rules of differentiation
DESCRIPTION
Rules of differentiation. REVIEW:. The Chain Rule. Taylor series. Approximating the derivative. Monday Sept 14th: Univariate Calculus 2. Integrals ODEs Exponential functions. Antiderivative (indefinite integral). Antiderivative (indefinite integral). - PowerPoint PPT PresentationTRANSCRIPT
Rules of differentiation 1
Sum rule: ( )' ' '
Product rule: ( )' ' '
Multiplicati
Pow
on by a constant: ( )' '
Lineari
er ru
ty
l
:
:
e d x xdx
f g f g
fg f g fg
af af
( )' ' ' af bg af bg
REVIEW:
The Chain Rule( ); ( )
'( ) '( )
y f x x g t
dy dy dx f x g tdt dx dt
Taylor series
0 0 0 0
0
(3) ( )2 30
0
( )
0
1 1 1( ) ( ) '( ) ''( ) ( ) ( )2! 3! !
where
OR:1( ) ( )!
n n
n nn
f x f x f x h f x h f x h f x hn
h x x
f x f x hn
Factorial function: ! ( 1)( 2) 1 0!=1 1!=1 2!=2 1=2
n n n n
3!=3 2 1=6 4!=4 3 2 1=24
Approximating the derivative
0 00
( ) ( )Centered difference: '( ) 2f x h f x hf x h
0 00
( ) ( )Forward difference: '( ) f x h f xf x h
0 0 00 2
( ) 2 ( ( )Centered 2nd derivative: ''( ) )f x h f x f x hf xh
Monday Sept 14th: Univariate Calculus 2
•Integrals•ODEs•Exponential functions
Antiderivative (indefinite integral)Suppose ' .g f
Antiderivative (indefinite integral)
' an antiderivative of
general antiderivative of
because ( )' ' ' ' 0
g f g f
g C f
g C g C g f
x1x 2x
Area nnf h
h
Area under a curve = definite integral
2
10 1lim ( )
N xn xh nf h f x dx
2
12
1
21
2 1
If ' , then
( )
'( )
( ) ( )|
xxxx
xx
g f
f x dx
g x dx
g x g xg
( )f x
nf h
x1x x2x 1x 2x
1Area
Nn
nf h
1
1
1 2 2
11
2
2 3 3
1Area ( )2
1 1 1 ( ( (2 2 2
1 1
) )
2 2
)
Nn n
n
Nn N
n
f h f h f h
h
f f h
f f f
f f h f h
h
Integrating data: the trapezoidal rule
Very similar!
nf h
112( )n nf f h
Example: integrating a linear function
x1x x
y mx b
212antiderivative: ( )
( )ydx mx b dx mx bx c
g x
1
1
21 1
2
1
21 1
21 12 2
12
( ) ( )
( )
) (
(
( )
) |
xx
xx
mx bx c
Area x f
mx bx c
m x x b x x
x dx
g x g x g
Another angle: the upper limit as an argument
11( ) ( ) ( )
( )
xx
g x g x f d
dg f xdx
1x x
Another angle: the upper limit as an argument
11( ) ( ) ( )
( )
xx
g x g x f d
dg f xdx
1x x
Another angle: the upper limit as an argument
11( ) ( ) ( )
( )
xx
g x g x f d
dg f xdx
1x x
Differential equationsAlgebraic equation: involves functions; solutions are numbers.
Differential equation: involves derivatives; solutions are functions.
2e.g.: 4 0x
11( ) ( )
'
( )x
xg x g x f
g
d
f
INITIAL CONDITION
x
( )y g x
( ) slopef x
1x
1 1( )g g x
e.g. dead reckoning
1
1
1 1
( ) ( ) (
' ; ( )
)x
x
g f g x g
g x g x f d
ExampleODE: ' cos( )
sin( )
Initial condition: (0) 0
0 sin(0) 0 sin( )
g x
g x C
g
CC
g x
Classification of ODEs
2''' 3 0 linear''' 3 0 nonlinear' '' 0 nonlinear' 2 1 / mondo nonlinear!f
f ff ff f ff f
2''' 3 0 homogeneous''' 3 0 homogeneous' '' 1 nonhomogeneous
f ff ff f f
2' 0 1st order
''' 3 0 3rd order' '' 0 2nd order' 2 1 / 1st orderf
f gf ff f ff f
Linearity:
Homogeneity:
Order:
Superposition(linear, homogeneous equations)
( ), ( ) solutions
( ) ( ) solution for all , .
f x g x
af x bg x a b
''' 3 0 (1) ''' 3 0 (2)
?
(1) (2):
(1): ''' 3 0 (2): ''' 3 0
sum: ( )''' 3( ) 0
is
linear, homogeneou
a solut on
s
i
f fg g
af bg
a b
a af afb bg bg
af bg af bg
af bg
Can build a complex solution from the sum of two or more simpler solutions.
Superposition(linear, inhomogeneous equations)
( ), ( ) solutions
( ) ( ) solution for all , ?
f x g x
af x bg x a b
''' 3 1 (1) ''' 3 1 (2)
?
(1) (2):
(1): ''' 3 (2): ''' 3
sum: ( )''' 3( ) ( )
is NOT a solu
linear, inhomogeneous
tion
f fg g
af bg
a b
a af af ab bg bg b
af bg af bg a b
af bg
Superposition(linear, inhomogeneous equations)
( ), ( ) solutions
( ) ( ) solution for all , ?
f x g x
af x bg x a b
''' 3 1 (1) ''' 3 1 (2)
?
(1) (2):
(1): ''' 3 (2): ''' 3
sum: ( )''' 3( ) ( )
is NOT a s
linear, homogeneo
olution
usf fg g
af bg
a b
a af af ab bg bg b
af bg af bg a b
af bg
Superposition(nonlinear equations)
( ), ( ) solutions
( ) ( ) solution?
f x g x
af x bg x
2
2
2
2
2
2
2
''' 3 0 (1) non''' 3 0 (2)
?
solution would require: ( )''' 3( ) 0(1) (2):
(1): ''' 3 0 (
linear
2):
, homog
''' 3 0 sum: ( )''' 3( ) 0
is
eneou
NOT a so
sf fg g
af bg
af bg af bga b
a af afb bg bg
af bg af bg
af bg
lution
ORDINARY differential equation (ODE): solutions are univariate functions
PARTIAL differential equation (PDE): solutions are multivariate functions
x
y
1 slope=1
ODE: '( ) ( );
IC: (0) 1
E x E x
E
Exponential functions: start with ODE
Qualitative solution:
Exponential functions: start with ODEODE: '( ) ( );
IC: (0) 1
E x E x
E
(3) ( )2 31 1 1( ) (0) '(0) ''(0) (0) (0)2! 3! !n nE x E E x E x E x E xn
' , (0) 1 '(0) 1, E E E E
Analytical solution
(3) ( )2 31 1 1( ) ''(0) (1 1 0) (0)2! 3! !n nE x x E x E x E xn
Exponential functions: start with ODEODE: '( ) ( );
IC: (0) 1
E x E x
E
2 31 1 12! 3! !( ) 1 n
nE x x x x x
.
Differentiate again:
'' ', '(0) 1 ''(0) 1,
''' '' 1, ... etc
E E E E
E E
Analytical solution
(3) ( )2 31 1 1( ) ''(0) (1 1 0) (0)2! 3! !n nE x x E x E x E xn
(3) ( )2 3
2 3
1 1 12! 3! !
1 1 12! 3! !
( ) ( ) '( ) ''( ) ( ) ( ) ( ) 1
( ) ( )
( ) ( ) ( )
{ }
n n
n
n
n
E x y E x E x y E x y E x y E x y
E x y y y y
E x E y
E x y E x E y
Rules for addition, multiplication, exponentiation
(3) ( )2 3
2 3
1 1 12! 3! !
1 1 12! 3! !
( ) ( ) '( ) ''( ) ( ) ( ) ( ) 1
( ) ( )
( ) ( ) ( )
Generalization: set : (2 ) ( ) (
{ }
n n
n
n
n
E x y E x E x y E x y E x y E x y
E x y y y y
E x E y
E x y E x E y
y x E x E x E
2
2 3
3
) ( ) 2 : ( 2 ) ( ) (2 ) ( ) ( ) ( )
or: (3 ) ( )
In general ( ) ( ) , or
x E xy x E x x E x E x E x E x E x
E x E x
E x E x E
( ) ( )x E x
Rules for addition, multiplication, exponentiation
Differentiation, integration( ) ( )
( )( ) ( )( )
d E x E xdx
d xd dE x E xdx d x dx
( )d xdx ( )E x
( ) ( )
1( ) ( )
d E x E xdx
E x dx E x C
(chain rule)
Properties of the exponential function
1
2 31 1 1 12! 3! 2! 3!1 , 1 1 2.71828
,
( ) , with special case 1/ ,
,
.
x
x y yx
x x x x
x x
x x
e x x x e
e e e
e e e e
d e edx
e dx e c
Sum rule:
Power rule:
Taylor series:
Derivative
Indefinite integral
All implicit in this: '( ) ( ); (0) 1E x E x E
0
00 0
( )
( ) ( )
:
.
'
const
Newton s law of cooling
dT k T Tdt
d T TT k T Tdt
0
1 0
1 0
1
0
,
) .
(
(
0)
x x kt
kt
d e e T T Cedx
TT T CT
T T T T e
Cooling
0T
( )T t
0( )dwm m m g kwdt kw
mg
0m g
( ) 0w t m0(1 / ) reduced gravity
dw k w gmdtg g m m
( )k k kt t tm m md dw kwe e w emdt dt
( )k k k kt t t tm m m mdw k de w e we g emdt dt
k kt tm mmwe g e Ck
(0) 0 (1 ) k tmmw w g ek
k tme
Sinking
(0) 0 (1 ) k tmmw w g ek
mw g k
w
kw
mg
0m g
( ) 0w t m
Homework:
Do exercises for section 2.6, 2.8 and 2.9. Omit 2.9, #1.This will include:
• Exercise with antiderivatives and classifying ODEs.
• Carbon dating (for Thursday field trip)
• Derive further well-known functions from f’’=-f