Addition formula for f(x+y) using mean-value theorem

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Homework Help Overview

The discussion revolves around the application of the Mean Value Theorem to derive addition formulas for two differentiable functions, f and g, under specific conditions. The original poster presents a function F defined in terms of f and g, and seeks to compute its derivatives and subsequently analyze another function E derived from F.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the correct treatment of variables when differentiating F with respect to x while holding y constant. There are questions about the treatment of f(y) and g(y) as constants during differentiation.
  • Some participants express confusion regarding the complexity of the derivatives involved and seek clarification on specific steps in the computation.
  • There is a suggestion to simplify expressions using known relationships between f and g, particularly in relation to their derivatives.

Discussion Status

The discussion is ongoing, with participants providing feedback on each other's computations and suggesting simplifications. Some guidance has been offered regarding the application of the Mean Value Theorem and the treatment of derivatives, but multiple interpretations and approaches are still being explored.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can share or the methods they can use. The original poster has expressed uncertainty about their computations, indicating a need for peer review and clarification.

mathmathmad
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Homework Statement


suppose f and g are two differentiable functions with f(0)=0, g(0)=1 and f'(x)=g(x) and g'(x)=-f(x). For a fixed y in R put

F(x) = f(x+y) - f(x)f(y) - g(x)f(y)

Compute F'(x) and F''(x)
Then let

E(x) = [F(x)]^2 + [F'(x)]^2


Apply Mean Value Theorem to E and hence prove the addition formulae for f(x+y) and g(x+y)

Homework Equations





The Attempt at a Solution


I don't think what I've computed it correct, can anyone kindly please check?
http://img192.imageshack.us/img192/5237/rimg0036.jpg
 
Last edited by a moderator:
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First thing I notice is that you take a derivative with respect to y in F'(x). You are holding y fixed... that is treat it like a constant. It is equivalent to taking the partial derivative w.r.t. x.

Hope this gets you somewhere.
 
that's what I thought but I was unsure about it
does that mean f(y) and g(y) will be treated as constants?
and is it correct that derivative of [F(x)]^2 is 2F(x).F'(x)?
 
Yes to all of the above :).
 
I'm still stuck :( got loads of fs f's f''s with x y @_@ squares and all.. T_T
 
mathmathmad said:
I'm still stuck :( got loads of fs f's f''s with x y @_@ squares and all.. T_T

Post your stuff. I'll take a look.
 
http://img641.imageshack.us/img641/3946/rimg0001m.jpg
http://img291.imageshack.us/img291/5116/rimg0003a.jpg
it's messy though...
 
Last edited by a moderator:
I would say everything looks great after the first page you posted.

Before getting started on the second page, have a look at the last line on your first page. You see the expression:

(F(c) + F''(c)) ?

Using the fact that f'(x) = g(x) and g'(x) = -f(x), and considering your formulas on lines 1 and 3 of the first page, you should be able to simplify this expression considerably.

Once you move on from here, you should consider taking a = 0 for your MVT and then use your information about f(0) and g(0) to further simplify the problem.

Unfortunately, I have to run! Good luck :).
 
http://img52.imageshack.us/img52/5700/rimg0001.jpg
http://img188.imageshack.us/img188/5554/rimg0002q.jpg

still stuck :(
 
Last edited by a moderator:
  • #10
You should be able to find (provided you follow the steps given above) that E(b) = 0. Are you able to get this?
 
  • #11
yeah.. so the addtion formula of f(x+y) = f(x)g(y) + f(y)g(x) right? =)
 

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