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Exercise 1: Approximating Polynomials. By Hand

A

Find for each of the following functions their approximating polynomials of first and second degree with development point $x_0=0\,.$

1.$ $ $f(x)=\e^x\,,\,\,x\in \Bbb R\,$

2.$ $ $f(x)=\cos (x)\,,\,\,x\in \Bbb R\,$

3.$ $ $f(x)=\e^{\sin( x)}\,,\,\,x\in \Bbb R\,$

%\item $f(x)=\sqrt{x+1}$, %\item $f(x)=\frac{1}{1-x}$, %\item $f(x)=\frac{1}{\sqrt{1-x}}$, %\item $f(x)=\sqrt{1-x^2}$, %\item $f(x)=\cosh (x)$.

B

The function

$$\,\displaystyle{f(x)=\frac 1 x\,,\,\,x>0}\,$$

cannot of course be expanded in Taylor polynomial from the development point $x_0=0\,.$ So in stead you are asked to determine the approximating polynomial of first and second degree for $\,f\,$ with the development point $x_0=1\,.$

Exercise 2: Introduction to Maple

The mathematcs program Maple is one of several mathematics program that are in use in education and research at DTU . Here is a short introduction to the most important aspects. It is important you use some time to get aquainted with Maple, because you shall use it a lot in Math 1.

Where Do I Get Maple You download Maple to your own computer from this site. Log in with your DTU-ID, find Maple2020 , and choose the right installation file for your operating system. Note the installation key (Activation key Stand alone), since you will need this when you install the program.

Configuration of Maple When you start using Maple, there are two options you must consider: 1) Do I wish to work in worksheet-mode or in document-mode, and 2) Do I wish to write command lines using MapleNotation or 2D-Math Notation. In the education at DTU the combination worksheet-mode and MapleNotation is used. If you configure your Maple to this combination, which e.g. are used in the MapleDemos in the course, you shall follow this recipe:

The First Time I Use Maple

  • Go to Tools $\rightarrow$ Options $\rightarrow$ Interface $\rightarrow$ Default format for new worksheets and choose Worksheet. End by Apply Globally.
  • Go to Tools $\rightarrow$ Options $\rightarrow$ Display $\rightarrow$ Input display and choose Maple Notation. End by Apply Globally.
  • Restart Maple – now we are ready! $ $

The Weekly Test Uses Maple Notation When you enter answers in the Weekly test you must use the MapleNotation. Knowledge of the MapleNotation is an important part of the Maple Maple-curriculum of the course! Of the highschools that have started to use Maple, many recommend 2D-Math Notation in stead of MapleNotation. One of the advantages, that it is easier to use palettes and right-click-menus. At DTU the raw text code, i.e. MapleNotation, is considered to be more scientific, since the documentation of is more clear. You are of course wellcome to experiment with your own Maple-configuration.

Pixi Tutorial in Maple Start Maple. Maple functions according to the principle with questions (input) and answers (output): You ask questions on a command line (to the right of the red > sign) as input to Maple, and Maple answers with output, centered and in blue such that answers can clearly be distinguished from questions. If you want Maple to accept an input, but without showing the answer, you write a colon after the command.

As the very first input in every Maple-sheet you write > restart; Maple execute the command, when you press Enter. This command will initialize the memory of the sheet, and since you often execute all the commands in the sheet one after the other more times, it is necessary to initialize at the top of the sheet. Note that there is no answer from Maple regardless whether or not you end the command by a colon.

Now try the following calculation: > 2 + 2; Maple can of course function as a calculator! But the more intersting qualities of the program lie in its ability to execute symbolic operations. E.g. you differentiate $\sin x$ with respect to $x$ with the command > diff(sin(x) , x); Try this. In this case you use the command diff. To this command two arguments: sin(x) and x, separated by a comma, belong.

Maple can also plot functions. The simplest plot-command is plot. The command has as at least two arguments: the expression that is to be plotted and the limits of the independent variable. If you e.g. wish to see the function $\sin x$ in the interval between 0 and 5, you use the command > plot(sin(x), x=0..5); Intervalls are always stated using two consecutive punctuation marks. To get the same units on the axes you write in addition scaling=constrained as the third argument, that is: > plot(sin(x), x=0..5, scaling=constrained); Can you see the difference? there is a plentitude of optional argument and possibilities for combinations to the plot-command. To find the one that best suits your needs, you can either use Maples help function in the menu line \includegraphics[scale=.8]{maplehelp.PNG}, or you can write > ? plot; To write a question mark in front works for all commands. Here is a further example of a plot, now with more functions and more arguments. Try to guess what they all do or look up in the help function of Maple under the plot-command. > plot([sin(x),x^2],x=0..5,y=-2..2,color=[red,blue],scaling=constrained); You can make powers by pressing $ \wedge $ key.

Besides: Use Maple’s help function. It is awesome!

Exercise 3: Illustrations of Taylor Approximations

A

Plot some of the four functions from Exercise 1 together with their approximating polynomials of first and second degree.

Exercise 4: Investigations of Approximating polynomials. Enjoy!

A

Use Maple to find the approximating polynomial of degree 9, $P_9(x)$, with the development point $x_0=0$ for the function $\sin(x)$. Draw in the same coordinate system $\sin( x)$ and $P_9(x)$. How far out to the sides can you get the approximating polynomial to follow the function? (Eksperiment with the degree of the the polynomial).

Exercise 5: Assessment of Errors by Approximation

A function $\,f:\Bbb R\rightarrow \Bbb R\,$ is given by

$$f(x)=\sqrt{2x-1}.$$
A

Determine the domain Dm$(f)\,$ for $\,f\,.$

B

Determine the approximating polynomial $\,P_3(x)\,$ of degree $3$ for $\,f\,$ with the development point $\,x_0=1\,.$

C

Explain that the remainder function $\,R_3(x)\,$ corresponding to $\,P_3(x)\,$ can be expressed by

$$ R_3(x)=-\frac 58\cdot \frac 1{(2\xi-1)^{7/2}}\cdot (x-1)^4 $$

for a $\,\xi\,$ between $\,1\,$ and $\,x\,.$ Show by assessment of the remainder function that the numerical value of the error you commit by using $\,\displaystyle{P_3\left(\frac 32\right)}\,$ in stead of $\,\displaystyle{f\left(\frac 32\right)}\,$ is less than or equal to $\,\displaystyle{\frac 5{2^7}}\,.$

Exercise 6: Introduction to complex Numbers with Maple

Download and work through the MapleDemo about complex numbers. In the demo the relevant maple commands for this exercise is presented. In this exercise you shall use Maple to solve the following exercises. You have earlier solved similar exercises using paper and pencil, reflect on the answers given by Maple. Do you have the complete answer or should you elaborate on the output from Maple.

A

What is $i^2$, $i^3$, $(-i)^4$ og $(-i)^{-5}\,?$

B

Determine the real part and the imaginary part of

$$\frac{-2+3i}i$$

and write the result in rectangular form.

C

Given $w=1-i\,$. 1.$ $ Determine $|\,w\,|$ and $\arg(w)\,$. 2.$ $ Determine $|\,\e^w\,|$ and $\arg(\e^w)\,$.

D

Write the following complex numbers in rectangular form: 1.$ $ $\e^{i\frac{\pi}{2}}$ 2.$ $ $3\e^{1+\pi i}$

E

Find the solutions to the equation

$$z^2+(2+2i)z-2i=0\,.$$

F

Find for every $\,t\in \reel\,$ the differential quotients of the following functions:

$$ \begin{aligned} f_{1}(t) &= t^{2} + i \, \sin(t) \\\\ f_{2}(t) &= 1+it^5\\\\ f_{3}(t) &= t^5-i\\\\ f_{4}(t) &= 3\, \e^{it}\\\\ f_{5}(t) &= i\, \e^{2t+3it} \end{aligned} $$

Exercise 7: Approximation of Complex Functions of a Real Variable

Approximating polynomials for complex functions of a real variable are stated using the same formula as real functions of a real variable. In what follows we consider the function:

$$f(x)=2\cos(x)+i\,\sin(2x)\,,\,\,x\in \Bbb R\,.$$
A

Determine the approximating polynomial $\,P_3\,$ of degree three at the most for $\,f\,$ with the development point $\,x_0=0\,.$

B

Determine, readily using Maple’s mtaylor, the approximating polynomial $\,Q_3\,$ of degree three at the most for $\,f\,$ with the development point $\,x_1=\frac{\pi}2\,.$

C

The number 1 lies closer to $\,x_1=\frac{\pi}2\,$ than to $\,x_0=0\,.$ Why is it still a much better idea to use $\,P_3\,$ than $\,Q_3\,$ if you need an approximated value for $\,f(1)\,?$

%####### begin:question %Hvilken grad er den mindste et approksimerende polynomium for $\,f(x)\,$ med udviklingspunktet $\,x_0=0\,$ skal have for at afstanden mellem $\,f(1)\,$ %og polynomiets værdi i 1 er mindre end en hundrededel? %####### end:question

%####### begin:answer %$\,P_7\,$ klarer det job. %####### end:answer

Exercise 8: Taylor’s Limit Formula. By Hand

This exercise gives a method for the computation of the limit value of a fractionfor which both the numerator and the denominator goes towards zero.

A

State for the function $\ln(1+x)$ Taylor’s limit formula with the development point $x_{0}=0$ for degree 1, 2 and 3.

B

Which of the result from question a) can not be used to find the limit value:

$$\lim_{x\rightarrow 0}\frac{\ln(1+x)-x}{x^2}$$

C

Can Maple calculate the limit value directly?

D

Advanced: Now compute using Taylor’s limit formula the following limit value:

$$\lim_{x\rightarrow 0}\frac{x(\e^x+1)-2(\e^x-1)}{x^3}.$$