Fi & Lu Area Algebra Trigonometry Functions Complex Numbers Calculus
Grandpa felt it was time for Fi and Lu to learn about functions. After all, their uncle was about their age when Grandpa taught him. “You see functions all the time. A function is anything that associates each thing in a certain collection of things with some other thing from perhaps a different collection of things. We say that the function maps the thing to the other thing. A dictionary is a function that maps a word to information about that word, usually its definition and perhaps its pronunciation and etymology. Addition is a function that maps a pair of numbers to their sum. Multiplication maps a pair of numbers to their product. The Fibonacci sequence is a function that maps an integer, n, to the nth Fibonacci number, \(f_n\).
“When a function maps a single number to another number, we often visualize that function with a two-dimensional plot. We usually use the first number (the argument) as the x coordinate (the abscissa) and the other number (the value) as the y coordinate (the ordinate) to locate the point that shows the mapping of the argument to the value. Typically, when we plot such a function, we also draw the x axis (the line of points with zero ordinate) and the y axis (the line of points with zero abscissa). For example, here’s the plot of the sine function:
“Some of the simplest mathematical functions we encounter are polynomials. We usually use the letter \(x\) to represent the argument. Unsurprisingly, a polynomial is a sum of monomials. ‘What is a monomial?’ you ask. It’s a number times a nonnegative integer power of \(x\). For example, \(7x^5\).
“The simplest polynomial is just a constant, such as \(\frac{1}{2}\), which plots as a horizontal straight line. Next is a linear polynomial such as \(\frac{1}{6}x-\frac{1}{2}\), which also plots as a line. A quadratic polynomial, like \(\frac{1}{16}x^2+\frac{1}{2}x\), plots as a parabola. Higher degree polynomials can wiggle, like this cubic polynomial \(\frac{1}{16}x^3-\frac{1}{8}x^2-\frac{1}{16}x+\frac{1}{8}\).
“It’s not hard to see that the plot of a polynomial can only change direction fewer times than its degree. To get an infinite number of wiggles like the sine function, you need a power series. In fact, \(\sin x = x-\frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\), otherwise written as \(\sum_0^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}\). And \(\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots = \sum_0^\infty \frac{(-1)^n x^{2n}}{(2n)!}\). To try this out for yourself, click here.
“There’s a particularly important function that maps polynomials to polynomials. It’s called the derivative, for reasons we’ll see when we talk about calculus. The derivative of \(p\) is designated by \(p^\prime\), pronounced pee prime. It’s a linear function, which means that (\(kp)^\prime = kp^\prime\) for any constant \(k\) and polynomial \(p\), and (\(p+q)^\prime = p^\prime+q^\prime\) for any pair of polynomials \(p\) and \(q\). So we can define it by just describing what it does to monomials: \((x^n)^\prime = nx^{n-1}\). Note that this means that the derivative of a constant is zero. It’s not hard to show that \((pq)^\prime = p^\prime q + pq^\prime\)—it’s clearly true for monomials, and linearity and distributivity complete the proof. One of the consequences of this is that if a polynomial has a repeated root, that root is also a root of its derivative.”
The degree of a monomial is its exponent, and the degree of a polynomial is the maximum degree of its monomials; the zero polynomial has no monomials and so its degree is \(-\infty\). For example, linear polynomials like \(x+3\) have degree 1, quadratic polynomials like \(3x^2+2x+1\) have degree 2, cubic polynomials like \(5x^3-7x^2+9x-11\) have degree 3, and all nonzero constant polynomials like \(2\) have degree 0.
With this definition, it’s clear that the degree of a product of polynomials is the sum of their degrees.
Note that we’ve only been talking about polynomials in one variable. For multivariable polynomials, the degree of a monomial is the sum of its exponents. For example, \(x^2y^3z\) has degree 6, since 2+3+1=6.
A root of a polynomial is an argument for which the polynomial’s value is zero: \(t\) is a root of polynomial \(p\) if \(p(t)\)=0. It’s easy to see that if \(t\) is a root of \(p\) then \(x-t\) divides \(p\): by polynomial division, we know that \(p = (x-t)q+r\) for some quotient polynomial \(q\) and remainder \(r\), and so \(0 = p(t) = (t-t)q(t)+r = r\), i.e., the remainder is zero. A root \(t\) of \(p\) is a repeated root if \((x-t)^2\) divides \(p\).