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

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 nn

f h

h

Area under a curve = definite integral

2

10 1lim ( )

N xn xh n

f h f x dx

2 211

2 1

If ' , then

( ) ( ) ( ) |x x

xx

g f

f x dx g x g x g

( )f x

x1x x2x 1x 2x

1Area

Nn

nf h

1

11

12

1Area ( )21 1 2 2

Nn n

nN

n Nn

f f h

f f hh f h

h

Integrating data: the trapezoidal rule

Very similar!2 2

112 1

If ' , then

( ) ( ) ( ) |x x

xx

g f

f x dx g x g x g

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.

2 4 0x

00( ) ( )

'

( )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( ) ( )

'

( )x

xg x g x f

g

d

f

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

f x g x

af x bg x

''' 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?

f x g x

af x bg x

''' 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

2 31 1 12! 3! !( ) 1 n

nE x x x x x

.'' ', ''' '', etcE E E E

Analytical solution

(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 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

Examples

Add examples 6, 7 from notes.

Homework:

Read examples 6 and 7 in text. (Should do in lecture)

Do exercises for section 2.6, 2.7 and 2.8. This will include:

• Exercise with antiderivatives and classifying ODEs.

• Derivation of ex via compound interest.

• Carbon dating (for Tuesday field trip)

• Derive further well-known functions from f’’=-f