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Taylor polynomials
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Taylor Polynomial Remainder
3 Activities
Find the degree 4 Taylor polynomial for
f
(x) = sin(3
x
) centered at
x
=
π
/
6.
x=\pi/6.
x
=
π
/6.
What is the lowest error bound within
[
0
,
π
/
6
]
[0,\pi/6]
[
0
,
π
/6
]
?
E
=
243
π
6
6
6
6
!
E=\frac{243\pi^6}{6^6 6!}
E
=
6
6
6
!
243
π
6
E
=
−
243
π
6
6
6
6
!
E=-\frac{243\pi^6}{6^6 6!}
E
=
−
6
6
6
!
243
π
6
E
=
243
π
6
(
36
)
(
6
!
)
E=\frac{243\pi^6}{(36) (6!)}
E
=
(
36
)
(
6
!)
243
π
6
E
=
−
243
π
6
(
36
)
6
!
E=-\frac{243\pi^6}{(36) 6!}
E
=
−
(
36
)
6
!
243
π
6
I don't know
Check Submission
More Taylor Polynomial Remainder Questions:
Let
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
f(x)=\cos{(3x)}+x-1
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
. Approximate
f
(
π
3
+
1
5
)
f(\frac{\pi}{3}+\frac{1}{5})
f
(
3
π
+
5
1
)
using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Use the second order Taylor polynomial for
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
to approximate
17
\sqrt{17}
17
. What is a good bound on the error in this approximation?
Estimate the size of the error made in the linear approximation to estimate
99.9
\sqrt{99.9}
99.9
.
Let
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
f(x)=\cos{(3x)}+x-1
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
. Approximate
f
(
π
3
+
1
5
)
f(\frac{\pi}{3}+\frac{1}{5})
f
(
3
π
+
5
1
)
using linear approximation. Is this overestimation or underestimation of the actual value?
Find the second order Taylor polynomial
T
2
(
x
)
T_2(x)
T
2
(
x
)
for
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
. Use this to approximate
17
\sqrt{17}
17
. What is a good bound on the error in this approximation?
Consider
f
(
x
)
=
1
+
x
f(x) = \sqrt{1+x}
f
(
x
)
=
1
+
x
.
Concept Clarifier
Find the fifth order Taylor polynomial for
y
=
sin
x
y=\sin\ x
y
=
sin
x
centred about
x
0
=
π
x_0=\pi
x
0
=
π
. What is the minimum error bound for such Taylor polynomial in the interval
x
∈
[
0
,
3
π
]
x \in [0,3\pi]
x
∈
[
0
,
3
π
]
?
If
f
(
x
)
=
e
−
2
x
+
3
x
2
f(x)=e^{-2x}+3x^2
f
(
x
)
=
e
−
2
x
+
3
x
2
approximate
f
(
0.1
)
f(0.1)
f
(
0.1
)
? Find the maximum error for approximating positive numbers?
Find the second order Taylor polynomial
T
2
(
x
)
T_2(x)
T
2
(
x
)
for
f
(
x
)
=
x
f(x)=\sqrt{x}
f
(
x
)
=
x
at
x
=
16
x=16
x
=
16
. Use this to approximate
17
\sqrt{17}
17
. What is a good bound on the error in this approximation?
Let
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
f(x)=\cos{(3x)}+x-1
f
(
x
)
=
cos
(
3
x
)
+
x
−
1
. Approximate
f
(
π
3
+
1
5
)
f(\frac{\pi}{3}+\frac{1}{5})
f
(
3
π
+
5
1
)
using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Consider
f
(
x
)
=
1
+
x
f(x) = \sqrt{1+x}
f
(
x
)
=
1
+
x
.