Question

how to use linear approximation when given a table of values?

my assignment asks this. In typical math department fashion there is no example in the notes or textbook

Answer 1

so the first row are the y values and the first column is the x values. How would I estimate the value at a point given only this info?\[\left[\begin{matrix}&0 & 1 & 2 & 3 &4 \\1& 2 & 0 & -3 & -6 & -5 \\1.5&3 &1 &-4 & -8 &-6 \\2&4&3&0&-5&8\\2.5&5&5&3&-1&-4\\3&7&8&6&3&0\end{matrix}\right]\]

Answer 2

What do the values inside the table represent? I having trouble figuring out what your table means.

Answer 3

what does given a 'point' mean? is that an X or Y point?

Answer 4

Or is it a two dimensional point?

Answer 5

(X,Y), either way please specify what point...

Answer 6

This is something = f(x,y).
Take a suitable step size h in each direction to get a linear approx near to a point.

Answer 7

alright so the inside values is the altitude given by their x and y component. The y components are the headers on top (0-4) and the x components are the headers on the left (1-3). What I need to do is find out how to estimate the altitude at a point that is not on this table. Like at point (2.75, 1.5)

Answer 8

Pick a point from the table nearest to the one you want to estimate, then take a linear approximation in the usual way (f(x+h) etc with a suitable h, maybe 0.5 or 0.25) and use that approximation for the point you want to estimate.

Answer 9

but how do I know what f is?

Answer 10

I don't have a function

Answer 11

The table is the function, that's what a function is, a list of matching values....
Perhaps you should go back and study single variable linearization first.

Answer 12

eg from your table f(2,3) is -5
You should select a point nearest to the one you want to estimate and linearize from there.

Answer 13

U want limit as h tends to 0 f(x+h,y) f(x,y)/h and do the same in the y etc but if you have no idea what I am talking about you need to go back and study some, OK?

Answer 14

alright well in single variable linerial approximation there is not a single example where I was given a list of values and had to approximate from that. I know how to use least-squares to fit data from statistics but I still have no idea what to do in this example.

Answer 15

it always gives me a function like a continous function that I can differentiate and then I just use that to approximate the value

Answer 16

Yes, I see, they probably gave you the function upfront as a formula rather than as a table.
Then again, in real life, that's all you will get.....:-)

Answer 17

ok so why wouldn't I just use the slope given by the two points the x is between and the two points the v is between instead of differentiating the function?

Answer 18

If you can get hold of a good calculus text (Stewart, for example) they will take you all through this....

Answer 19

alright thanks a lot. I use edwards and penny and every time I open stewart everything mkes sense. I'll go take it out!