Note DERIVE demo version (30-day free trial) at

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Spring ht vs. mass:

mass ht y

2

5

25

0

 

Linear function. Easily graph the two points:

wpe2.jpg (13240 bytes)

Slope from (2,5) to (25,0) is `dy / `dx = -5 / 23 = -.22 approx.

It turns out that the straight line between points is height = -.217 * mass + 5.44, approx.

 

 

wpe3.jpg (15534 bytes)

 

We'll see shortly how to get the equation.

A series of observations might give us a table and a graph something like

 

mass

ht y

2

5

5

4.1

10

2.9

15

2.05

20

0.93

25

0

wpe4.jpg (10737 bytes)

 The 'best-fit' straight line for this data is a little different than for the 2-point data:

wpe5.jpg (13460 bytes)

Note that the 'best-fit' line doesn't go through any of the points.  For real-world data this is usually the case.  The line comes as close as possible, on the average, to all the points.

Back to the original 2-point graph:  How do we get the equation of a line through two points?

Substitute coordinates of the two points into the form y = mx + b to get

If we subtract 2d eqn from first we have

which gives us

This is the same as the slope we got before.  Note that we coulda just substituted this slope into y = mx + b to get y = -.217 + b, then just substituted the coordinates of one point.

Now substitute m = -.217 into the first of the two equations to get

and solve for b to get

Thus our equatoin is

meaning

 The data set

mass

ht y

2

5

5

4.1

10

2.9

15

2.05

20

0.93

25

0

gave us the best-fit line

 To judge quality of fit we find the predicted height for each of the observed masses:

mass

ht y

pred ht

deviation

2

5

4.8082

-0.1918

5

4.1

4.168

0.068

10

2.9

3.101

0.201

15

2.05

2.034

-0.016

20

0.93

0.967

0.037

25

0

-0.1

-0.1

Standard deviation is square root of average of squared deviations:

squared deviations are -.1918^2 = .04 approx, .068^2 = .0046 approx., etc.

We average these squares then take the square root to get std dev.

If we have 12.4 kg of water what is ht?

Cups to bring end to ground: