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Mathematical Scratchpad

Mathematical Scratchpad

Semester Review

The Big Picture:

Chapter 5: Presents the basics of the theory of integration

Chapter 6: Standard applications of definite integrals

Chapter 7: How to find antiderivatives

Chapter 8: Sequences and Series


The Medium Picture:

Chapter 5

The basic theory of integration  

Chapter 6

Standard Applications of Definite Integrals  

Chapter 7

Methods of Integration  

Chapter 8

Infinite sequences and series  

More Detailed Outline

Chapter 5: The fundamentals of integration

Discrete Domain Functions

Interval Domain Functions



Chapter 6: Applications of definite integrals


Chapter 7: Methods of Integration

Chapter 8: Sequences and Series

Tests for Convergence of åk¥ak  

Absolute and Conditional Convergence  

Power Series

Taylor Series and Maclaurin Series


Analogies between Sequences/Series and Functions/Integrals



Dk[ kp] = pkp-1     
 d

dx
[ xn] = nxn-1
Dk[ k-p] = -p( k+1) -p-1
 d

dx
[ x-n] = -nx-n-1
Dk[ rk] = ( r-1) rk
 d

dx
[cx] = ln( c) cx
Dk[ A( k) ] = a( k) ® å
a( k) = A( k) +C
 d

dx
[ F(x) ] = f( x) ® ó
õ
f( x)dx=F( x) +C
å
kp=  1

p+1
kp+1+C
ó
õ
xndx=  1

n+1
xn+1+C
å
k-p=  1

-p+1
( k-1) -p+1+C, if p ¹ 1
ó
õ
x-ndx=  1

-n+1
x-n+1+C,  if n ¹ 1
å
 1

k1
=H( k) +C   Harmonic Series
ó
õ
 1

x
dx=ln| x| +C
å
rk=  1

r-1
rk+C,  r ¹ 1
ó
õ
rxdx=  1

ln( r)
rx+C,  r ¹ 1
å
1k=k+C
ó
õ
1dx=x+C
n
å
k=m 
a( k) = A( k) ê
ê
n+1
m 
=A( n+1) -A( m)
1 FT
ó
õ
b

a 
f( x) dx=F( x) ê
ê
b
a 
=F( b) -F( a)
Dk é
ë
k-1
å
j=m 
a( j) ù
û
=a( k)
2 FT
 d

dx
ó
õ
x

a 
f( t)  dt=f( x)
Dk[ ukvk] = ukDk[ vk] +vk+1Dk[ uk]
 d

dx
[ uv] = u  dv

dx
+v  du

dx
n
å
k=0 
Uk vk=UkVk ê
ê
n+1
0 
- n
å
k=0 
Vk+1uk
ó
õ
b

a 
u dv=uv ê
ê
b
a 
- ó
õ
b

a 
v du
¥
å
k=m 
a( k) =
lim
n® ¥ 
n
å
k=m 
a( k)
ó
õ
¥

a 
f( x)dx=
lim
b® ¥ 
ó
õ
b

a 
f( x) dx
0 £ a( k) £ b( k) and ¥
å
k=m 
b( k) conv.
             Þ ¥
å
k=m 
a(k) conv.
0 £ f( x) £ g( x) and ó
õ
¥

a 
g( x) dx conv.
             Þ ó
õ
¥

a 
f(x) dx conv.
0 £ a( k) £ b( k) and ¥
å
k=m 
a( k) div.
             Þ ¥
å
k=m 
b(k) div.
0 £ f( x) £ g( x) and ó
õ
¥

a 
f( x) dx div.
             Þ ó
õ
¥

a 
g(x) dx div.
      

lim
n® ¥ 
an ¹ 0 Þ ¥
å
k=1 
ak diverges

lim
x® ¥ 
f(x) = c ¹ 0 Þ ó
õ
¥

1 
f( x) dx div.
Fns as series ( f( x) = ¥
å
k=1 
akxk )
Fns as integrals (G( x) = ó
õ
¥

0 
tx-1e-t dt )
¥
å
k=0 
 1

k!
f( k) ( c) ( x-c) k  (Taylor Series )




File translated from TEX by TTH, version 3.04.
On 7 Dec 2004, 14:04.