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So length can be measured with some error.
Let's say error one.
Height can be measured with some error two.
And the question was ...
... if you think about those two errors effecting the ...
... error in my computation of the area ...
... which one is effecting the most?
Can you give me any thoughts?
So what do you think?
Student: [...] since the height is smaller proportional to the length [...].
So you are saying that this error ...
... influences the area more than that one.
Student: Assuming that they are approximately the same [...].
Well yes, so those are relatively small ...
... with respect to any actual measurement.
NB: Dave, did you have the same part? Student: No.
Why?
Student: [...].
Height is smaller as more effect on the area.
OK.
Any opposite? Yeah.
Student: [...].
Because you multiply the length contributes more to the product.
So may be the error in the length contributes more to the error.
So that's another valued thought.
Right? So er ...
How to think about it?
Well the ...
... way to think about it would be to say that ...
We have one variable -- length.
And another variable -- height.
And then what I want to compute is a function of those two.
And in this case it's a very simple function. It's x times y.
And then ...
I'm really interested in computations ...
... that happen very close to some actual values of length and height.
And I don't know those values exactly.
But I have ...
I probably have a rough idea of what those values are.
So er ...
Well let's say ...
... just to be specific this length is two hundred ...
... units. And that height is ...
... equal to one hundred units.
And then ...
What I'm interested in is behavior of this expression of this function ...
... close to the value x equals two hundred, y equals one hundred.
Right? So I'm interested in ...
... f ...
Er ... Well in values of f.
At points ...
... close to the point two hundred, one hundred.
So now we are coming slowly to the setup of doing approximations.
Right? We have a point of major interest.
And the idea now is to replace that relatively complicated function with a linear function.
So how do you replace ...
... a function ...
... x times y with a linear function?
Student: Take partial derivatives.
Take partial derivatives.
So what we did to replace with a linear function ...
... with linear thing is ...
... not a function itself but ...
... something like expression.
Right? So ...
What I'm actually interested in is ...
... the error.
The difference between the value at x, y ...
... that is close to two hundred, one hundred ...
... and the value at that point.
Er ...
So ... how do we do linearization?
Well I have to take partial derivatives.
We have to take partial with respect to x.
It is y.
And the partial with respect to y.
Student: x. NB: It is x.
And what is it that we make?
Student: [...].
Well we ...
We put them together ...
... after evaluating, right?
f_x at that point of interest ...
... is what?
One hundred. And f_y ...
... at that point ...
... is two hundred.
And then we use those coefficients.
So f_x at that point should be multiplied by ...
... x minus two hundred.
And then this number two hundred ...
... should be multiplied by y minus one hundred.
What is that?
Student: [...]. NB: What is it that we are making here?
Well it's a linear expression.
It's a line but so far there is no equation.
But we don't expect equation. We expect approximate equality. Right?
So this is approximately equal to what?
Student: [...].
You think so?
Well you see if ...
Suppose I made no error at all.
So that x equals two hundred.
And y equals one hundred.
Then the whole value is zero.
So that is approximately the error, right?
This quantity.
It's approximately f of ...
... x, y minus f of ...
... two hundred, one hundred.
So ...
Can we make a conclusion at this point?
Does it help at all?
Because I did something. Right?
What's the relevance?
Student: [...].
So this is the ...
... error one.
And that is the ...
... error two. Right?
So the total error in the area ...
... is approximately -- not exactly but approximately ...
... equal to one hundred times error one ...
... plus two hundred times error two.
And now you should tell me which one is more important.
Student: Height.
The height.