Question

What is a linearization function?

Answer 1

The idea behind linearization is that the math is often simpler if we can deal only with linear functions, and it is possible to substitute a linear function for some other function by determining the equation of the tangent line at a point on the original function. We then use the equation of the tangent line instead of the original function, but we have to stay "close" to the point where we found it.

One very frequent linearization is the substitution of \(\theta\) for \(\sin \theta\) in problems involving harmonic motion. For example, at \(\theta = 0.25\) (radians), \(\sin \theta = 0.2474\) which is only a 1% error. I've attached a graph showing \(\theta, \sin \theta, \sin\theta - \theta\) so you can see that for small values (staying "close" to the spot where we linearized), it's a very good approximation.

Answer 2

Blue line = \(\theta\)
Purple line = \(\sin\theta\)
Olive line = \(\theta - \sin\theta\) (the error in the approximation)

Answer 3

Thank you

Answer 4

In my problem i'm given sqrt(64.04) and sqrt(63.95). I'm asked to find a linearization function. What would that be?

Answer 5

I believe you would find the slope of the line connecting \(y=\sqrt(x)\) at x=64.04 and x = 63.95 and write an equation for a line that went through those points. (crude drawing just to give a general idea, not represent this function necessarily)

Answer 6

Ok. I'll give it a try. Thank you for your help.