Calculus! #6: Early Transcendentals 1.2D

Hooo doggie. This chapter has a bunch more descriptions of functions. Let’s see if I can get through all of them in one nice post.

Subsection: Power Functions

A power function is simply a function of the form f(x) = x^a, where a is some constant.

What’s that you say? “This is bullshit! That’s just a polynomial of order a!” FOOL! Recall that a polynomial has it’s variables only raised to non-negative integer powers. The constant here can be anything.

With that in mind, the book presents a few important cases.

Case 1: In which a is an integer

This is simply a polynomial of degree a. This is a niiiice easy to work with function, which we discussed in greater detail in the last section.

Case 2: In which a is of the form “1/n,” where n is a positive integer. Also known as a “Root function.”

Why is this called a root function? Because x^(1/2) is the square root. x^(1/3) is the cube root, et cetera.

Case 3: In which a = -1. Also known as the “Reciprocal function.”

This one can also be represented as f(x) = x^-1.

___

These are all just particular families of equations, but they’re types you should be intuitively familiar with. Polynomials, we discussed last time. Root functions tend to rise quickly, then taper off in growth. The higher the root, the more true this is. Reciprocal functions start high, zoom down low, then slowly approach zero.

Having an intuitive grasp of these functions will help you in understanding problems and thinking about models. For example, a root function might describe bacteria growth in a petri dish. It grows very quickly at the outset, then slows down as the container fills.

Subsection: Algebraic Functions

I’ll give you the textbook version here: “A function f is called an algebraic functino if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots).

So, for example, \dfrac{\sqrt{2x}}{x^2 + x^3}.

You can probably see how Algebraic functions could be a bit hard to work with. But, that’s also why they’re powerful. A lot of the stuff we deal with in this book will be either algebraic, or of the type I talk about next…

Subsection: Trigonometric Functions

We went over most of what you need to know here a while back when I talked about trig. You’ve got to understand trig if you want to deal with any kind of periodic functions. And, since much of the stuff you would ever want to model in reality is periodic, it’s valuable. The big thing to remember here is that at its core, trig is simple. It all comes back to sine. Cosine is just sine shifted over by pi/2. Tan is just sine/cosine. All the other fancy stuff is just modifications thereof.

Subsection: Exponential Functions

Exponential functions are of the form f(x) = a^x. This might seem easy to confuse with power functions (which are f(x) = x^a) but it won’t be once you get into Calc more. Why? Power functions are easy to deal with. Exponential functions are less so. But, we’ll get into that more later.

In terms of graphs, exponential functions tend to grow a lot faster, after starting out a little slower. This is obvious if you think about it a little. 10^2 is a whole lot smaller than 2^10. In power functions, the base grows. In exponential functions, the power grows.

Subsection: Logarithmic Functions

The book says there’ll be more on this later, so I won’t get in too deep now. If you don’t know, a log is basically the operation that undoes exponents. You should already know logs from earlier courses, but if you don’t, refresh yourself.

The log is to the exponential function what the root is to the power function. The log helps you deal with the base of the exponential function. The root helps you deal with the power in the power function. That might seem a little weird at first, but you should make sure you understand it. Logs are how you’re going to take some ugly functions and get them to look pretty.

Graphically speaking, the log function is similar to the root function (in the same way power and exponential functions are similar). It rises quickly, then tapers off. The one big difference is that the log of 1 (regardless of the log’s base) is always zero.

Subsection: Transcendental Functions

We get into these in more depth later, so I’ll keep it brief. The basic idea is that a transcendental function is a function that can’t be expressed as a finite number of algebraic operations. Later, we’ll learn how to express transcendentals as infinite sequences of algebraic operations. Then, you’ll get your badass card.

For our purposes now, transcendentals include the following types: trig functions, inverse trig functions, log functions, and exponential functions. So, you see transcendentals aren’t necessarily complicated, though they often are. But, there are plenty of simple ones too.

End of Chapter.

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7 Responses to Calculus! #6: Early Transcendentals 1.2D

  1. mmm says:

    I think you’d want to say that a polynomial (or monomial in one variable) is of the form x^n, where n is a non-negative integer

  2. Dudley Grant says:

    You can probably differentiate any differentiable function, so a question for you:
    Derive the formula for the derivative a^x where a =/= e.

    Maybe you know it instantly, but for many, it is one of those ‘pff that’s eas-… bwuh!?’ questions.

    Also in general; huge respect for being able to change your life and then switching from biology to physics “B-)”

  3. nisbahmumtaz says:

    Uhh…I’m not so much a genius, let alone an intellectual, but

    “Reciprocal functions start high, zoom down low, then slowly approach zero.”

    They don’t really approach zero, do they? I thought it forms an asymptote. If anything, ‘slowly’ is used veeeerryy lightly here.

    • ZachWeiner says:

      I didn’t say they touch zero :) And yeah, slowly is decriptive, but I think that’s clear when you end a word with “ly.”

  4. Pingback: » An update, in point form. Because I’m a busy person (aka lazy) like that. Domomo

  5. BaloogyMcBoy says:

    “For example, a root function might describe bacteria growth in a petri dish. It grows very quickly at the outside, then slows down as the container fills.”

    Bacteria grows very quickly indeed in the outside, but I think you meant outset (start)

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