HUT – DEPARTMENT OF MATH. APPLIED

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MATH 1

LEC 02: FUNCTION

Instructor: Dr. Nguyen Quoc Lan (October, 2007)

NOTION OF FUNCTION ---------------------------------------------------------------------------------------------------------------

--------------------

Some quantity A changes and depends on

another quantity B Function: A = f(B).

Example: The human pouplation depends

on the timeYear Population (million)

1910 1750

1920 1860

1930 2070

1940 2300

1950 2560

1960 3040

HISTORY ---------------------------------------------------------------------------------------------------------------

--------------------

Mid – eighteenth

century, Euler: By

alphabet y = f(x)

1786,

Scotland: The

Commercial

an Political

Atlas,

Playfair.

Graph used

to compare

exports,

imports by

England to

Denmark …

f Function

x Input Computer y Output

MATHEMATICAL DEFINITION ---------------------------------------------------------------------------------------------------------------

--------------------RX

RY A function y = f(x): X R Y

R: Rule associates each x

X unique output (exactly

one) y Y. Variable x,

function (value) y.One x Two

different y: It’s not a

function Vertical

test (for a graph)

Domain D = {x| f(x)

defined}Range Imf: y =f(x),

xDf y = sinx D= R, Imf =

[–1, 1]

THE VERTICAL LINE TEST ---------------------------------------------------------------------------------------------------------------

--------------------

The Vertical Line Test: a curve in the xy-

plane is the graph of a function of x if and

only if no vertical line intersects the curve

more than once

(a,b)

x=a

a

(a,b)

(a,c)

a

x=a

This is the

graph of a

function of x

This is not the graph

of a function of x

GIVE A FUNCTION: TABLE OF ITS VALUE ---------------------------------------------------------------------------------------------------------------

--------------------

Verbally and table of it values: The natural

way to represent the function C(w)

expressing the cost of mailling first class

letter is using a table of valuesw (ounces) C(w) (dollars)

0<w1 0.34

1 <w2 0.56

2 <w3 0.78

3 <w4 1.00

4 <w5 1.22

…

GIVE A FUNCTION: GRAPH ---------------------------------------------------------------------------------------------------------------

--------------------

Plot a picture (graph): For the function P(t)

expressing the dependence of human

population in time, one can express it by

table of values, then construct a graphP(t)

0

1000

2000

3000

4000

5000

6000

7000

1880 1900 1920 1940 1960 1980 2000 2020

P(t)

The

graph

of this

function

is a

scatter

plot

FUNCTION DEFINED BY ALGEBRAIC FORMULA --------------------------------------------------------------------------------------------------------------------

-----------------------

Explicit form: y =

f(x)Example: y = x2, elementary

functions

tyy

txx:form Parametric

Example: x = 1 + t, y = 1 – t

Line

: 1 t 1 (x,

y)

Example: x = acost, y = asint

Circle

Implicit form F(x, y) = 0 y

= f(x)Example: x2 + y2 –

4 = 0,

01916

22

yx

Formul

a:

MAPLE ---------------------------------------------------------------------------------------------------------------

--------------------

(Declare a function) p := x^3 +

x^2 + 1; (Evalue its value)

subs(x=1, p); (Evalue its limit) limit(

sin(2*x)/x, x = 0) ; (Evalue its derivative) diff(p, x) ; (2nd order)

diff(p,x$2) (Graph) plot(sin(x), x = 0..Pi); (Many graphs)

plot( [sin(x),cos(x)],x = 0..2*Pi, color =

[red,blue]); (Parametric curve) plot( [31*cos(t)-

7*cos(31*t/7), 31*sin(t)-7*sin(31*t/7), t =

0..14*Pi] ); plot( [17*cos(t)+7*cos(17*t/7), 17*sin(t)- …, t

= 0..14*Pi] );

MATHEMATICAL MODEL: RADIOACTIVE DECAY -----------------------------------------------------------------------------------------------------------------

---------------

Radioactive elements disintegrate continuously

in a process called radioactive decay.

Experimentation has shown that the rate of

disintegration is proportional to the amount of

the element present. Find the rule of the

radioactive decay processSolution: Suppose that m = m(t): the amount at

time t The rate of disintegration is dm/dt. The

assumption above gives: minus issign the,:m(t) As constant. positive :k kmdtdm

00.lnln mCtCetmCktmkdtmdm kt

MATHEMATICAL MODEL: RADIOACTIVE DECAY -----------------------------------------------------------------------------------------------------------------

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By the expression of m(t), all radioactive

elements have a common special property: after

some constant period of time, its original amount

will reduce by half the half – life. Every

radioactive element has a specific half – life, and

it depends only on decay rate, not on the initial

amountThe half – life of radioactive carbon C – 14 is

about 5730 years. Find the expression m(t) of C

– 14?Solution: T – the half – life The amount: m0/2

at time T: T

kkTeRR kT 2ln

2ln2 0

0 teRtRT 000121.005730

THE SHROUD OF TURIN -----------------------------------------------------------------------------------------------------------------

---------------

In 1988 the Vatican authorized the British

Museum to date a cloth relic called the Shroud

of Turin, possibly the burial shroud of Jesus of

Nazareth. This cloth founded in 1356 and

contains the negative image of a human body.

From the British Museum: the fibers in the cloth

contained between 92% and 93% of their

original carbon C – 14. Conclusion?Solution:

From

teRtR 000121.0

0

0

ln000121.0

1RtR

t

R/R0: 0.92 0.93

60093.0ln&68992.0ln 21 tt

The test: 1988 The shroud was between 600 –

688 years old!