best linear approximation (1 Viewer)

[xanny]

L(x)=cos x =>

girve the best approximation of the form f(x)=a(e^x)+bx+c

Thanks for your helps!

 

[Trev]

Erm, where is this question from?

 

[Stan..]

L(x) = f(a) + (x-a)f'(x)
L(x) = -xSinx + aSinx + Cosa
I guess you could expand sin and cos from here.

 

[SeDaTeD]

Um, at x=A
L(A) = cosA
L'(A) = -sinA
L''(A) = -cosA

Now to match the value of the function and also the 1st and 2nd derivatives.
let:
f(A) = a(e^A)+bA+c = L(A) = cosA
f'(A) = a(e^A) + b = L'(A) = -sinA
f'(A) = a(e^A) = -cosA
=> a = -cosA*e^-A
b = -sinA - a(e^A) = -sinA + cosA
c = cosA - a(e^A) - bA = cosA + cosA -A(-sinA + cosA) = 2cosA - A(cosA - sinA)

so f(x) = -cosA*e^-A*e^x + (cosA - sinA)x + 2cosA - A(cosA - sinA)
= - cosA*e^(x-A) + (cosA - sinA)x - A(sinA - cosA)
as an approximations of L(x) = cosx, at x=A, in the form a(e^x) + bx + c.

Though that's not a linear approximation, as the title suggests.