Proportionality and Linear Functions - Infinity is Really Big

Calculus 1, Lectures 2A and 2B

Quantities that are directly proportional to each other arise everywhere in science. Examples include distance traveled with speed, volume with temperature for an ideal gas, and force with acceleration.

If the two quantities are thought of as variables, then the direct proportionality between them leads to an equation belonging to a special class of functions: the linear functions.

In calculus, linear functions are those whose formulas are of the form $y=f(x)=mx+b$ for some constants $m$ and $b$ . The constant $m$ is called the slope of the linear function. And the constant $b$ is called the y-intercept the linear function.

For quantities $x$ and $y$ which are directly proportional, the value of $b=0$ . This means the graph of such a function passes through the origin $(x,y)=(0,0)$ .

In Lecture 1 of my series of Calculus 1 lectures at Bethel University in the fall of 2019, I give a broad overview of the big ideas of calculus. But, in lecture 2, I start getting into details of the course by delving into the properties of proportionality and linear functions.

Lecture 2A

I begin Lecture 2A by talking about the importance of linear functions. Then I consider examples of directly proportional quantities, focused on unit conversions. From there, I move on to a discussion of fundamental properties of proportionality functions, including visuals. Finally, I look at examples of proportional quantities.

Calculus 1, Lecture 2A: Importance of Linear Functions, Direct Proportionality Examples and Facts

Fundamental Facts

Two main abstract facts of proportionality functions $y=f(x)=kx$ are worth emphasizing and being able to derive.

If you multiply the input by a given factor, the output will be multiplied by the same factor.
The constant $k$ is the rate of change (slope) of the graph.

Verification of Fundamental Property 1

Property 1 is verified by the following symbolic calculation. The letter $a$ represents the factor you are multiplying by. If the input is doubled, for example, then $a=2$ .

$f(ax)=k(ax)=(ka)x=(ak)x=a(kx)=af(x)$

As I emphasize in the lecture, it is important to be clear about the symbols and concepts. The letter $f$ is the name of the function. Therefore, the symbolic expression $f(ax)$ should be read as “ $f$ of $a$ times $x$ .”

On the other hand, the letter $k$ is the name of a constant (number). Therefore, the symbolic expression $k(ax)$ should be read as “ $k$ times the quantity $a$ times $x$ .”

You should also be able to identify where the associative and commutative properties of multiplication are used in the symbolic calculation above.

Verification of Fundamental Property 2

Property 2 is verified by the following symbolic calculation. The quantities $\Delta x$ and $\Delta y$ represent, respectively, the “change in x” and the “change in y“. The conclusion is that the ratio of these two numbers is the constant $k$ . Therefore, the slope of the graph is constant and equal to $k$ .

$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{k(x+\Delta x)-kx}{\Delta x}=\frac{kx+k\cdot \Delta x-kx}{\Delta x}=\frac{k\cdot \Delta x}{\Delta x}=k$

Lecture 2B

I start lecture 2B by answering a couple student questions from the break, and then doing some problem solving with direct proportions. From there, I move on to a discussion of linear functions in general, including their slopes and intercepts. Finally, I summarize, in an intuitive way, some calculus-related properties of linear functions. The most difficult concept of this portion of the lecture is the idea of a limit.

Calculus 1, Lecture 2B: Proportionality Examples and Linear Functions

Infinite Limits at Infinity for Linear Functions

Let’s end this blog post by elaborating on the meaning of the limit discussed in the lecture.

We’ll take a particular example. Let $y=f(x)=2x-8$ . I claim that $y$ “goes to plus infinity“ as $x$ “goes to plus infinity“.

Saying these things does not mean I am treating infinity as a number here.

In symbols, this is written as

$\displaystyle\lim_{x\rightarrow +\infty}f(x)=\displaystyle\lim_{x\rightarrow +\infty}(2x-8)=+\infty$

What does this really mean? It means the output $y=f(x)$ can be made as large as we like, as long as $x$ is sufficiently large. In other words, the graph of $y=f(x)$ can get above any horizontal line $y=M$ , no matter how big $M$ is, as long as $x$ is big enough (positively).

For example, if $M=10^{6}=1000000$ , how big should $x$ be so that $f(x)>M$ ?

To answer this, just solve the inequality $f(x)>M$ , though technically the logic in the solving needs to be reversible, which it is. Here are the details:

$f(x)>M\Leftrightarrow 2x-8>1000000\Leftrightarrow 2x>1000008\Leftrightarrow x>500004$ .

Therefore, if $x>500004$ , then $f(x)>1000000$ .

The same kind of calculation can be done for any value of $M$ . It could even be $M=1000^{1000^{1000}}$ or some other huge monster of a number. The abstract calculations are on the line below:

$f(x)>M\Leftrightarrow 2x-8>M\Leftrightarrow 2x>M+8\Leftrightarrow x>\frac{1}{2}M+4$ .

The limit concept is at the heart of calculus. It’s what gives calculus its power and is what makes calculus fundamentally different than topics like algebra and discrete mathematics. Make sure you work hard at understanding it as you go through your own study of the big ideas of calculus.