From Förberedande kurs i matematik 1

Theory

Exercises

5.1 Individual assignment and group project

When you have answered correctly all questions in both the basic and the final tests, you should start to work with your individual assignment. A link to this can be found in the "Student Lounge."

After you submit your assignment, you will automatically be grouped together with three other students in the class who have recently submitted their assignments. Within this group you will be able to review each other's assignments, and your group will have access to its own discussion forum.

Your group's task is to look at and discuss all the individual submissions by members of the group, and then agree on a common 'best' solution to each of the individual projects. Your group should then make a joint submission. This, together with an account of the contributions of each member of the group, will make up your group assignment.

The objective here is that you discuss your work with the other members of your group, and work together to decide what should be included in your final submission. It is this final submission that will be reviewed and commented on by your teacher. The teacher will communicate with the entire group, and if there is anything that you have missed, you will have the opportunity to make any necessary changes and re-submit your project.

To pass the group assignment you need to participate actively in the project, for example by asking questions and helping with work on the final submission. The individual submissions do not have to be perfect as it is within the group that the final solutions are worked out. If you get stuck, or are uncertain of something, you can always send questions to your teacher who will help you. Our goal is for everyone to pass all parts of the course - and that in doing so you should feel well prepared for your university studies.

5.2 Mathematical text

Explain your solution

The main advice is: Explain your solution.

The solution must not only be a statement of the formulas that you used, but also a description of how you reasoned. Use words to do this! To achieve the appropriate level of the solution: Imagine you are explaining the solution for a class mate who has difficulty in keeping up with all the steps. You need not explain every little calculation but nonetheless do not skip over important steps. If you just follow the above advice, you will have done 80% of what is required to provide an adequate solution.

Write good English

Although this is not an assignment in English and, of course, it is the mathematical content that is the most important, you should nonetheless think about things like typos, grammatical errors, etc. If your solution has too many language errors it can give a very negative impression and affect the credibility of the solution. Impression is important!

Treat formulas as part of the text

It is important that you write your solution in a way that makes it easy for others to follow. To help you we will present some don'ts and dos below to illustrate some tips and common errors that can occur when you mix formulas and text.

Formulas should not be seen as something that is extraneous to the text (or vice versa), but both text and formulas are to be integrated together in a linear flow. Therefore don´t write the text inside brackets behind the formulas. Instead, write the explanatory text on the previous line.

Formulas can be either written as part of the text or as separate formulas. When formulas are separated from the text they appear on their own line and are either centered or slightly indented.

As a formula is to be part of the text, it must be treated as part of the sentence. Think therefore about the punctuation. For example, do not forget the full stop after the formula.

A common mistake is to use a colon in front of all displayed formulas.

A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You need seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.

Sometimes one wants to refer back to a separate formula or equation, and in this case it can be given a number (or star) in brackets in the right or left margin.

5.2 Advice

Make a clean copy of the solution

After you have solved the problem, you should rewrite the solution. Then you can rather concentrate on the presentation of the solution and this even may lead to improving your original solution. A tip is to ask someone else to read your solution to detect ambiguities. It is better to postpone the presentation phase for a later date so that when you solve the problem the first time you are able to work freely and not commit yourself too early to a specific way to solve the problem. When you enter the solution create a text document and not as screen dumps from your word processor. While it may be easier to write the solution on your own computer using your favorite program, you should remember that in the next phase your solution is to be included as part of a group project and thus it is important that the solution is in editable state.

A clear answer

Write a clear answer at the end. This is especially important if the solution is long and the answer is scattered in various parts of the text. However, there are problems where the actual solution constitutes the answer (e.g. "Show that ...") and of course no separate answer at the end is required.

Simplify the answer as far as possible.

Try and check sub-steps and answers

It can happen that when you solve some equations so called spurious roots turn up as a consequence of the method of solution that is being used. In these cases, explain why spurious roots may have appeared and test the solutions to see which are real solutions and which are spurious roots.

An important part of the solution is to think out reasonable methods to check the answer. For example, inserting the solution of the equation and make sure that it really is a solution because one may well have calculated wrongly (do not confuse this with the investigation of spurious roots). This may also be done for the sub-steps in a solution. Another point is to assess if the answer is reasonable. Insert values for some of the parameters and ensure that you get the right answer (what happens if \displaystyle a = 0, \displaystyle a = 1 or a goes to infinity?)

Draw clear figures

A figure may explain introduced symbols or reasoning many times better than text, so please use figures. Bear in mind to draw them clearly and do not overload a figure with too many things. It may be better to have several nearly identical figures where each illustrates one point rather than have a great combination-figure which illustrates everything.

5.3 Common errors

Be careful with arrows and similarities

There is a difference between \displaystyle \Rightarrow (implication arrow), \displaystyle \Leftrightarrow (equivalence arrow) and \displaystyle = (equals sign). For two equations that are known a priori to have the same solutions one uses the equivalence arrow \displaystyle \Leftrightarrow to represent this.

However, if we write Equation 1 \displaystyle \Rightarrow Equation 2, it means that all solutions that Equation 1 has, Equation 2 also has, (but Equation 2 may have more solutions).

One often does not bother to write the symbol \displaystyle \Leftrightarrow between the different steps in a solution when they are on different lines (and thus the equivalence is implied). It is also often better to use explanatory text instead of arrows between the different steps in the solution. Do not use the implication arrow as a general continuation symbol (in the sense "The next step is").

The equal sign (\displaystyle =) is commonly used in two senses, firstly between things that are identical, eg \displaystyle (x - 2)^2 = x^2-4x + 4 which is true for all \displaystyle x, and secondly in equations in which both sides are equal for some \displaystyle x, such as \displaystyle (x - 2) ^2 = 4, which only is satisfied if \displaystyle x = 0 or \displaystyle x = 4. You should not mix these two different uses of the same symbol.

(There is also a third use of the equals sign, which occurs when defining an expression or for example an operation.)

Simple arrow (\displaystyle \rightarrow) is used in mathematics often to handle different kinds of limits: \displaystyle a \to \infty means that a increases without limit (goes towards infinity). You will probably not need to use a simple arrow in this course.

Do not be careless with parenthesis

Since multiplication and division have higher priority than addition and subtraction, one must use brackets when addition and/or subtraction is to be carried out first.

When dealing with algebraic expressions one usually omits the multiplication sign. For example, one almost never would write \displaystyle 4\times x\times y\times z but rather \displaystyle 4xyz.

This omission of the multiplication gives precedence over other multiplication and division (but not exponentiation). When one therefore writes \displaystyle 1/2R it means \displaystyle 1 / (2R) and not \displaystyle (1 / 2) R. Since this can be a source of misunderstanding, it is not entirely unusual to print the brackets in both situations.

Arguments to the basic elementary functions are written without parentheses. Therefore, you should not write

\displaystyle \cos (x), \displaystyle \sin (x), \displaystyle \tan (x), \displaystyle \cot (x), \displaystyle \lg (x) and \displaystyle \ln (x)

but

\displaystyle \cos x, \displaystyle \sin x, \displaystyle \tan x, \displaystyle \cot x, \displaystyle \lg x and \displaystyle \ln x.

In fact you should write \displaystyle \cos 2x and not \displaystyle \cos (2x) (since the argument \displaystyle 2x is tightly linked together via a juxtaposition), but brackets are necessary when you write \displaystyle \sin (x + y); \displaystyle \sin(x / 2) or \displaystyle (\sin x)^2 (which you, alternatively, can write as \displaystyle \sin ^2\!x).

Beware of the following common mathematical errors

Spurious roots (squaring can give false roots, multiplying equations with factors containing \displaystyle x can give false roots, roots outside the functions definition volume are false). ...

Lost solutions. Eg a factor on both sides of the equation is cancelled out and one does not realize that the equation obtained by setting this factor to zero provides additional solutions.

5.4 Writing formulas in TeX

Common mistakes

Miscellaneous

• Forgotton <math> and </math> tags.

• No \ in front of commands. Beware that command must be written with a backslash (\) in front, so you should write \sin x and not sin x.

• No space after command. For example, \sinx instead of \sin x.

• Writing * instead of \displaystyle \times (\times) for the multiplication sign.

Exponents and indices

• Leaving out the braces around the exponent or index. Instead of writing a^22 and a_22 you should write a^{22} and a_{22}. Leaving out the braces only works when the exponent or index is one symbol.

• Dropping ^ for exponents and _ for indices. Don't write \sin2 2x (\displaystyle \sin2 2x) when you really mean \sin^2 2x (\displaystyle \sin^2 2x), and x1 (\displaystyle x1) when you mean x_{1} (\displaystyle x_{1}).

• The degree sign is written 30^{\circ} (\displaystyle 30^{\circ}) and not 30^{o} (\displaystyle 30^{o}) or 30^{0} (\displaystyle 30^{0}).

Brackets

• Forgotton end bracket. Make sure brackets, braces and parenteses are balanced: (1-(1-x) should be written (1-(1-x)).

• Using parenthesis instead of braces in commands. It is wrong to write \sqrt(a+b) and \frac(a)(b). You should write \sqrt{a+b} and \frac{a}{b}. Don't overdo and write \sqrt{(a+b)} (\displaystyle \sqrt{(a+b)})

• Wrongly sized parenthesis. If the expression a pair of parenthesis enclose is large you should use extensible parentheses. Write \left(\dfrac{a}{b}+c\right) (\displaystyle \left(\dfrac{a}{b}+c\right)) and not (\dfrac{a}{b}+c) (\displaystyle (\dfrac{a}{b}+c)).

Fractions

• Wrongly sized fractions. A rule of thumb is that fractions where the numerator and denominator are a few digits whould be written as a small fraction (i.e. with \frac), while other fractions should be large (i.e. with \dfrac).

Therefore, \frac{1}{12} is ok, but \frac{1}{12} is probably not ok (unless surrounding expressions are equally large). You should write \dfrac{a+b}{c} and not \frac{a+b}{c}; if the vetical space is tight you can write (a+b)/c (don't forget the parenthesis!).

• Fractions in exponents and indices are not so great. The expressions a^{\frac{b}{c}} and a_{\frac{b}{c}} are better written as a^{b/c} and a_{b/c}.