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Differential
calculus - derivatives
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Applications
of differentiation
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Taylor's
theorem (Taylor's
formula) - The extended mean value theorem
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Suppose
f
is continuous
on the closed interval [x0,
x0
+ h]
with continuous
derivatives to
(n
-
1)th
order on the
interval and its nth
derivative defined on
(x0,
x0
+ h)
then, |
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is called Taylor's theorem.
To prove Taylor's theorem we
substitute x0
= a,
x0
+ h = b
and introduce the function |
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where Q
is still undetermined and p
> 0.
Since j
(b) = 0
we
can define
Q
by setting j
(a)
= 0
too, |
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| then
we compute j
' (x)
to apply Rolle's
theorem. Note
that successive terms cancel (by the product rule) when differentiating j
(x), so we get |
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By Rolle's theorem there is some intermediate point c
inside the interval [a,
b]
such that j'
(c)
= 0, |
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which
gives |
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| By
plugging the Q
back into above equation j
(a)
= 0
while returning substitutions, |
| a
= x0,
b
= x0
+ h,
b
-
a
= h
and where |
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obtain Taylor's theorem to be proved. The
last term in Taylor's formula |
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called the remainder and denoted Rn
since it follows after n
terms. |
| By
plugging, a)
p
= n into
Rn
we get |
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the
Lagrange
form of the remainder, |
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while if b)
p
= 1
we get |
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the
Cauchy
form of the remainder. |
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| Thus,
by substituting x0
+ h = x
obtained is |
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| Taylor's
formula with Lagrange form of the remainder. |
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Therefore, Taylor's
formula gives values of a function f
inside the interval [x0,
x0
+ h]
using its value and the values
of its derivatives to
(n
-
1)th
order at the point x0 in the form |
| f (x) =
Pn -
1(x
-
x0) + R n |
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where,
Pn
-
1(x
-
x0)
is
(n
-
1)th
order Taylor polynomial for f
given by the first n
terms in the above
formula and Rn
is one of the given remainders. |
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Maclaurin's
formula or Maclaurin's
theorem
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formula obtained from Taylor's
formula by setting x0
= 0 |
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holds in an open neighborhood of the origin, is called Maclaurin's
formula or Maclaurin's
theorem. |
| Consider
the polynomial
fn(x)
= anxn
+ an -
1xn -
-
-
1 + · · · + a3x3
+ a2x2 + a1x
+ a0, |
| let
evaluate the polynomial and its successive derivatives at the origin, |
| f
(0) =
a0, f '(0) = 1· a1,
f ''(0) = 1· 2a2,
f '''(0) = 1· 2· 3a3, . . .
, f (n)(0) =
n!an |
| we
get the coefficients, |
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| Therefore,
the Taylor polynomial of a function f
centered at x0
is the polynomial of degree n
which has the same
derivatives as f
at x0,
up to order n. |
| If
a
function f
is infinitely differentiable on an interval about a point x0
or the origin, as are for example ex
and sin
x
then, P0 (x)
= f (x0), |
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P1 (x) =
f (x0)
+ (x
-
x0)
f ' (x0), |
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| where
P0,
P1,
P2,
. . . is a sequence of increasingly approximating polynomials for f. |
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Contents
K
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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