# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 17

*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 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 strictly decreasing and therefore injective (with the image interval ). 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*

### Exercise (4 marks)

Find the Taylor polynomials in up to degree of the function

### Exercise (5 marks)

Let . Determine a polynomial of degree , which in the two points and has the same linear approximation as .

### Exercise (4 marks)

Discuss the behavior of the function

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise (4 marks)

Discuss the behavior of the function

concerning zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise (4 marks)

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.

### Exercise (6 marks)

For let be the area of a circle inscribed in the unit regular -gon. Prove that .

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