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. 130. Integrate the approximation sin t ≈ t − t 3 6 + t 5 120 − t 7 5040 evaluated at π t to approximate ∫ 0 1 sin π t π t d t .
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. 130. Integrate the approximation sin t ≈ t − t 3 6 + t 5 120 − t 7 5040 evaluated at π t to approximate ∫ 0 1 sin π t π t d t .
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.
130. Integrate the approximation
sin
t
≈
t
−
t
3
6
+
t
5
120
−
t
7
5040
evaluated at
π
t
to approximate
∫
0
1
sin
π
t
π
t
d
t
.
What are the taylor polynomials p4 and p5 centered at pi/6 for f(x)=cos(x)?
Let f(x)=8x+2ln(x). Find a second degree Taylor polynomial about c=1 and use P2(x) to approximate 16+2ln(2) with determining a bound for the error of the approximation.
Compute the Taylor polynomial T5 (x) and use the Error Bound to maximize possible size of the error.
f(x) = cos(x), a=0, x = 0.15
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