5. Curves and Curve Fitting

7/18/2019 5. Curves and Curve Fitting

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Worked examples and exercises are in the textSTROU

PROGRAMME 12

CURVES AND

CURVE FITTING


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Introduction

Standard curves

Asymptotes

Systematic curve sketching, given the euation o! the curve

"urve !itting

Method o! #east suares

Programme 12$ "urves and curve !itting


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Introduction

Standard curves

Asymptotes


"urve !itting




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Introduction


The purpose of this Programme is to develop methods for establishing the

relationship between two variables whose values have been obtained by

experiment. Before doing this it is necessary to consider the algebraic

relationships that give rise to standard geometric shapes.


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Introduction

Standard curves

Asymptotes


"urve !itting




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Standard curves

Straight line

Second-degree curves

Third-degree curves

Circle

Ellipse

Hyperbola

Logarithmic curves

Exponential curvesTrigonometric curves



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Standard curves

Straight line


The equation of a straight line is a first-degree relationship and can always

be expressed in the form:

where m =dy/dxis the gradient of the line

and cis the y value where the line

crosses they-axis the vertical intercept.

y mx c= +


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Standard curves



The simplest second-degree curve is expressed by:

!ts graph is a parabola" symmetrical aboutThe y-axis and existing only fory#.

y$ ax%gives a thinner parabola if a& ' and

a flatter parabola if # ( a( '. The general

second-degree curve is:

where a" band cdetermine the position")width*

and orientation of the parabola.

%y x=

%y ax bx c= + +


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Standard curves

Second-degree curves (change of vertex


!f the parabola:

is moved parallel to itself to a vertex

position +%" ," for example" its equation

relative to the new axes is

where !$y , and"$x %.elative to the original axes this gives

%y x=

%! "=

% / 0y x x= +


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Standard curves



#ote: !f:

and a( # then the parabola is inverted.

1or example:

%y ax bx c= + +

%% 2 3y x x= + +


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Standard curves

Third-degree curves


The basic third-degree curve is:

which passes through the origin.

The curve:

is the reflection in the vertical axis.

,y x=

,y x=


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Standard curves

Third-degree curves


The general third-degree curve is:

4hich cuts thex-axis at least once.

, %y px $x rx s= + + +


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Standard curves

Circle


The simplest case of the circle is with centre at the origin and radius r.

The equation is then % % %x y r+ =


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Standard curves

Circle


5oving the centre to +h" % gives:

where:

The general equation of a circle is:

% % %" ! r+ =

" x h

! y %

=

=

% %

% %

% % #

centre + " - radius

x y gx fy c

g f g f c

+ + + + =

+


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Standard curves

Circle


Exercise

'.6o the equation x%7 y%7 %x 2y -'3 $ # represent a circle with

centre .... and radius ......

%.x%7 y%$ '%.%3

,.x%7 y%$ '##

.x%7 y% x 7 2y , $ #

3.%x% ,x 7 y 7 %y%$ #


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Standard curves

Ellipse


The equation of an ellipse is:

!f a& bthen ais called the semi-ma8or

axis and bis called the semi-minor axis.

% %

% % 'x y

a b+ =


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Standard curves

Hyperbola


The equation of an hyperbola is:

4heny = #"x$ 9 a and

whenx = #"y%$ b%and the curve

does not cross they-axis.

#ote: The two opposite arms of the

hyperbola gradually approach two

straight lines +asymptotes.

% %

% % 'x y

a b =


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Standard curves

Logarithmic curves


!f y$ logx" then when:

so the curve crosses thex-axis at

x$ '

lso" logxdoes not exist for realx; #.

' then log' #x y= = =


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Standard curves

Exponential curves


The curvey$ excrosses they-axis atx$ #.

6ometimes called thegro&th curve'


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Standard curves

Exponential curves


The curvey$ excrosses they-axis aty$ '.

6ometimes called the decay curve'


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E%ponentia# Graph &ogarithmic Graph

Graphs o! inverse

!unctions arere!#ected a'out the

#ine y( x


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Standard curves

Trigonometric curves (see slides


The sine curve is given as:

+a- sin where

,2# Period " amplitude

+b- sin where

% Period " amplitude

y ) nx

)n

y ) t

)

=

= =

=

= =

o


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Standard curves

Trigonometric curves (see slides



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Introduction

Standard curves

Asymptotes


"urve !itting




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Symmetry

*ntersection &ith the axes

Change of origin

)symptotes

Large and small values of x and y

Stationary points

Limitations



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Symmetry


!nspect the equation for symmetry:

+a !f only even powers ofyoccur" the curve is symmetrical about thex-axis

+b !f only even powers ofxoccur" the curve is symmetrical about they-axis


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*ntersection &ith the axes


Points at which the curve crosses thex- andy-axes:

>rosses thex-axis: Puty$ # and solve forx

>rosses they-axis: Putx$ # and solve fory

1or example" the curve

>rosses thex-axis atx$ ?'#>rosses they-axis aty$ % and

?3

% , % @y y x+ = +


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Change of origin


Aoo for a possible change of origin to

simplify the equation. 1or example" if" for

the curve

The origin is changed by putting !$y7 ,

and"$x " the equation becomes that of

a parabola symmetrical about the !axis:

%/+ ,- + /-y x+ =

%/! "=


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Stationary points


6tationary points exists where:

!f further:

#dy

dx

=

%

%

%

%

%

%

# the stationary point is a maximum

# the stationary point is a minimum

# with a change in sign through the stationary point

then the point is a point of inflexion

d y

dx

d y

dxd y

dx

=


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Introduction

Standard curves

Asymptotes


"urve !itting




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"urve !itting


eadings of two related variables from a test or experiment normally

include errors of various inds and so the points that are plotted from the

data are scattered about the positions they should ideally occupy.

Caving plotted the points a graph is then drawn through the scattered

points to approximate to the relationship between the two variables. 1rom

the parameters involved in this graph an algebraic form for the

relationship is then deduced.


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"urve !itting

Straight-line la&


!f the assumption that the two variablesxandywhose values are taen

from experiment are linearly related then their relationship will be

expressed algebraically as:

where arepresents the gradient of the straight line and brepresents the

vertical intercept

1rom a plot of the data" a straight line is drawn through the data as the

)line of best fit*. The values of aand bare then read off from the graph.

y ax b= +


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Introduction

Standard curves

Asymptotes


"urve !itting




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+itting a straight-line graph


=rawing a straight line of best fit through a set of plotted points by eye

introduces unnecessary errors. To minimise errors the method of least

squares is used where the sum of the squares of the vertical distances from

the straight line is minimised.

ssume that the equation of the

line of best fit is given by:

y a bx= +


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The ithpoint plotted" +xi"yi" is a vertical distance from the line:

The sum of the squares of these

differences for all npoints

plotted is then:

i iy a bx

( )%

'

n

i i

iS y a bx

==


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The values of aand bmust now be determined that gives Sits minimum

value. 1or Sto be a minimum:

This yields the two simultaneous equations from which the values of a

and bcan be found:

# and #S Sa b

= =

' '

%

' ' '

n n

i ii i

n n n

i i i i

i i i

an b x y

a x b x x y

= =

= = =

+ =

+ =


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W k d l d i i th t tSTROU

&earning outcomes

=raw setch graphs of standard curves

=etermine the equations of asymptotes parallel to thex- andy-axes

6etch the graphs of curves with asymptotes" stationary points and other features

1it graphs to data using straight-line forms

1it graphs to data using the method of least squares


5. Curves and Curve Fitting

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