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Linear Interpolation

f(x) = 3tan(x/2) - x - 1 ... x is between plus minus pi

Given that f(x) = 0 has a root between 1 and 2, use linear interpolation once on the interval [1, 2] to find an approximation to this root.

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How to do this? I did f(1) and f(2) to form:

(x-1)/0.3601... = (2-x)/1.672...

but this gives x as 0.47... which is clearly wrong. Is there a different approach to it being in radians, etc?
(edited 11 years ago)
Reply 1
Avatar for valhalla92
valhalla92
2
Original post by Lunch_Box
f(x) = 3tan(x/2) - x - 1 ... x is between plus minus pi

Given that f(x) = 0 has a root between 1 and 2, use linear interpolation once on the interval [1, 2] to find an approximation to this root.

----
How to do this? I did f(1) and f(2) to form:

(x-1)/0.3601... = (2-x)/1.672...

but this gives x as 0.47... which is clearly wrong. Is there a different approach to it being in radians, etc?


It has to be in radians. I can't really tell what you're doing since you've only shown 1 line of working, but your first iteration should be 0=f(1)+[f(2)-f(1)]*(x-1)/(2-1), after which you solve for x.
Reply 2
Avatar for Lunch_Box
Lunch_Box
OP
15
Original post by valhalla92
It has to be in radians. I can't really tell what you're doing since you've only shown 1 line of working, but your first iteration should be 0=f(1)+[f(2)-f(1)]*(x-1)/(2-1), after which you solve for x.


I'm unaware of that formula...

I use this method of similar triangles:




I have done:

(x-1) / f(1) = (2 - x) / f(2)
(edited 11 years ago)
Reply 3
Avatar for Damask-
Damask-
16
The formula is af(b)−bf(a)f(b)−f(a)\frac{af(b)-bf(a)}{f(b)-f(a)}
(edited 11 years ago)
Reply 4
Avatar for Lunch_Box
Lunch_Box
OP
15
Original post by Damask-
The formula is af(b)−bf(a)f(b)−f(a)\frac{af(b)-bf(a)}{f(b)-f(a)}


This formula is awesome. And also, is this allowed for FP1 edexcel if you know by any chance lol
(edited 11 years ago)
Reply 5
Avatar for Damask-
Damask-
16
Original post by Lunch_Box
So I have to take into account the negatives? And also, is this allowed for FP1 edexcel if you know by any chance lol


Yes you do, and it's a perfectly valid way to go about solving the question, so you can use it in FP1. Our teacher got us to derive it in a lesson so we'd understand where it comes from.

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