Beta. Content is under active construction and has not been peer-reviewed. Report errors on
GitHub
.
Disclaimer
Theorem
Path
Curriculum
Paths
Demos
Diagnostic
Search
Quiz Hub
/
Taylor Expansion
Taylor Expansion
2 questions
Difficulty 4-4
View topic
Intermediate
0 / 2
2 intermediate
Adapts to your performance
1 / 2
intermediate (4/10)
state theorem
The Taylor expansion of
f
around
x
0
is
f
(
x
)
=
f
(
x
0
)
+
f
′
(
x
0
)
(
x
−
x
0
)
+
2
1
f
′′
(
x
0
)
(
x
−
x
0
)
2
+
…
. For the Lagrange form of the remainder, what is the key statement?
Hide and think first
A.
The remainder after
n
terms is bounded by the maximum of
f
(
n
+
1
)
on the interval times
(
x
−
x
0
)
n
+
1
/
(
n
+
1
)!
B.
The remainder is always zero if
f
is infinitely differentiable, so the Taylor series converges to
f
everywhere
C.
The remainder can be expressed as an integral of
f
(
n
)
times a polynomial kernel, which is unique and independent of
ξ
D.
The remainder decays exponentially with the order
n
, making Taylor series accurate for arbitrary
x
in the domain
Submit Answer