In the following exercises, verify that the given choice of n in the remainder estimate | R n | ≤ M ( n + 1 ) ! ( x − a ) n + 1 where M is the maximum value of | f ( n + 1 ) ( z ) | on the interval between a and the indicated point, yields | R n | ≤ 1 1000 . Find the value of the Taylor polynomial P n of f at the indicated point. 131. Integrate the approximation e x ≈ 1 + x + x 2 2 + ... + x 6 720 evaluated at −x 2 to approximate ∫ 0 1 e − x 2 d x .
In the following exercises, verify that the given choice of n in the remainder estimate | R n | ≤ M ( n + 1 ) ! ( x − a ) n + 1 where M is the maximum value of | f ( n + 1 ) ( z ) | on the interval between a and the indicated point, yields | R n | ≤ 1 1000 . Find the value of the Taylor polynomial P n of f at the indicated point. 131. Integrate the approximation e x ≈ 1 + x + x 2 2 + ... + x 6 720 evaluated at −x 2 to approximate ∫ 0 1 e − x 2 d x .
In the following exercises, verify that the given choice of n in the remainder estimate
|
R
n
|
≤
M
(
n
+
1
)
!
(
x
−
a
)
n
+
1
where M is the maximum value of
|
f
(
n
+
1
)
(
z
)
|
on the interval between a and the indicated point, yields
|
R
n
|
≤
1
1000
. Find the value of the Taylor polynomial Pnof f at the indicated point.
131. Integrate the approximation
e
x
≈
1
+
x
+
x
2
2
+
...
+
x
6
720
evaluated at −x2to approximate
∫
0
1
e
−
x
2
d
x
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Mathematics for Elementary Teachers with Activities (5th Edition)
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