Functions, Variables, Equations and Graphs Questions and Answers on the Background and Objectives in Calculus I by Eric Carlen Professor of Mathematics at Georgia Tech Q1: What does calculus mean? The word calculus has the same root as calcium this is because the Romans used little pebbles, generally of limestone, arrayed on boards to do their reckoning. During the middle ages, a calculus came to mean any method of reckoning or problem solving. The subject of this course was once called the calculus of innitesimals, and was one calculus among many such as the calculus of probabilities. It has proven to be so useful that nowadays it is simply called the calculus, although one does occasionally run into references to other calculi. Calculus is part of a branch of mathematics known as analysis. And the things one analyses in analysis are functions, and the solutions to equations. Q2: What are functions? A function is simply a rule that assigns a member of an output set, usually called the range, to each member of a given set of possible inputs, usually called the domain. In this class, the domain and range will both generally be subsets of the real numbers, but later in the course we will consider the complex numbers as well. There is no restriction; the input and output sets can be any sets at all. What is important though is that exactly one output is assigned to each input. Any relation between inputs and outputs that doesnt satisfy this requirement is just not a function. Q3: O.K., a function is a rule assigning outputs to inputs but how do we specify the rule? There are many ways to specify functions. If the set of possible inputs the domain is a small enough nite set, one could just list the outputs associated to a given input. For example, here is a function f with domain {0, 1, 2, 3} and range {0, 2, 4, 6} given explicitly by f (0) = 0 f (1) = 2 f (2) = 4 f (3) = 6 The input is shown inside the parentheses, and the output is specied to the right of the equal sign. Q4: What if my domain is innite? If there are innitely many inputs, I would need an innitely long list. If there are innitely many inputs, one denitely cant use a list. Instead, one has to specify what it is that the function does to its inputs. 1
You can see what the function that we specied by a list just above is doing it just doubles the input. That makes sense for any real number as input, and by specifying this action of the function we can avoid an innite list! To do this, we turn to variables and operations. A real variable is simply a named container for a real number. Think of it as a box, with a label on front, and a real number inside. We specify variables by name the label on the box. This picture represents a variable x, with the current contained value being = 3.14169...
= 3.14159...
X
It is standard to use letters like x, y and z for the names of variables. There is actually an interesting story behind the use of x as the standard name for a variable. An operation is something you can do the value; i.e, the real number, inside the box. For instance, you can double it, or you can square it. When we write 2x, this means take the value in the box labeled x, and double it. When we write x2 , this means take the value in the box labeled x, and square it. And so on. This way of thinking about variables and operations is familiar to you if you have done any computer programming, especially in a language like C. In this setting, a variable has a name and a type. The name is associated with an address in memory, and the type tells how big a block of memory, starting from the specied address, is used to hold the value. When you call the variable latter in the program, you get the value stored in that block of memory. We can now specify a function f by giving its domain say, all of the real numbers and the sequence of operations it performs on a given variable x, which could be holding any of the values in the domain. For example, f (x) = 2x or f (x) = x2 or f (x) = 2(1 + x2 )
The last example was the only one involving more than a single operation. Here, there were three: rst we squared the value in x, then added 1 to it, and then doubled that. 2
Q4: Cant I just think of the action of f in the last example as one bigger operation? Yes, you can. But actually it wil turn out to be very useful in general to think of functions as an assembly line: We can think of this function f as an assembly line: A box x comes in at the left containing some value, it is opened and sent to three successive stations, where operations are applied to it. Let the three stations be given by the elementary functions p(x), q(x) and r(x) where p(x) = x2 q(x) = 2 x r(x) = 1 + x
x
f(x)
x
p(x)
q(p(x))
r(q(p(x)))
A basic strategy in the analysis of functions in all of science for that matter is to take complicated functions apart into simpler pieces, and then to analyse them in terms of these simpler constituents. We will do this again and again here. The divideand-conquer principle is fundamental to mathematics and much else besides warfare. The assembly line analogy leads us straight into another important notion the composition of functions. If f and g are two functions, and the domain of g contains the range of f , then we get a new function g f , called g composed with f by dening g f (x) = g(f (x)) . This means take the value in x, put it through f , and then take what comes out, and put it through g. 3
This only makes sense if the output values of f are possible input values for g, so the requirement that the domain of g contains the range of f is crucial. However, in many cases the domain of g will contain all real numbers, and there is no problem. Or, if there is a problem, the domain of f can be restricted so it doesnt produce any output values outside the domain of g. Q5: Is composition the only way to build complicated functions out of simple ones? No, composition is just one way to combine a pair of functions to produce a new function. There are others, more closely related to the familiar arithmetic operations we can perform on numbers. For instance, we can add up the outputs of two functions. The net result can be viewed as a new, more complicated function.
x
f(x)
+
x g(x)
(f + g)(x)
More generally, if f and g are two functions with the same domain, we dene new functions by (f + g)(x) = f (x) + g(x) (f g)(x) = f (x) g(x) f g(x) = f (x)g(x) If moreover 0 is not in the range of g, we dene f f (x) (x) = . g g(x) In this way some very complicated functions can be built out of very simple building blocks. For example, with f (x) = x + 1
2
g(x) = 1/x
h(x) = x2 and
j(x) = sin(x)
1 + sin2 (x2 ) = g f h j f + h j h 1 + sin (x + 1) 4
Q6: O.K., so now I know what functions are, and I understand that being able to take them apart into simple pieces is supposed to help me analyze them, but exactly what sort of analysis will I be doing? Suppose I have a function to analyze. Which specic questions will we be asking about it? There are many kinds of questions, and it is not possible to list them all now. But here are two examples that we can talk about at this time. (1) The maximization question: Given a function f , is there an input x0 so that f (x0 ) f (x) for all other x in the domain of f ? If so, what is f (x0 ), the largest output, and what are all of the input values that produce it? (2) The equation solving question: Given a function f and a value a, are there any input values x so that f (x) = a? That is, are there any solutions of the equation f (x) = a? If so, what are they? The rst of these is an optimization problem, as would be the corresponding question about smallest values. If the function has a nite domain, and is given in the form of an explicit list, as in our rst example, then the problem is solved simply by running down the list. But if the domain is innite, we cannot use a list. We must instead analyse the operations, or assembly line steps, out of which the function is built. Calculus provides methods for this. Likewise with the second problem. If f is given by a list, just look down the list of output values and see if you see a. In the special case a = 0, the solutions are called roots of f . We can always reduce to this case by dening a new function g(x) = f (x) a. Then solutions of f (x) = a are roots of g. Calculus provides powerful methods for nding roots. Q7: What kinds of methods will we use besides taking functions apart into simple pieces? Analysis is part of guring things out, and guring things out is quite literally part of analysis. Drawing gures and using geometrical insight will basic to our strategy of analysis. We can bring in a geometric perspective by turning to the graphs of our functions. The range, or some piece of it, is conventionally drawn on the vertical axis, and the domain, or some piece of it, on the horizontal. For each point in the domain of f , draw the vertical line though that input value on the horizontal axis. Then draw the horizontal line through the corresponding output value on the vertical axis. These two lines meet at a single point, which is a point on the graph of f . The graph of f consists of all the points that obtained are in this way. As a subset of the plane, a graph can be drawn on a sheet of paper, or a computer screen. One way of drawing a graph is to compute the points on it for a large but nite collection of input values, and then to connect the dots. This is tedious to do by hand, but easy on a computer. 5
We can do this with Maple very easily. We will just need a few simple commands. (In fact, a few simple commands will get you very far in this whole course. A tuttorial on these commands, written especially for this course, is available on the web.) The function we will take as our example is: f (x) = x (1 + x2 ) . (1 + x4 )
The following three commands dene this function for further use, graph it on the range 0 x 2, and evaluate it at x = 0.5.
> f:= x*(1+x^2)/(1+x^4); f := > plot(f,x=0..2,y=0..1.5); x ( 1 + x2 ) 1 + x4
1.4
1.2
1
0.8 y
0.6
0.4
0.2
0
0.5
1 x
1.5
2
> subs(x=0.5,f); .5882352941 >
6
Now lets zoom in on the part of the graph near the point (0.5, 0.58823) which is almost on the graph as we found by using the substitution command in Maple. The thing to notice is that now we just see something very simple: a straight line.
0.6
0.55
0.5
0.45
0.4
0.45
0.5 x
0.55
0.6
This is an absolutely central point in this whole course: Up close, the graphs of most reasonable functions look like lines. The escape clause about most reasonable functions is necessary. Try for example f (x) = |x| (which you would type into maple using the absolute value function which is denoted by abs(x)) at the point (0, 0). As you zoom in around this point, the graph doesnt even change; it keeps its kink forever. But at any other point, it does become linear if we zoom in. 7
Q7: Its nice that the functions look like lines up close, but whats so good about lines? Lines are very simple they are described by simple equations. Simplicity is not just good its great!. Any line that is not exactly vertical has a nite slope and y-intercept, and is the graph of a linear function f (x) = mx + b . Here m is the slope, and b is the y-intercept. These two numbers m and b completely specify the linear function f , and the corresponding line that is its graph. Another way to specify a line is to give a point (x0 , y0 ) on the line and the slope m. The function f is given by f (x) = m(x x0 ) + y0 . Notice that f (x0 ) = y0 , as must be the case. Also m(x x0 ) + y0 = mx + (y0 mx0 ) so that the y intercept b is y0 mx0 . Now, consider any point (x0 , y0 ) on the graph of f i.e., any point (x0 , y0 ) with y0 = f (x0 ). Let m be the zoomed in slope i.e., the slope of the line we see when we zoom in on (x0 , y0 ). Then the graph of f (x) = m(x x0 ) + y0 is a line which passes through (x0 , y0 ) and ts the graph of f itself there as much as possible. This line is called the tangent line to the graph of f at (x0 , y0 ). The problem of nding the formula for this line which amounts to the problem of nding the zoomed in slope m since x0 and y0 are known is called, naturally enough, the tangent line problem. It is the one of the most central problems of this course. Q8: Whats so important about this tangent line problem? It is not at all obvious that this should be such a central problem. The greeks missed this point completely. So lets recapitulate. Weve seen that up close most functions at most points, anyhow look like linear functions, and linear functions are very simple. This simplicity in the small is the guiding light in the innitessimal calculus. To see how one could use this simplicity in the small to analyse a problem, lets look at an example. Lets try to answer question (1) for the function f (x) = x 1 + x2 1 + x4
on the domain 0 x 2. The graph appears a bit back in this section of notes. You can see a hump near x = 1 where f takes on the value 1. Zoom in on (1, 1) either using Maple, or the graph a few pages back together with your imaginiation) until you see a line. What is its slope? 8
You presumably found a horizontal line; i.e., zero slope. There is exactly one point on the graph at which the zoomed in slope is zero. And as we will see later, it is only at such points that we have to look for maxima and minima. So nding the zoomed in slope will be the key to solving optimization problems. How about the second problem. Lets try to nd the roots of f (x) = x3 2x 5 . Here is a graph centered on x = 2.
15
10
5
01
1.5
2 x
2.5
3
-5
9
There is exactly one root visible, and it is not too far away from x = 2. However, at x = 2, one easily computes that y = f (x) = 1. Now, the slope of the tangent line to the graph y = f (x) at the point (2, 1) turns out to be 10. (Notice the dierent scales on the x and y axes!) You will soon learn how to compute such slopes, but now we want ot explain what they are good for once you have computed them. Using the above information, one easily works out that the eqaution of the tangent line is y = 10x 21 . Now lets graph the function and the tangent line together on the same graph:
15
10
5
01
1.5
2 x
2.5
3
-5
-10
Now, since the graphs of f and its tangent line are close together near x = 2, the place on the x-axis where the tangent line crosses it must be close to the root of f that we are looking for. The later is hard to nd, since f is cubic, but it is easy to nd the place where the tangent line crosses the x-axis: This is given by solving the linear equation 10x 21 = 0, or x = 21/10 = 2.1. so a pretty good approximate solution of our equation is given by x = 2.1, and ...