PG demo Content

Question 1

A smooth function \(f\) of one variable fulfills that \(\,f(2)=1\,\) and \(\,f'(2)=1\,.\) Its approximating second-degree polynomial with development point \(\,x_0=2\,\) fulfills \(\,P_2(1)=1\,.\) Determine \(\,P_2(x)\,\).

Hint for question

In this exercise we are testing in particular the following differentiation rules:

• The product rule used when differentiating the product of two functions.

• The chain rule used when differentiating a composite function (“a function within a function”).

\[\begin{gather*} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather*}\]

Answer

\[ \int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15} \]

We will compute the volume of the parallelepiped that is spanned by three geometric vectors in a standard coordinate system in 3D space

Important

Remember to readTheorem 10.54 in eNote 10.

Question 2

A smooth function \(f\) of one variable fulfills that \(\,f(2)=1\,\) and \(\,f'(2)=1\,.\) Its approximating second-degree polynomial with development point \(\,x_0=2\,\) fulfills \(\,P_2(1)=1\,.\) Determine \(\,P_2(x)\,\).

Hint for question

In this exercise we are testing in particular the following differentiation rules:

• The product rule used when differentiating the product of two functions.

• The chain rule used when differentiating a composite function (“a function within a function”).

\[\begin{gather*} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather*}\]

Answer

\[ \int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15} \]