Y

X

The derivative

Lecture 5Handling a changing world

Y

X

x2-x1

y2-y1

The derivative

x2-x1

y2-y1

xy

xxyyslope

12

12

xxfxxf

xxxfxfslope

)()()()( 11

12

12

x1 x2

y1

y2

xxfxxfslope x

)()(lim 0

xxfxxfyxf

dxdy

x

)()(lim)(' 0

The derivative describes the change in the slope of functions

Aryabhata (476-550)

Bhaskara II (1114-1185)

The first Indian satellite

-10

-5

0

5

10

-4 -2 0 2 4

YX

)()()( ufbudxxdfbauuy

u

0)2(2

2)2(

bdxxdf

bay

( * ) ' '* * 'f g f g f g ( ( )) ' '* 'f g f g

( ) ' ' '( ) ' ' 'f g f gf g f g

'

2

'* * 'f f g f gg g

Four basic rules to calculate derivatives

b

Local minimum

0)( dxxdf Stationary point,

point of equilibrium)(')()( cf

abafbf

Mean value theorem

0

1

2

3

0 5 10 15 20

Y

X

0510152025303540

0 5 10 15 20

Y

X

y=30-10

x=15-5

25101030lim 0

xy

dxdy

x

The derivative of a linear function y=ax equals its slope a

xy 2

y=0

0lim 0 xy

dxdy

x

The derivative of a constant y=b is always zero. A constant doesn’t change.

2y

aybaxy '

0

5

10

15

20

25

30

0 1 2 3 4

Y

X

xx edxdyey

dy

dx

xeydxdy

The importance of e

ax

x exa

1lim

ex

x

x

11lim

)ln(xy

xedxdye

dydx

dydxdxdy

exxy

yy

y

11/1)ln(

-2

-1

0

1

2

3

4

0 1 2 3 4

Y

X

)ln(xy xy 1

1)ln()ln()ln()ln(

)ln()ln(

)1)ln(00())'ln()(ln(''

bbxbaxbau

uxbab

abxxbax

xbxexbaeuey

eeaxy

baxy

xxxxu

ubxax

bbababxbabbxaabuey

eeaby

)ln()ln()0)ln(10())'ln()(ln(''

)ln()ln(

xaby

xxx

xxxxxxy

xxxxy

xx

x

)sin(lim)cos()sin()sin()cos()cos()sin(lim'

)sin()sin(lim'

00

0

)sin(xy

)cos()(sin'1)sin(lim 0 xxxx

x

)sin()(cos' xx

0

5

10

15

20

25

30

-2 -1 0 1 2 3 4 5

Y

X

-25

-20

-15

-10

-5

0

5

-2 -1 0 1 2 3 4 5

Y

X

-20-15-10-505101520

-2 -1 0 1 2 3 4 5Y

X

Stationary points

Minimum MaximumHow to find minima and maxima of functions?

0

5

10

15

20

25

30

-2 -1 0 1 2 3 4 5

Y

X

f’<0 f’>0 f’<0f’=0

f’=0

f(x)

f’(x)

f’’(x)

86''283'

10242

23

xyxxy

xxxy

387.2;279.0910

342830' 212,1

2 xxxxxy

0

5

10

15

20

25

30

-2 -1 0 1 2 3 4 5

Y

X

Maximum and minimum change

Point of maximum changePoint of inflection

f’=0

f’=0

Positive sense

Negative sense

At the point of inflection the first derivative has a maximum or minimum.To find the point of inflection the second derivative has to be zero.

34860''

86''283'

10242

23

xxy

xyxxy

xxxy

4/3

Series expansions

)(),(0

xfixgn

i

xxaaxaxaxaxan

n

1)1(...1

32

Geometric series

We try to expand a function into an arithmetic series. We need the coefficients ai.

......)( 44

33

2210 n

nxaxaxaxaxaaxf

333

433

2222

4322

1113

42

3211

0

32)0(...)1)(2(...43232)(

2)0(...)1(...43322)(

)0(......432)(

)0(

afxnannxaaxf

afxnanxaxaaxf

afxnaxaxaxaaxf

af

nn

nn

nn

i

i

in

n

xifx

nfxfxfxfxffxf

0

44

33

22

1

!)0(...

!)0(...

!4)0(

!3)0(

!2)0()0()0()(

McLaurin series

...)(...)()()()()( 44

33

2210 n

n bxabxabxabxabxaaxf

333

433

2222

4322

1113

42

3211

0

32)(...)()1)(2(...)(43232)(

2)(...)()1(...)(43)(322)(

)(...)(...)(4)(3)(2)(

)(

abfbxnannbxaabf

abfbxnanbxabxaabf

abfbxnabxabxabxaabf

abf

nn

nn

nn

i

i

in

n

bxibfbx

nnfbxbfbxbfbfxf )(

!)(...)(

!)(...)(

!2)())(()()(

1

22

1

Taylor series

00

04

03

02

000

!1

!...

!...

!4!3!2 i

i

i

nx

ie

ixx

nexexexexeee

iin

i

nnnnn xaninxannnxannxnaaxa

0

33221

)!1(!!...

!3)2)(1(

!2)1()(

Binomial expansion

iin

i

n xain

xa

0

)( Pascal (binomial) coefficients

in

Series expansions are used to numerically compute otherwise intractable functions.

xy sin

i

i

in

n

xifx

nfxfxfxfxffxf

0

44

33

22

1

!)0(...

!)0(...

!4)0(

!3)0(

!2)0()0()0()(

....!7!5!3

...!4)0sin(

!3)0cos(

!2)0sin()0cos()0sin()sin(

753432 xxxxxxxxx

Fast convergence

Degrees Radians Sin 1 2 3 4 5 Sum30 0.523599 0.5 0.523599 -0.02392 0.000328 -2.14072E-06 1.55678E-08 0.545 0.785398 0.70711 0.785398 -0.08075 0.00249 -3.65762E-05 3.98984E-07 0.7071160 1.047198 0.86603 1.047198 -0.1914 0.010495 -0.000274012 3.98534E-06 0.8660390 1.570796 1 1.570796 -0.64596 0.079693 -0.004681754 0.00010214 0.99995

Summands

1

15432

)1(...5432

)1ln(i

ii

ixxxxxxx

Taylor series expansion of logarithms

In the natural sciences and maths angles are always given in radians!

Very slow convergence

Sums of infinities

xxfxxfyxf

dxdy

x

)()(lim)(' 0

The antiderivative or indefinite integral

)(')( xfdxxdf

dxxfxF )()(

Cxdxx

Cxdxx

Caa

dxa

Cxa

dxx

Cxdxx

Cea

dxe

Caxdxa

xx

aa

axax

)sin()cos(

)cos()sin(

)ln(111

ln1

1

1

dxxgdxxfdxxgxf

dxxfadxxaf

)()()()(

)()(

Integration has an unlimited number of solutions. These are described by the integration constant

dxxdf

dxdC

dxxdf

dxCxfd )()())((

0 1000N

Assume Escherichia coli divides every 20 min. What is the change per hour?

1

2

3

1000*2*2*21000*2*2*2*2*2*2

1000*2 tt

NN

N

3 30 1 0 0

3 3 3 31 2 1 1

3 3( 1) 31 1 1

1000*2 1000 (2 1)

1000*2 *2 1000*2 (2 1)

1000*2 1000*2 (2 1)t tt t t t

N N N N

N N N N

N N N N

How does a population of bacteria change in time?

11 ttt rNNNNFirst order recursive function1

trNtN

Difference equation

rNdtdN

tN

t

0lim

Differential equations contain the function and some of it’s derivatives

rtrtC KeeeN

CrtCN

rdtNdN

rdtNdNrN

dtdN

21)ln(

rt

r

eNN

KKeNt

0

000

Any process where the change in time is proportional to the actual value can be described by an exponential function.

Examples: Radioactive decay, unbounded population growth,First order chemical reactions, mutations of genes,speciation processes, many biological chance processes

Allometric growth

In many biological systems is growth proportional to actual values.

A population of Escherichia coli of size 1 000 000 growths twofold in 20 min. A population of size 1000 growths equally fast.

2000100020000001000000

10

10

PPNN

100020000001000000 1

PNN

PP

NN

PPz

NN

PdPz

NdN

1)ln()ln( cPzNPdPz

NdN

zzc cPPeN 1

Proportional growth results in allometric (power function) relationships.

Relative growth rate

0

20

40

60

80

100

0 2 4 6 8 10

N

t

The unbounded bacterial growth process

2ln002tt eNNN

How much energy is necessary to produce a given number of bacteria? Energy use is proportional to the total amount of bacteria produced during the growth process

8

20

8

2

2t

t

tt

tt NN

What is if the time intervals get smaller and smaller?

Gottfried Wilhelm Leibniz (1646-1716)

Archimedes (c. 287 BC – 212 BC)

Sir Isaac Newton (1643-1727)

The Fields medal

0

20

40

60

80

100

0 2 4 6 8 10

N

t

2ln002tt eNNN

tf(t)

bt

at

bt

att ttfN )(

The area under the function f(x)

ttftFttF

tft

tFttF

tfdtdF

t

t

)()()(lim

)()()(lim

)(

0

0

bt

at

bt

att

bt

att

bt

att tFttFtFttFN )()(lim)]()([lim 00

0

20

40

60

80

100

0 2 4 6 8 10N

t

2ln002tt eNNN

tf(x)

)1())1((

))(())1((...)3()4()2()3()1()2(

)()(lim 0

recFnrecFN

nrecFnrecFrecFrecFrecFrecFrecFrectFN

tFttFN

bt

att

bt

att

bt

at

bt

att

bt

att

b

a

bt

atit

bt

atit

dttfdtafdtbfN

aFbFN

)()()(lim

)()(lim

0

0

bt

at

bt

att ttfN )(

b

a

bt

att dttfttfArea )()(lim 0

Definite integral

)()()( aFbFFdttfArea b

a

b

a

0

20

40

60

80

100

0 2 4 6 8 10N

t

2ln002tt eNNN

tf(x)

0000

8

20

8

20 559.363

2ln252

2ln4

2ln256

2ln22 NNNNNNNt

ttotal

What is the area under the sine curve from 0 to 2p?

011)0cos()2cos()cos()sin( 2

0

2

0

ppp

xdxxA

4)0cos(4)2/cos(4)cos(4)sin(4 2/

0

2/

0

ppp

xdxxA

0

20

40

60

80

100

0 2 4 6 8 10N

t

a

b

What is the length of the curve from a to b?

dxdxdydcL

dxdxdy

dxdxdydxdydxdc dydxdydxdydxdc

2

2

0,2

222

0,22

0,0

1

1lim)(lim)(limlim

What is the length of the function y = sin(x) from x = 0 to x = 2p?

p2

0

2)cos(1 dxxL

c

x

y

2)cos(1 xLNo simple analytical solution

22016011

248262)cos(1

7532 xxxxdxx

625526.7

]22580480

11215362482

2[4)cos(147532/

0

2

ppppp

dxx

We use Taylor expansions for numerical calculations of definite integrals.

Taylor approximations are generally better for smaller values of x.

1 2.2214412 -0.456773 0.1408784 0.000828

Sum 1.9063814 times 7.625526

What is the volume of a rotation body?

y y

x

x

b

a

b

adx dxxfdxxfV 22

0 )()(lim pp

)(

)(

22

1

)(bfy

afy

dyygV p

What is the volume of the body generated by the rotation of y = x2 from x = 1 to x = 2

44 2

2

11

7.5 23.562

V y dy ypp p

What is the volume of sphere?

34

322)

3(22

33

0

32

0

222 rrxxrdxxrVrr pppp

y

x