Aus Online Mathematik Brückenkurs 1

Theorie

Übungen

Hinweise

Erläutern Sie ihre Lösung

Der wichtigste Hinweis ist:

Erläutern sie stets hinreichend ihre Lösung.

Die Lösung eines Problems oder einer Aufgabe darf nicht nur darin bestehen, die verwendete Formel zu erwähnen, sondern muss eine Beschreibung enthalten, wie gedacht wurde. Verwenden Sie Worte, um dies zu tun! Stellen Sie sich vor, Sie würden die Lösung einem Klassenkameraden erklären, der Schwierigkeiten damit hat, die einzelnen Schritte zu begreifen. Sie brauchen nicht jede kleine Rechnung zu erklären, jedoch dürfen Sie keinen wichtigen Schritt auslassen. Wenn Sie diesen Rat befolgen, werden Sie 80% dessen erreicht haben, was nötig ist, um eine angemessene Lösung zu liefern.

Schreiben Sie gutes Deutsch

Obwohl dies keine Hausaufgabe im Fach Deutsch ist und der mathematische Inhalt selbstverständlich am wichtigsten ist, sollten Sie trotzdem stets auf Ausdrucksweise und grammatikalische Sauberkeit etc. achten. 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!

Make a clean copy of the solution

After you have solved the problem, you should rewrite the solution. Then you can concentrate on its presentation; this even may lead to improvements in the solution itself. 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 solution method.

When you enter the solution, be sure to enter it as text, rather than (say) a screen capture from a word processor. It may be easier to write the solution on your own computer using your favourite program, but 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 can be edited, which a screen capture cannot.

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 then 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.

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.

An important part of the solution process is to think of reasonable methods to check the answer. For example, one might substitute the solution of an equation back into that 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. Eg 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 details. 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.

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.

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

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 a formula if it ends a sentence.

A bad habit is excessive numbering. For example, to put a number in front of each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You seldom need to 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.

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 brackets

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).