Use a transformation to solve the equation. f ∙ 8 = 40 a f=5 b f=32 c f=48 df=320

Using Taylor expansion, show that

f0

(x0) = f (x0 + h) - f (x0)

h - h

2

f00 (ξ),

for some ξ lying in between x0 and x0 + h.

Solution: We expand the function f in a first order Taylor polynomial around x0:

f (x) = f (x0) + (x - x0) f0

(x0) + (x - x0)

2 f00 (ξ)

2,

where ξ is between x and x0. Let x = x0 + h:

f (x0 + h) = f (x0) + hf0

(x0) + h2

2 f00 (ξ).

Solving for f0

(x0), we obtain:

f0

(x0) = f (x0 + h) - f (x0)

h - h

2

f00 (ξ)