Homework Assignment 2

Homework Assignment 2#

Deadline: March 23, 2025. Your answer to the assignment must be written by hand and uploaded as a pdf to “Assignments” on Learn. Read the rules here: Homework Assignments. You may use a calculator or similar tools for simple calculations, e.g., calculating \(P_4(1/2)\).

Remember, all answers must be justified and well-reasoned.

Exercise 1: Computation of Cosine Values#

In Python the value of e.g. \(\cos(\sqrt{17})\) is computed approximatively with the call:

import math

print(math.cos(math.sqrt(17)))

But how does a computer actually find this approximation? A computer either uses a precomputed lookup table or approximating polynomials or a mix. In this exercise we will study how the computer would carry this out using Taylor polynomials.

We consider a function \(f:\mathbb{R} \rightarrow \mathbb{R}\) given by the expression:

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

Question a#

Determine the approximating polynomial \(P_4 = P_{4,f,x_0}\) of degree (no higher than) 4 with expansion point \(x_0 = 0\) for the cosinus function.

Question b#

Use the polynomial \(P_4\) you have found in the previous question to give an approximated value of \(\cos(1/2)\), and estimate how far the found value is from the exact value.

Question c#

Let \(I = [-\pi/2, \pi/2]\). Use Theorem 4.3.3 to show that

\[\begin{equation*} \lvert \cos(x) - P_4(x) \rvert \le \frac{(\pi/2)^5}{5!} < \frac{2^5}{5!} \end{equation*}\]

for all \(x \in [-\pi/2, \pi/2]\). Explain in your own words what the number

\[\begin{equation*} \max_{x \in [-\pi/2, \pi/2]} \lvert \cos(x) - P_4(x) \rvert \end{equation*}\]

describes.

Question d#

Show that the \(K\)’th-degree Taylor polynomial \(P_K\) with the expansion point \(x_0 = 0\) for the cosinus function fulfills

\[\begin{equation*} \lvert \cos(x) - P_K(x) \rvert < \frac{2^{K+1}}{(K+1)!} \end{equation*}\]

for all \(x \in [-\pi/2, \pi/2]\). Determine \(K\) such that the error is smaller than \(10^{-10}\). By “error” we mean the numerical difference between \(P_K(x)\) and \(\cos(x)\) for \(x \in [-\pi/2, \pi/2]\).

Hint

You do not need to calculate \(P_K(x)\).

When you are trying to determine \(K\), it is enough to look at the right-hand side \(\frac{2^{K+1}}{(K+1)!}\). You might need a calculator or similar tool.

Question e#

We can now with fairly good accuracy compute \(\cos(x)\) for any \(x \in [-\pi/2, \pi/2]\) by use of the Taylor polynomial \(P_K\) with expansion point \(x_0 = 0\) (for a fixed, but sufficiently large chosen \(K\)). But how do we calculate an approximated value of \(\cos\) on the rest of the real line, meaning when \(x\) is outside the interval \([-\pi/2, \pi/2]\)?

In order to answer this question we let \(x \in [-\pi/2, \pi/2]\) be given and assume that the value of \(\cos(x)\) is known (i.e. calculated approximately as above). Explain how you for an arbitrarily given \(y \in \mathbb{R}\) now can find the value of \(\cos(y)\).

Hint

You only need some properties of the cosine function. Draw the graph of \(\cos\) over \([-2\pi,2\pi]\). Use the fact that \(\cos\) is periodic and “odd” (not even) around \(x = \pi/2\).

Exercise 2: L’Hôpital’s Rule and Taylor’s Limit Formula#

L’Hôpital’s rule is used for determination of limits of the type \(\lim_{x \to a} \frac{f(x)}{g(x)}\) where \(f(a)=0\) and \(g(a)=0\). The limit value is in this case said to be of the type \(\frac{0}{0}\). The rule is stated as follows:

Theorem (L’Hôpital’s Rule)

Let \(I \subseteq \mathbb{R}\) be an open interval, let \(a \in I\), and let \(f\) and \(g\) be differentiable on \(I\setminus \{a\}\). Assume the following:

  1. \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\)

  2. \(g'(x) \neq 0\) for \(x \in I \setminus \{a\}\)

  3. The limit \(\lim_{x \to a} \frac{f'(x)}{g'(x)}\) exists

Then it holds that:

\[\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}. \end{equation*}\]

Consider the limit:

\[\begin{equation*} \lim_{x \to 0} \frac{\ln(1 + x)}{1 - \sqrt{1 + x}}. \end{equation*}\]

Question a#

Show that this limit is of the type \(\frac{0}{0}\).

Question b#

Check that \(g'(x) \neq 0\) for \(x\) close to \(0\), so that we can use the rule.

Question c#

Use L’Hôpital’s rule to determine the limit value.

Question d#

Determine this same limit limit using Taylor’s limit formula for \(\ln(1 + x)\) and \(\sqrt{1 + x}\) up to a fitting order.

Question e#

Prove L’Hôpital’s rule using Taylor’s limit formula for general functions \(f(x)\) and \(g(x)\) that fulfill the assumptions in the theorem.

Hint

Use a Taylor expansion of \(f(x)\) and \(g(x)\) around \( x = a \) up to the first order with epsilon functions:

\[\begin{equation*} f(x) = f(a) + f'(a)(x-a) + \epsilon_f(x-a)(x-a), \end{equation*}\]

and

\[\begin{equation*} g(x) = g(a) + g'(a)(x-a) + \epsilon_g(x-a)(x-a). \end{equation*}\]

Update March 12

You may assume that \(g'(a) \neq 0\). This is often the case (for example, in our application), and it simplifies the proof.

Hint

Use what you know about \(f(a)\) and \(g(a)\).

Exercise 3: Image Set of a Continuous, Differentiable Function#

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

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

\[\begin{equation*} f(x_1 ,x_2) = x_1^2​ + x_2^2 + x_1​ + 1. \end{equation*}\]

State the image set of \(f\).

Exercise 4: Local Extrema#

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

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

Question a#

Find all stationary points for \(f\).

Question b#

State for each stationary point whether it is a local maximum, a local minimum or a saddle point.

Question c#

For each stationary point, plot the function in Python along with the approximating polynomial \(P_{2}\) of degree (no high than) 2 expanded from the stationary point.

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