Determine the Taylor polynomial of degree
4
{\displaystyle {}4}
R
⟶
R
,
x
⟼
sin
x
cos
x
,
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin x\cos x,}
at the zero point.
Determine all the Taylor polynomials of the function
f
(
x
)
=
x
4
−
2
x
3
+
2
x
2
−
3
x
+
5
{\displaystyle {}f(x)=x^{4}-2x^{3}+2x^{2}-3x+5\,}
at the point
a
=
3
{\displaystyle {}a=3}
Let
∑
n
=
0
∞
c
n
(
x
−
a
)
n
{\displaystyle {}\sum _{n=0}^{\infty }c_{n}(x-a)^{n}}
f
(
k
)
(
a
)
{\displaystyle {}f^{(k)}(a)}
Let
p
∈
R
[
Y
]
{\displaystyle {}p\in \mathbb {R} [Y]}
g
:
R
+
⟶
R
,
x
⟼
g
(
x
)
=
p
(
1
x
)
e
−
1
x
.
{\displaystyle g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto g(x)=p{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}.}
Prove that the derivative
g
′
(
x
)
{\displaystyle {}g'(x)}
g
′
(
x
)
=
q
(
1
x
)
e
−
1
x
,
{\displaystyle {}g'(x)=q{\left({\frac {1}{x}}\right)}e^{-{\frac {1}{x}}}\,,}
where
q
{\displaystyle {}q}
We consider the function
f
:
R
+
⟶
R
,
x
⟼
f
(
x
)
=
e
−
1
x
.
{\displaystyle f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto f(x)=e^{-{\frac {1}{x}}}.}
Prove that for all
n
∈
N
{\displaystyle {}n\in \mathbb {N} }
n
{\displaystyle {}n}
f
(
n
)
{\displaystyle {}f^{(n)}}
lim
x
→
0
f
(
n
)
(
x
)
=
0
.
{\displaystyle {}\operatorname {lim} _{x\rightarrow 0}\,f^{(n)}(x)=0\,.}
Determine the Taylor series of the function
f
(
x
)
=
1
x
{\displaystyle {}f(x)={\frac {1}{x}}\,}
at point
a
=
2
{\displaystyle {}a=2}
4
{\displaystyle {}4}
4
{\displaystyle {}4}
2
{\displaystyle {}2}
Determine the Taylor polynomial of degree
3
{\displaystyle {}3}
f
(
x
)
=
x
⋅
sin
x
{\displaystyle {}f(x)=x\cdot \sin x\,}
at point
a
=
π
2
.
{\displaystyle {}a={\frac {\pi }{2}}\,.}
Let
f
:
R
⟶
R
,
x
⟼
f
(
x
)
,
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),}
be a differentiable function with the property
f
′
=
f
and
f
(
0
)
=
1.
{\displaystyle f'=f{\text{ and }}f(0)=1.}
Prove that
f
(
x
)
=
exp
x
{\displaystyle {}f(x)=\exp x}
a
∈
R
{\displaystyle {}a\in \mathbb {R} }
Determine the Taylor polynomial up to fourth order of the inverse of the sine function at the point
0
{\displaystyle {}0}
remark .
Hand-in-exercises
Find the Taylor polynomials in
0
{\displaystyle {}0}
4
{\displaystyle {}4}
f
:
R
⟶
R
,
x
⟼
sin
(
cos
x
)
+
x
3
exp
(
x
2
)
.
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin {\left(\cos x\right)}+x^{3}\exp {\left(x^{2}\right)}.}
Discuss the behavior of the function
f
:
[
0
,
2
π
]
⟶
R
,
x
⟼
f
(
x
)
=
sin
x
cos
x
,
{\displaystyle f\colon [0,2\pi ]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin x\cos x,}
concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.
Discuss the behavior of the function
f
:
[
−
π
2
,
π
2
]
⟶
R
,
x
⟼
f
(
x
)
=
sin
3
x
−
1
4
sin
x
,
{\displaystyle f\colon [-{\frac {\pi }{2}},{\frac {\pi }{2}}]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin ^{3}x-{\frac {1}{4}}\sin x,}
concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.
Determine the Taylor polynomial up to fourth order of the natural logarithm at point
1
{\displaystyle {}1}
remark
from the power series of the exponential function.
![{\displaystyle {}p\in \mathbb {R} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c31604bad592d5f545212100558efcdb81e66835)
![{\displaystyle f\colon [0,2\pi ]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin x\cos x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0634f89c847364eaa6f2a291345fa99a4bfde03a)
![{\displaystyle f\colon [-{\frac {\pi }{2}},{\frac {\pi }{2}}]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=\sin ^{3}x-{\frac {1}{4}}\sin x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7713929cf399cf4357dc45d82dc22e71615529)
