I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.
Test 2
The second test is on Tuesday 27 November at 8.20 pm sharp.
I will give ye a sample on Thursday 22 November.
Schedule
Tuesday 20 November: Finish off Chapter 3 and start looking at Multiple Integration (line integrals).
Thursday 22 November: Tutorial for Test 2
Tuesday 27 November: Test 2. Start double integrals.
Tuesday 4 December: Tutorial for line integrals & double integrals
Thursday 6 December: Finish double integrals and Triple integrals
Wednesday 12 December: Possibly finish off multiple integration; Exam Format Review Lecture; Tutorial.
2 comments
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November 19, 2012 at 8:44 am
Student 25
Hi JP
I am going over the notes and have a question, On p101 the formula’s at the top of the page which are called normal equations, what is the relationship between them and are they a part of the normal equations on p102 part 3.2.2 and on page 104 where does the sum of C be got from?
November 19, 2012 at 9:02 am
J.P. McCarthy
Student 25,
If you are given a set of data
you might want to fit a curve to the data.
Suppose you want to fit a curve of the form
to the data. Here
is the output, 
is the input and 
& 
are functions/formulae in terms of the input 
.Finally 
and 
are constants.
We define the curve of best fit as the one that minimises the sum of the squared errors:
Each pair of values
gives rise to one curve of the form (*). Each pair of values 
gives rise to a sum of squared errors 
. Therefore we can use the calculus of two variables to find the values of 
and 
such that 
is minimised.
It turns out that the curve of best fit is found by solving the simultaneous equations
where the sum is over the
data points 
. We call these the normal equations for the curve(s) 
.
These equations are found by starting at the curve
multiplying across by the multiple of
and the multiple of 
and summing:
For the example of fitting a line
so we multiply everything by
and sum and we also multiply everything by the multiple of 
: one; and sum:
So yes the normal equations on page 101 belong with the normal equations on page 102. Remember though that you have
curve
normal equations for that curve
Now to do a sum of the form
. Suppose we call the constant by 
. We want to find
Now the sum is over the
data points so really what we have is
This means “add up the pattern
from 
up to 
“… add up 
s:
Regards,
J.P.