Cambridge Additional Mathematics

0

1

2

3

4

5

6

7

1750 1800 1850 1900 1950 2000 2050

year,x

populationP(billions)

“”Best fit line

exponential model

0

0.5

1

1.5

2

2.5

1750 1800 1850 1900 1950 2000 2050

year,x

lnP

-0.5

The “best fit line” is

not a perfect fit because

we are using real data.

Straight line graphs (Chapter 7) 193

The population data is presented on the graph below:

The “best fit line”, P =0: 0222 x¡ 39 : 6 , does not fit the data very well. Instead, we try to fit an

exponential curve of the form P=aemx.

Taking the natural logarithm of both sides, we have lnP=mx+lna, which is the equation of a

straight line.

We now plot lnP againstx:

The equation of this “best fit line” is lnP=¡ 15 :5+0:008 55x.

Using our model this means that lny=¡ 15 :5+0:008 55x,

) the data can be modelled by P=e¡^15 :5+0:008 55x.

This is shown as a dashed line on the original graph. This is not a

perfect fit either, but is a considerable improvement on the original

straight line graph.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_07\193CamAdd_07.cdr Friday, 4 April 2014 11:50:30 AM BRIAN