In these pages we would like to tell you what we would like our entering students to know. There are a number of topics that you might not have studied. Do not worry! What we outline below is what we consider the core of mathematical culture, and here we emphasize the word culture; it contains many topics that you need for your daily life but also some topics that have a more artistic flavor and should help you to appreciate the beauty of Mathematics. Note that the number of topics below is fairly large. If you have not yet learned some of them, please do not get discouraged.
As we said we do not expect you to know everything, but you must be prepared to learn some of this while being at Georgia Tech in addition to the Calculus material. You may notice that we do not talk about the Calculus in the list of topics below. Calculus in high school is not a required course; we teach it here at Georgia Tech. The most important thing is to know some mathematics to which you can apply Calculus. For instance, if you aren't acquainted with some interesting functions, what is the point in knowing what a derivative is?
General remarks about Mathematics: Mathematics can only be learned by doing it. In this respect it is just like playing a sport. To be good you have to practice, and you have to practice a lot. As a high level skill, you will want to pay attention to many aspects of mathematics. If you have trouble concentrating and you make lots of mistakes (silly or not, they are mistakes), you have trouble getting answers to problems. Math is all about concepts, and you need to think critically about them. Every time you hear a definition, ask yourself why is it useful. Even the idea of making definitions is important in mathematics. It is much more than just giving a name to something. The great applied mathematician James Glimm had a wonderful insight: 'Every definition should be the hypothesis of a good theorem'. The word 'good' can have a number of interpretations, such as that it is useful, or that it is amazing and completely unexpected.
Logic: It is most important that you understand some simple logic. A mathematical statement cannot be true and false at the same time. It can't just be mostly true. In logic, truth is absolute.
You should come to grips with statements like 'if and only if'. What do we mean by this?
The following rule should strike you as reasonable: If the truth of a statement A implies that statement B is true and the truth of statment B implies that statement C is true then A being true implies that C is true.
How about: 'A is true implies that B is true' if and only if 'B is false implies that A is false'.
You should be able to understand what a hypotheses is, what a definition is, what theorems are.
One cannot overstate the importance of understanding definitions. Take, as an example from calculus, the definition of the derivative as a limit of difference quotients. This is it! All the other wonderful rules are derived from this definition. If you get stuck in trying to calculate the derivative you can always go back to that definition.
To stay with the calculus, here is an example which is often misunderstood. What is the definition of an integral? For many the answer is 'The integral of a function is an antiderivative'. Unfortunately this is a bad answer. It is bad not because it is completely wrong but it is bad because it gives the wrong impression of what the integral is all about. If this were really the definition, then why do we have to give the antiderivative another name like the 'integral'? Also, the Fundamental Theorem of Calculus would appear to be a definition, so what is all the fuss about it?
The point is that the integral can be found by making a very difficult calculation of the area underneath the graph of a function by exhausting it by a large number of rectangles and then passing to the limit where the base of the rectangles tend to zero, provided, of course, that this limit exists. This is very hard to do, but that is the definition of the integral. The beauty of all this is, that should you know the antiderivative of the function, then the integral is easy to calculate and this is the Fundamental Theorem of Calculus! So, all the difficulty is now shifted to finding an antiderivative and that can be achieved in many cases!
How important are theorems? We have just seen that the Fundamental Theorem of Calculus is really important; it helps you solve a problem that is very difficult to solve otherwise. So theorems are important! The problem is that we tend to forget them. The only way I know how to remember them is by understanding why they are true. Understanding a theorem is synonymous with understanding the proof!
Numbers: You should have a good feeling for natural numbers and integers. You should have no trouble executing the four fundamental operations of arithmetic with rational numbers. You should have a solid understanding of decimal fractions and some understanding of real numbers.
Important is the principle of mathematical induction, which is a fancy but very useful way of counting, and the fundamental theorem of arithmetic which says that every integer can be written uniquely as a product of prime numbers. A little bit of elementary number theory is quite helpful, like division of two integers with remainder and Euclid's algorithm to find the greatest common divisor of two integers. With this you can already solve some simple Diophantine equations, i.e., equations where one looks for solutions in the realm of integers.
Finally, knowing complex numbers and their geometric meaning is of great help.
Algebra: Linear functions and their graphs. These are the simplest functions whose evaluation only involves the four fundamental operations. They are the main actors in Calculus I, II and III. How to calculate with polynomials. Being able to divide two polynomials with remainder. Roots of simple polynomials, solving quadratic equations, not just knowing the formulas, but understanding the process of completing the square. You should be able to graph the function x^a for various values of a. Likewise you should be able to graph simple polynomials, rational functions.
A really important formula is the binomial formula! This cannot be emphasized enough. You should be able to answer simple combinatorial questions, like in how many ways can we choose k objects from n with repetition and what is the answer without repetition.
Functions: The concept of a function is central to all of science. You must understand the definition of a function as a rule assigning a definite output to each input. You should know and understand a collection of functions such as exponential functions and the logarithms. Change of basis for logarithms, trig functions, polynomials and rational functions. You should be able to graph these functions qualitatively, i.e., without a pocket calculator.
Geometry: his is an important topic, because a lot more information can be stored in a good picture (a picture is more useful than a thousand words). You should know about triangles, inscribed and circumbscribed circles, elementary constructions with compass and ruler, bisections of angles and segments. Pythagoras' theorem and the law of cosines. You should also know the definitions of the sine and cosine and tangents functions and should know some values for special angles. These functions will be the cast of characters in any halfway serious calculus course.
You also should have a very good feeling for the circle, its area, its circumference, the number pi is here the star.
You should know a little about conic sections, i.e., ellipses, hyperbolas and parabolas. Planetary orbits are, to a very good approximation, conic sections. It was one of the great achievements of Isaac Newton to derive this from his gravitational force law.
Finally, wouldn't it be nice if you knew a little about solid geometry, volume of balls and cones, surface areas of spheres?
Applications: You should be able to understand simple applications, like calculating simple areas and volumes in geometry, but also work out compound interest formulas and annuities. The hope is that if you take a loan which you have to pay off in a number of years, you should be able, based on the interest, to figure out what the monthly payment is. Applications of mathematics depend on other domains of knowledge, but, and this cannot be emphasized enough, it is through the applications that most of us get a real feeling how certain mathematical concepts work.
Problem Solving: How do you solve a problem? First you have to understand what is given and what is asked. You have to understand what these words mean and then apply what you know. With this you should able to solve simple problems. It is good to know a few simple tricks like the formula (a-b)(a+b) = a^2-b^2. The point is not that you just remember it, but that you can see in certain circumstances that this solves a problem for you.
Things get a bit more sophisticated once you still cannot do the problem. It does not help to stare at what is given and what is asked if you have understood that. It does not help if you roam around in your bag of knowledge and you cannot come up with anything useful. Problem solving is a vast field. Here is one thing you have to do, create a hypothesis whose truth might bring you closer to a solution of the problem. Create simple statements related to the problem, and show that they are true or false. Even if a guess of yours is false, once you know it you have learned something, you have gained experience. If you are interested in these kind of things we recommend George Polya's book "How to solve it" very enthusiastically.
Please take these comments seriously. While it is true that you do not have to show such skills in a multiple choice test, we certainly expect that you start learning some of these here at Georgia Tech. After all, engineering and science are all about problem solving.
Calculators: Calculators are wonderful tools in the sciences and in mathematics but you have to know what you are doing! The fact is that we make lots of mistakes when typing numbers or doing some programming. Thus, the only way to ensure that you get a reasonably correct result is that you can roughly follow the calculation on a sheet of paper or in your head so that you know whether the answer you get on the machine is reasonable or not. This means that you have to know how to calculate, at least orders of magnitude. Thus, it is a big mistake to use the calculator all the time. You have to have a good feeling for numbers and that you can only learn by doing computations.