is the radius of convergence
n ->0
Anti
Alternating Series Test : let an be a
decreasing sequence with ant0 eg .
n
Then
*
-1 an =
a
,
-an , as is convergent
Eg
.
for what
+1
values of does converge ?
(x -
3)n
need to examine
Lim
limkc-3 k-3 when KC-31-131
n+ 1
= =
converges ,
247 the end limits
o
n> ok-3n
n +1
= 2 -
decreasing series with limit tending too-converges by alternating series test
= harmoniseries -
divesis
converges for 2
, Differentiating a Power Series differentiate a
: we can power series (f(x) =
do + ap + az" +... + and" +... ) to
get the power series approximation for f'bs) (f'() a + 2023 3933 +... nanah" +... ) We can apply
= +
, + .
the ratio test again to check the radius of convergence
-
the series will converge for KCKRf ·
Hence for
convergence we require lim (n lan <I +
Kim n mman
nanech-
the first limit
im nim
can be rearranged to get
Combining these limits in the ratio test
. 1 bcR Re =
Differentiating a
convergent power series keeps it convergent with the
original radius of convergence R .
Isame is true for integrating)
Changing the interval : eg. F =
- +
ux
...
UX
- U = > 13 when u= 1x = 1
,
u= 10 = 3
use coordinates and straight line equation (-1 ,
-1) (1 , 3) = u
= then replace u in the series
-
1) -
k) = 3 -
1x3
-
2
Products of Power Series : the product of two functions is the product of their individual power series . The
expansion of the product of the series gives the new power series .
. functions of 1 variable
2
Definition 1: A function is a that assigns to each element in set A one element in set B
rule . The set
A is called the domain (possible inputs) and the set of all possible values of f(x) is the range (possible
outputs) .
Definition 2 : the graph of the function f is defined to be the Cartesian graph of y
= f(x)
Notation : x + (a b] ,
= a(x> b
x + (a . b) = acx(b
Limit of a function & Continuity : a limit of a function is the value that the function f(x) approaches as
approaches a
given value.
Suppose that f(x) is defined over the open interval (a b) , ,
and f(x) = 1 . So is the value approaches and
L is the value of the function at > =o .
The function f(x) is continuous at c=
if ( + = -
= L
The limit from the left is 2- = lim
x x
f(x) and we can
say f(x) = L or limf(x + h)
n >0
= L
L
-
L+ (then f(xo)
-
= = =
(
The limit from the right is L += lim f(x)
x
# L-* L , then the
x-
function is discontinuous .
limit can have indeterminate form eg f(x)
A .
=
E is not defined at = 0
,
but the limit as -o does exist
This limit can be found using ('Hopitals rule or a
power series for sinc. Thus we can make the
function continuous by modifying the definition : f(x)
=
