Foundations of Data Science

f(x)
a
c
b
Figure 12.3: Illustration of the mean value theorem.
One application of the mean value theorem is with the Taylor expansion of a function.
The Taylor expansion about the origin of f(x) is
f(x) = f(0) + f ′ (0)x + 1 2! f ′′ (0)x 2 + 1 3! f ′′′ (0)x 3 + · · · (12.3)
By the mean value theorem there exists c, 0 ≤ c ≤ x, such that f ′ (c) = f(x)−f(0)
x
f(x) − f(0) = xf ′ (c). Thus
or
xf ′ (c) = f ′ (0)x + 1 2! f ′′ (0)x 2 + 1 3! f ′′′ (0)x 3 + · · ·
and
f(x) = f(0) + xf ′ (c).
One could apply the mean value theorem to f ′ (x) in
Then there exists d, 0 ≤ d ≤ x such that
f ′ (x) = f ′ (0) + f ′′ (0)x + 1 2! f ′′′ (0)x 2 + · · ·
xf ′′ (d) = f ′′ (0)x + 1 2! f ′′′ (0)x 2 + · · ·
Integrating
1
2 x2 f ′′ (d) = 1 2! f ′′ (0)x + 1 3! f ′′′ (0)x 3 + · · ·
Substituting into Eq(12.3)
f(x) = f(0) + f ′ (0)x + 1 2 x2 f ′′ (d).
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