Linearizing a system using a taylor expansion

[Giuseppe]

Homework Statement

Linearize the system operator illustrated below by applying a Taylor series expansion.

f(t) ----> e^f(t) -----> g(t)

Homework Equations

I only find the general form of a taylor series relevant.

g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n!
The system is centered at 0, so a=0.

The Attempt at a Solution

My question is that I'm not sure if I'm looking into the problem too closely, but is it just asking to find the taylor expansion of e^x? I know in that case the answer should be

g(t)= sum(0,infinity) t^n/n!

 

[christianjb]

My guess

e^x=1+x/1!+1/2! x^2 +1/3! x^3 + a/4 x^4!+...

A linear approximation is given by the first two terms
e^x f(x)=[approx]=(1+x) f(x)

 

[HallsofIvy]

christianjb is correct: a "linearization" is just a linear approximation. If you already have the Taylor's series for the function at a given point, you can just take the first two (linear) terms to get a linear approximation. Of course, that is exactly the same as finding the tangent line at the point.

I'm not clear on exactly what your function is. You seem to be writing
e^f(x). How you would linearize that, or how you would find its Taylor series depends on what f(x) is! If you just want e^f(x) as a linear function of f(x), that would be 1+ f(x).
(christianjb thinks you mean e^x*f(x).)

 

[Giuseppe]

I agree with what you're saying. That's why I didn't think the problem was as hard as it I was thinking. No function is explicitly defined.

 

[SGT]

The first derivative of e^{f(x)} is e^{f(x)}.f´(x), so your linearization would be

e^{f(0)} + e^{f(0)}.f´(0).x