Scientists: Which scientific advance has had the most impact on people’s everyday lives? #3: Darwin’s theory of evolution

Scientists: Which scientific advance has had the most impact on people’s everyday lives?

#3: Darwin’s theory of evolution



#2: Einstein’s relativity (cold war)



#2: Einstein’s relativity (cold war)

#1: Newton’s Calculus

Calculus: the Science of ChangeNo single thing abides,but all things flow - Heraclitus

Example: Climate change

• Global mean temperature– Rate of change consistent with natural causes?– OR is human activity implicated?

• What else changes due to global warming?– Sea ice extent– …?

??

Think of a quantity you might measure

Think of a quantity you might measure

• How fast is it changing:– over a decade?– a year?– a month?– a day?– a second?– right now?

Monday Sept. 13Univariate Calculus 1

• Derivative: the RATE OF CHANGE• Taylor series approximations• Differentiating data

Calculus: the Science of Change

Functions as models of changing quantities

0

0

0

212

population growth:

radioactive decay:

gravity:

rt

Rt

P P e

N N e

h h gt

x

y y mx b 2 1

2 1Slope:

y y ym x x x

Univariate function and slope

change in slope = change in

yx

x

y

( )y f x

y mx b 2 1

2 1Slope:

y y ym x x x

Univariate function and slope

Slope ?

x

y( )y f x

“Slope” of a function: the tangent line

( )y f x

x x h

( )f x h

( )f x

0( ) ( )'( ) lim

hf x h f xf x

h

The derivative as a limit

( )y f x

x x h

( )f x h

( )f x

0( ) ( )'( ) lim

hf x h f xf x

h


( )y f x

x x h

( )f x h

( )f x

0( ) ( )'( ) lim

hf x h f xf x

h


( )f x

x

'( )f x

( )y f x

0( ) ( )'( ) lim

hf x h f xf x

h


0( ) ( )'( ) lim

= "little change in"

hf x h f x dff x

h dx

d

Alternative symbols

Examples

2

2

3

1

2 2

0

2 2

2 2

( ) ,

( ) ( ) ( )

( )

'

; '( ) ?

?

lim

; '(

.

( )

)( )

h

f x x

f x h f x x hx h x x hh h

x xf h x

f f xx

f x

x

f x x

Rules of differentiation

31

RULE

Power rule:

EXAMPLE

Sum rule: ( )' ' '

?

d dx xdx

x

f g f g

dx

3 4

3

2

2 7

( )'

Product rule: ( )' ' '

Multiplication by a constant: ( )' ' (2 )'

Linearity: ( )' ' '

(

?

?

(2 3

?

2 )

?)

1

'

x x

fg f g fg x

af af x

af bg af bg x x

d xdx

The Chain Rule:Latitude

1o = 110km

30o

60o

Unit conversion

100km hr

1degree degrees100km 0.91hr 110km hr

dydt

ddt

Unit conversion

100km hr

1degree degrees100km 0.91110km hr hr

dydt

ddt

d dydy dt

The CHAIN RULE

The Chain Rule( ); ( )

'( ) '( )

y f x x g t

dy dy dx f x g tdt dx dt

2 22 2

1

2 7

; ( )(1 ) i.e. ( ) 1

(1 )

(4 5

? ?

)

?

?

y f x x dyd t x g t tdt dt

d tdt

d t tdt

EXAMPLES:

Higher-order derivatives

2

2

'( )

''( )

dff xdx

d f dfdf xdx dx dx

(3)

2

1

'( )

''(

?

?

?

?

)

( )

'( ) 2

''( ) 2

( ) (1

)

)

(

f x

f x

f x

f x x

f x

f x x

f x x

EXAMPLES

maximum ' 0; '' 0 f f

minimum

' 0; '' 0 f f

inflectionpoint

'' 0f

Extrema

Taylor series

0

0 0 0 0 0(3) ( )2 31 1 1( ) ( ) '( ) ''( ) ( ) ( )

2! 3! !n nf x f x f x h f x h f x h f x h

n

h x x

Factorial function: ! ( 1)( 2) 1 1!=1 2!=2 1=2 3!=3 2 1=6

n n n n

4!=4 3 2 1=24

(3) ( )0 0 0 0

2 30

1 1 12

( ) '( ) ''( ) (! 3! !

( () ) ) nnh h h hf x f x f x f x xn

f fx

constant derivativeat x0

powerof h=x-x0

Taylor series example(3) ( )2 3

0 0 0 0 01 1 1( ) ( ) '( ) ''( ) ( ) ( )2! 3! !

n nf x f x f x h f x h f x h f x hn

01

2 3 4

( )

( )

2 3

Expand: ( ) (1 )

'( ) (1 ) ; ''( ) 2·(1 ) ; '''( ) 3·2·(1 )

(0) 1; '(0) 1; ''(0) 2; '''(0) 3·2

(0) !

(0) 1!

( )

about 0

1

n

n

f x x

f x x f x x f x x

f f f f

f n

fn

f x x

x

x x

Taylor series example

2 31Example: ( ) 11

f x x x xx

2 31 x x x

21 x x

1 x

1

Taylor series example(3) ( )2 3

0 0 0 0 01 1 1( ) ( ) '( ) ''( ) ( ) ( )2! 3! !

n nf x f x f x h f x h f x h f x hn

02

21 122!

Expand: ( ) (1 )

'( ) 2(1 ); '(0) 2

''( ) 2; ''(0) 2

''

about 0

(0) 1

'( ) 0; '''(0) 0 etc.

( ) 1

2

xf x x

f x x f

f x f

f x f

f x x x

f

3

2

03!

1 2 0

x

x x

Negating the argument

(3) (4) ( )2 3

(3) (4) ( )2 3

4

4

2

1 1 1 1( ) (0) '(0) ''(0) (0) (0) (0)2! 3! 4! !

1 1 1 1( ) (0) '(0)( ) ''(0)( ) (0)( ) (0)( ) (0)( 1)2! 3! 4! !

1 1 (0) '(0) ''(0)2! 3!

n n

n n n

f x f f x f x f x f x f xn


f f x f x f

(3) (4) ( )3 4 ( 1)1(0) (0) (0)4! !

nn nx f x f x

n

02Expand: ( ) (1 about 0)g x x x

EXAMPLE:

Negating the argument

(3) (4) ( )2 3

(3) (4) ( )2 3

4

4

2

1 1 1 1( ) (0) '(0) ''(0) (0) (0) (0)2! 3! 4! !

1 1 1 1( ) (0) '(0)( ) ''(0)( ) (0)( ) (0)( ) (0)( 1)2! 3! 4! !

1 1 (0) '(0) ''(0)2! 3!

n n

n n n



f f x f x f

(3) (4) ( )3 4 ( 1)1(0) (0) (0)4! !

nn nx f x f x

n

0

0

2

2

2

2

about 0

about 0

Expand: ( ) (1 )

Recall: ( ) (1 )

( ) 1 2

g( ) 1 2

g x x

f x x

f x x x

x x x

x

x

EXAMPLE:

Approximating the derivativeObservational data analysisNumerical modeling

0

0

(3) (4)2

0

3 30 0 0 0 0 0

(3) (4) 3200 0 0

1 1 1( ) ( ) '( ) ''( ) ( ) ( )2! 3! 4!

Solve for

1 ''

'( )

( ) ( ) 1 1( ) ('( ) ( ) ) 3! 4!2!

f x f x f x h f x h f x h f x h

f x

f x h ff x hf x

h

f hx xx

hh f

Forwarddifferenceapproximation

Error ~ O(h)

Differentiating data:Forward difference example

x f=x3 f’ 3x2

1 1 (8-1)/1 = 7 3

2 8 (27-8)/1 = 19 12

3 27 (64-27)/1 = 37 27

4 64 (125-64)/1 =61 48

5 125 75

0 00

( ) ( )' ) (

f x h f xf x

h

1h

Approximating the derivativeObservational data analysisNumerical modeling

Backwarddifference

(3) (

0 0

0 00

4 300

)20

( ) ( )

( ) ( )' 1 ''(( 1 1( ) ( )

3! ! !) )

42f x

h h

f x h f x h

f x h f xh

f x f x hf

hhx

(3) (4)2 3 30 0 0 0 0 0

(3) (4) 30 0

200

00

1 1 1( ) ( ) '( ) ''( ) ( ) ( )2! 3! 4!

1 1( )( ) 1 ''(

(( )

3)

2! !)

!)

4'(

f x

f x f x f x h f x h f x h f x h

f x hh f x

f f

h

hh

xx x h f

Forwarddifferenceapproximation

Error ~ O(h)

y( )y f x

FD

actual

x hx

BD

x h

Forward and backward differences

Approximating the derivative

20 00

(3) (4) 30 00

1 1(( 1 ''( ) ) (

))

3( )

! 4!2)

!'( f x h f xf x h h

f x h f xf x

h

Forwarddifference

Backwarddifference

0 0 (3) (4)0

30

200

1 ''( ) 1 1( ) ( )3! 4!

( ) ('( )

2!)

f x h ff x f

f xx h

hh

xhf x

SUM


Forwarddifference

Backwarddifference

SUM (3 20 )0

00

( ) ( )2 '( 2 ( )

3!0) 0

f x h f x hf x f x h

h

/2

20 00

(3) (4) 30 00

1 1(( 1 ''( ) ) (

))

3( )

! 4!2)


f x h f xf x

h

0 0 (3) (4)0

30

200

1 ''( ) 1 1( ) ( )3! 4!

( ) ('( )

2!)

f x h ff x f

f xx h

hh

xhf x


Forwarddifference

Backwarddifference

SUM (3 20 )0

00

( ) ( )2 '( 2 ( )

3!0) 0

f x h f x hf x f x h

h

/2 0 00

(3) 20

( ) ( )'( ) 1 ( )

3

2 ! f

f x h ff h

hx

x hx

Centereddifference

Error ~ O(h2)

20 00

(3) (4) 30 00

1 1(( 1 ''( ) ) (

))

3( )

! 4!2)


f x h f xf x

h

0 0 (3) (4)0

30

200

1 ''( ) 1 1( ) ( )3! 4!

( ) ('( )

2!)

f x h ff x f

f xx h

hh

xhf x

Centered difference example

x f=x3 f’ 3x2

1 1

2 8 (27-1)/2 = 13 12

3 27 (64-8)/2 = 28 27

4 64 (125-27)/2 = 49 48

5 125 (216-64)/2=76 75

6 216 (343-125)/2=109 108

7 343

0 00

( ) ( )'( )

2f x h f x h

f xh

Approximating the 2nd derivative(3) (4)

0 02 3

00 0

01 ''( ) 1 1( ) ( )

3! 4!( ) ( )

'(2!

) f x h ff xf x h f x

f xh

h x h

Forwarddifference

Backwarddifference

0 0 (3) (4)2 30 00 0

1 ''( ) 1 1( ) ( )3! 4!

( ) ('( )

2!)

f x h ff x f

f xx h

hh

xhf x

SUBTRACT

/h 0 0 0 (4)0 02

2( ) 2 ( ) ( )'' 2 ( )

4( )

!f x h f x f x h

f xh

f x h

Centereddifference

Error ~ O(h2)

(4)00

0 0 30

( ) 2 ( ) ( ) 20 ''( ) 2 0 !2!

( )4

f x h f x f x hf x h f x h

h

Application: error analysis

Floodwaters in the Kalama Gap

V?

1/2 2( ) ; 10 /

. .150 :

160

?

?

In general

:

...?

V gh g m s

e gh m V

h m V

x

( )y f x

R

2 3/2'' 1 ;

(1 ' )fK

Rf

Curvature

x

( )y f x

R

2 3/2'' 1 ;

(1 ' )fK

Rf

CurvatureGentle turn

Sharp turn

r

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Scientists: Which scientific advance has had the most impact on people’s everyday lives? #3: Darwin’s theory of evolution