Homework Assignment 2

Remember that all answers must be well-reasoned. An answer without reasoning or argumentation will give no points.

1. It is a good idea to create your answer directly as a Jupyter Notebook. You can download the ipynb file directly from this page. Here you can see the markdown syntax and for instance how to write mathematical equations using latex.

2. Your answer to the assignment must be delivered as a PDF file, which must be uploaded to DTU Learn. If you encounter issues with the export of your Jupyter Notebook to PDF, then contact the DTU Python Support.

Problem 1

A function \(f:\mathbb{R} \rightarrow \mathbb{R}\) is given by the expression:

\[\begin{equation*} f(x)=\cos (3x). \end{equation*}\]

a) Determine the approximating polynomial \(P_{4,f,x_0}\) of degree (at most) \(4\) with expansion point \(x_0 = 0\).

b) Use the polynomial you found to compute an approximate value of \(\cos (\frac{1}{2} )\), and estimate how far the found value is from the exact value.

We are given the following information about a function \(g\):

\[\begin{equation*} g(0)=1 \quad \text{and} \quad g^{(n)} (0)= \begin{cases} (-1)^{\frac{n}{2}}2^n & n \text{ is even}, \\ 0 & n \text{ is odd}. \end{cases} \end{equation*}\]

c) Determine the approximating polynomial \(P_{7,g,x_0}\) of \(g\) of degree (at most) \(7\) with expansion point \(x_0 = 0\). Give your best suggestion on which function \(g\) might be.

Problem 2

Let \(B\) be the set \(B=\left\{ (x_1 , x_2) \in \mathbb{R}^2 \mid x_1^2 + x_2^2 \leq 1 \wedge x_1 \geq 0 \right\}\).

A function \(f:B \rightarrow \mathbb{R}\) is given by

\[\begin{equation*} f(x_1 ,x_2 )=x_1^2 x_2^2 +3x_1 x_2 +x_2 -4. \end{equation*}\]

State the image of \(f\).

Problem 3

A function \(f:\mathbb{R}^2 \rightarrow \mathbb{R}\) is given by

\[\begin{equation*} f(x_1 ,x_2 )=x_1^2 - 2x_1 +3x_2^5 - 5x_2^3. \end{equation*}\]

a) Determine all stationary points of \(f\).

b) State whether there is a local maximum, a local minimum, or whether there is a saddle point at the stationary points.

c) Plot for each stationary point the function along with the approximating polynomial \(P_{2}\) of degree (at most) \(2\).

Problem 4

A function \(f:[0,5] \rightarrow \mathbb{R}\) is given by the expression \(f(x)=2x+3\).

a) Determine, using Python, a value of the Riemann sum over the interval \([0,5]\) with 30 subintervals, where firstly the left interval end-points of the subintervals are used. Repeat with the right interval end-points.

b) Determine exact values of the Riemann sum over the interval \([0,5]\) with \(n\) subintervals, where firstly the left interval end-points of the subintervals are used. Repeat with the right interval end-points.

c) Provide an exact expression for the maximal and minimal errors. Both expressions may only depend on \(n\). (We here consider the Riemann sum as an approximative value for the Riemann integral.)

d) Argue in this special case for the fact that the Riemann sum has the same limit value regardless of the choice of point in the subintervals.