Calculus is all about functions.
Now you're probably used to thinking about functions in terms of their graphs. The pair, x with f of x, for a function f. One can also take a more mechanistic view, where x is considered as an input to f, and f of x is considered as the output. Indeed, visualizing this as a machine is extremely helpful, where one feeds x into f. And receives, as an output, f of x.
With this in mind, certain terminology about functions becomes natural. For example, the domain of a function f, consists of all possible inputs. All the things you might put into the function. The range of a function, f, consists of all possible outputs, things you might receive from f.
Now this is single variable calculus, and because of that, our domain's in our ranges, are going to be relatively simple. There going to consist of either the real number line, or a certain sub intervals, there of.
Certain operatios on functions become critical. Perhap's the most important is that of composition. F composed with g, this is the function that takes, as it input, x and returns its output. Denoted f composed with g of x, and that is g of x fed into the function f. We say f of g of x. One can visualize this, as chaining together the functions f and g. And in the proper order, g comes first, and then f. For example, if one considers the function, square root of 1 minus x squared.
How can one decompose this into the composition of f and g? Well, f, in this case, would be the function that is on the outside, or that is done last. This is the square root function. G, that which comes first, is, what is on the inside of the square root function, mainly g of x equals, 1 minus x squared. One additional important operation on functions, is that of inverse. The inverse of a function f is denoted with an f with superscript negative 1.
Don't let that fool you, that does not mean you take the reciprocal of f of x, indeed f inverse of x is defined. As that function which takes as its input x, and returns as an output f inverse of x. Such that if one composes with f, one gets back the original value of x, for all such x. Now, one can also run this in reverse. If you do f first, and then f inverse, one again obtains x. that means that intuitively, f inverse is the machine that undoes whatever f does.
Let's look at a specific example. Where we take as f the function x cubed. What would be the inverse of this be? Well, it has to be a function that undoes whatever x cubed does. This is, as you may have guessed, x to the 1 3rd power, or the cube root of x. Now there are several ways to see why this is right. One is that, in taking the inverse we are reversing the role, the input and the output. Or geometrically, we are flipping the graph of this function along the diagonal line, where the input and the output are equal.
Of course, we can also think in terms of what the inverse has to satisfy, if we do x cubed and then take it's cubed root, we get x. Or the other way around, if we take the cube root of x and cube it, we get x.
Certain classes of functions wind up being extremely important throughout calculus. Perhaps the simplest such class, is that of the polynomials. That is, functions of the form a constant, plus a constant times x, plus a constant times x squared, all the way up to some finite degree, constant times x to the n. That largest power of x is called the degree.
There is a summation notation, that makes writing out polynomials very simple. We use the Greek capital Sigma, and right polynomial as the sum Sigma, as k goes from 0 to n of c sub k times x to the k. Here the c k's are coefficients, or constants.
Another class of commonly observed functions are the irrational functions. These are functions of the form P of x over Q of x, where P and Q are polynomials.
Simple examples, like 3x minus 1 over x squared plus x minus 6, are very common throughout mathematics and its applications. Rational functions are very nice to work with. You do, however, have to be careful of what happens in the denominator?
When you try to plus in a value of x, it evaluates the denominator to 0. Your function is not necessary well
defined, at such a point.
Other powers besides integer powers are important and prevalent. I'm guessing that you all know what x to the 0 is, that is, of course, equal to 1. What is x to the minus one half? And recall that fractional powers can note roots, and negative powers mean that you take the reciprocal. So that x to the negative one half is 1 over the square root of x. And what is x to the 22 7th's?
Well, break this up into pieces. First, we take x to the 22nd power, and then we take the 7th root of x.
Lastly, what is x to the pi? Well we're not going to answer that quite yet, but you may have a guess, especially considering the fact that there are rational numbers that are very close to the irrational number pi.
Trigonometric functions are extremely common, and important. You should already know a bit about sine, and cosine. Let's review. Perhaps the most important of the trigonometric identities, that is cosine squared plus sine squared equals 1. There's several ways to interpret this. You've seen the interpretation involving a right triangle, with hypotenuse equal to 1 and with angle set to theta. Then, in this case, the sine of theta is the length of the opposite edge to theta, and the cosine of theta, is the length of the adjacent edge. If we think of those as x and y coordinates respectively, then we see that there's a relationship between this trigonometric identity, and the equation for the circle, x squared plus y squared equals 1.
And indeed, if we think of what happens when we move a point along a unit circle, rotating it by an angle theta, from the positive x axis. Then the x and y coordinates, of that point on the unit circle, are precisely, cosine and the sine, respectively.
Other trigonometric functions are common and important. I'm sure you recall that tangent is the ratio of the sine to the cosine function, and it's reciprocal, the cotangent function.
One can also take reciprocals of cosine, obtaining the secant function, and of sine, obtaining the co-secant function, respectively.
Now all of these that involve ratios, wind up having vertical asymptotes in their graphs, that is, places where the function is undefined, and the denominator goes to 0. One of the things that you'll note about both the arcsin, and the arccos, is that they have a restricted domain. The domain must be the close general from negative 1 to 1.
Because of course, sine and co-sine can only take values in that interval. In contrast to this, the arc tangent function, does have an infinite domain. Its range however, is limited between negative pi over 2, and pi over two. These are all functions that you're going to want to be familiar with, for moving forward in calculus.
The last class of functions that is of critical importance, are the exponential functions. These are functions of the form e to the x. to the x being the canonical example of an exponential function.
It's inverse is the logarithm or more precisely, the natural logarithm, ln of x. You've seen these functions before, you know, because they're inverses, that their graphs have this symmetry about the diagonal.
So for example if e to the 0 is 1, as it must be, then log of 1 must be equal to 0, certainly. The question that is often unanswered in pre-calculus or even elementary calculus courses is, what is e? And why is it so important in this exponential function? Well one could say that e is that value whose logarithm is equal to 1. But since we've defined the natural logarithm in terms of base e, that's a bit of circular reasoning.
How do we reason about e? Well certainly e is a number. It is a particular location, on the real number line.
It has a decimal expansion, but being irrational it is a little bit hard to remember all of it. That doesn't really answer the question of what is e? Why is it so important? Before we get to the answer to that, let's review some of the algebraic properties associated with the exponential function.
I hope you remember that e to the x times e to the y is e to the x plus y. The exponents add and e to the x raised to the y power is e to the x times y. In your prior exposure to calculus, I'm sure that you've seen some of the differential and integral properties of this function. E to the x has this wonderful property that it is it's own derivative, and of course, but it is its own integral, at least up to a constant. These facts are easy to remember, but maybe not so easy to fully comprehend.
There is one last ingredient that we're going to need before we go deep into exploring what e to the x means. This is something called Euler's formula. This is simple looking. It states that e to the i times x, equals cosine of x plus i times sine of x.
This is a wonderful formula, but whatever it means. What does it mean? What is this i that is in here?
I am sure that you've seen before, the notation for the square root of negative 1, for the primal imaginary number, i. That is what is meant in Euler's formula, it has the property of course that, i squared is equal to negative 1.
Now you're probably used to thinking about functions in terms of their graphs. The pair, x with f of x, for a function f. One can also take a more mechanistic view, where x is considered as an input to f, and f of x is considered as the output. Indeed, visualizing this as a machine is extremely helpful, where one feeds x into f. And receives, as an output, f of x.
With this in mind, certain terminology about functions becomes natural. For example, the domain of a function f, consists of all possible inputs. All the things you might put into the function. The range of a function, f, consists of all possible outputs, things you might receive from f.
Now this is single variable calculus, and because of that, our domain's in our ranges, are going to be relatively simple. There going to consist of either the real number line, or a certain sub intervals, there of.
Don't let that fool you, that does not mean you take the reciprocal of f of x, indeed f inverse of x is defined. As that function which takes as its input x, and returns as an output f inverse of x. Such that if one composes with f, one gets back the original value of x, for all such x. Now, one can also run this in reverse. If you do f first, and then f inverse, one again obtains x. that means that intuitively, f inverse is the machine that undoes whatever f does.
Let's look at a specific example. Where we take as f the function x cubed. What would be the inverse of this be? Well, it has to be a function that undoes whatever x cubed does. This is, as you may have guessed, x to the 1 3rd power, or the cube root of x. Now there are several ways to see why this is right. One is that, in taking the inverse we are reversing the role, the input and the output. Or geometrically, we are flipping the graph of this function along the diagonal line, where the input and the output are equal.
Of course, we can also think in terms of what the inverse has to satisfy, if we do x cubed and then take it's cubed root, we get x. Or the other way around, if we take the cube root of x and cube it, we get x.
Certain classes of functions wind up being extremely important throughout calculus. Perhaps the simplest such class, is that of the polynomials. That is, functions of the form a constant, plus a constant times x, plus a constant times x squared, all the way up to some finite degree, constant times x to the n. That largest power of x is called the degree.
There is a summation notation, that makes writing out polynomials very simple. We use the Greek capital Sigma, and right polynomial as the sum Sigma, as k goes from 0 to n of c sub k times x to the k. Here the c k's are coefficients, or constants.
Simple examples, like 3x minus 1 over x squared plus x minus 6, are very common throughout mathematics and its applications. Rational functions are very nice to work with. You do, however, have to be careful of what happens in the denominator?
When you try to plus in a value of x, it evaluates the denominator to 0. Your function is not necessary well
defined, at such a point.
Well, break this up into pieces. First, we take x to the 22nd power, and then we take the 7th root of x.
Lastly, what is x to the pi? Well we're not going to answer that quite yet, but you may have a guess, especially considering the fact that there are rational numbers that are very close to the irrational number pi.
And indeed, if we think of what happens when we move a point along a unit circle, rotating it by an angle theta, from the positive x axis. Then the x and y coordinates, of that point on the unit circle, are precisely, cosine and the sine, respectively.
Other trigonometric functions are common and important. I'm sure you recall that tangent is the ratio of the sine to the cosine function, and it's reciprocal, the cotangent function.
One can also take reciprocals of cosine, obtaining the secant function, and of sine, obtaining the co-secant function, respectively.
Because of course, sine and co-sine can only take values in that interval. In contrast to this, the arc tangent function, does have an infinite domain. Its range however, is limited between negative pi over 2, and pi over two. These are all functions that you're going to want to be familiar with, for moving forward in calculus.
It's inverse is the logarithm or more precisely, the natural logarithm, ln of x. You've seen these functions before, you know, because they're inverses, that their graphs have this symmetry about the diagonal.
So for example if e to the 0 is 1, as it must be, then log of 1 must be equal to 0, certainly. The question that is often unanswered in pre-calculus or even elementary calculus courses is, what is e? And why is it so important in this exponential function? Well one could say that e is that value whose logarithm is equal to 1. But since we've defined the natural logarithm in terms of base e, that's a bit of circular reasoning.
How do we reason about e? Well certainly e is a number. It is a particular location, on the real number line.
It has a decimal expansion, but being irrational it is a little bit hard to remember all of it. That doesn't really answer the question of what is e? Why is it so important? Before we get to the answer to that, let's review some of the algebraic properties associated with the exponential function.
I hope you remember that e to the x times e to the y is e to the x plus y. The exponents add and e to the x raised to the y power is e to the x times y. In your prior exposure to calculus, I'm sure that you've seen some of the differential and integral properties of this function. E to the x has this wonderful property that it is it's own derivative, and of course, but it is its own integral, at least up to a constant. These facts are easy to remember, but maybe not so easy to fully comprehend.
This is a wonderful formula, but whatever it means. What does it mean? What is this i that is in here?
I am sure that you've seen before, the notation for the square root of negative 1, for the primal imaginary number, i. That is what is meant in Euler's formula, it has the property of course that, i squared is equal to negative 1.
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