- Exercises
Determine all the Taylor polynomials of the function
-
at the point
.
Write the polynomial
-
in the new variable 
, using two different ways, namely
a) directly by inserting,
b) via the Taylor-polynomial in the point 
.
Determine the
Taylor polynomial
of order 
for the function
in the point
.
Determine the Taylor series of the function
-
at point
up to order (Give also the Taylor polynomial of degree at point , where the coefficients must be stated in the most simple form).
Determine the
Taylor polynomial
of degree 
of the
rational function
-
in the point 
.
Determine the Taylor polynomial of degree of the function
-
at the zero point.
We consider the function
-
over the real numbers.
a) Determine the range of 
.
b) Sketch for 
between
and 
.
c) Determine the first three derivatives of .
d) Determine the Taylor-polynomial of order of in the point 
.
Determine the Taylor polynomial of degree of the function
-
at point
-
Let
be a function. Compare the
polynomial interpolation
for 
given point and the
Taylor-polynomials
of degree 
in a point.
Let
be an -fold
differentiable
function in the point 
. Show that the -th
Taylor polynomial
for in the point , written in the shifted variable 
, equals the -th Taylor polynomial of the function
in the zero point.
Let
be a function. Is it possible to get the -th
Taylor polynomial
of in the point 
from the -th Taylor polynomial of in the point .
Let
be polynomials of degree , let
be points and
natural numbers fulfilling
-
Suppose that the derivatives of
and 
coincide in den points 
up to the 
-th derivative. Show
.
Let
.
Determine a polynomial 
of degree 
, with the property that its linear approximation at the points
and
coincide with those of .
Let
.
Determine polynomials 
of degree , fulfilling the following conditions.
(a) 
coincides with at the points 
.
(b) 
coincides with in and in 
up to the first derivative.
(c) 
} coincides with in 
up to the third derivative.
Determine the
Taylor series
of the
der
exponential function
for an arbitrary point
.
Let
![{\displaystyle {}p\in \mathbb {R} [Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c31604bad592d5f545212100558efcdb81e66835)
be a polynomial and
-
Prove that the derivative 
has also the shape
-
where 
is a polynomial.
We consider the function
-
Prove that for all
the -th derivative 
satisfies the following property
-
Determine the
Taylor polynomial
of the third order of the function 
in the zero point, using the power series approach described in
remark *****.
Let
-
Because of
-
this function is on the open interval ![{\displaystyle {}]-1,1[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2192ff64c086e1d4d5d5f5b56cd3411774d7fb)
strictly decreasing and therefore injective
(with the image interval ![{\displaystyle {}]-2,2[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e9da57114a18b548a79a8e59f31526e203a84de)
).
Also,
.
Let
-
be the inverse function, which we want to understand as a power series. Determine from the condition
-
the coefficients 
.
Determine the Taylor polynomial up to fourth order of the inverse of the sine function at the point with the power series approach described in
an remark.
- Hand-in-exercises
Find the Taylor polynomials in up to degree of the function
-
Let
.
Determine a polynomial of degree , which in the two points
and
has the same linear approximation as .
Discuss the behavior of the function
-
![{\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)
concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.
Discuss the behavior of the function
-
![{\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)
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 
with the power series approach described in
remark
from the power series of the exponential function.
For
let 
be the area of a circle inscribed in the unit regular -gon. Prove that
.
