Built on Facts

Let’s start with a pretty simple function. It’s not this week’s official Sunday Function, but we’ll use it to get there.

Take the number 1, divide it by x. Pretty easy. Now imagine putting a dot at the point x = 1 on the x axis, and draw a line straight up from there to the curve formed by the graph of the function. Then do the same thing for another point on the x axis (with x > 1). Shade in the area in between the lines. Ok, that might be a little hard to visualize if you’ve never done it before. Let me pick the other number to be x = 4, and I’ll actually put the picture below the next paragraph.

Now you or I could have picked any number, not just 4. It stands to reason that the bigger the number we picked, the more area will be in the shaded region. The area of that shaded region is itself a function. We ought to give it a name. How about log(x)?

Log(x), huh? Isn’t there already a function with that name? Yes, the natural logarithm. It’s this very function. Our Sunday Function! Now let’s try to qualitatively determine some of its properties. As x grows larger, our function 1/x gets very small. That means we’re not adding area very rapidly as we extend the shaded region farther to the right. The log function is in fact very slowly growing. It never stops, and if you pick an x big enough log(x) will also get very large. But it does so quite slowly. If you want log(x) = 10, x is going to have to be more than 22,000.

Since this is a natural logarithm, we might expect the number e to be lurking about somewhere. It is. If the total shaded area is 1, the x you will have to have picked is e = 2.718…

Neat, huh? The world of mathematics yields some unusual jewels, and one of my favorites is this relationship between the humble 1/x and the dramatic and ubiquitous exponential function.

[Note: I have followed the standard convention that the natural logarithm is denoted as log(x). In introductory treatments it's usually denoted ln(x).]

Comments

#1 Winter Toad
March 8, 2009

One thing we liked about the 1/x function in school was its surface/volume of revolution. Rotate the 1/x function about the x-axis, and take the region from x=1 to x->inf. The resulting object has a finite volume but an infinite surface area, so while you could fill it up, you could never paint it.
#2 Len
March 8, 2009

But, Toad — what if I filled it up with paint?
#3 Hank Roberts
March 8, 2009

Does this mean you could paint the inside surface, but not the outside surface?
#4 cfcasper
March 8, 2009

Longtime Sunday Function fan, first-time commenter (I think). This was a particular highlight for me — I had never understood natural logarithms until just now. Great post.
#5 Michael F. Martin
March 8, 2009

More fun: consider the Fourier transform of 1/z.
#6 Comrade PhysioProf
March 8, 2009

Coolio! I wish I had access to these great syntheses when I was taking calculus in high-school. Just one question: Why didn’t you explicitly state that the area under the curve y=1/x is the integral?
#7 Matt Springer
March 8, 2009

“Why didn’t you explicitly state that the area under the curve y=1/x is the integral?”

Hard to do without getting carried away and turning it into a complete Calc 1 course.
#8 Comrade PhysioProf
March 8, 2009

Hard to do without getting carried away and turning it into a complete Calc 1 course.

Go for it, dude! Seriously, I bet you could write a really useful big-picture short calculus book that would be useful as an accompaniment to the usual detailed problem-based textbooks.
#9 joemac53
March 8, 2009

Another great post. I have my calc students make sums from 1 to 2 with 4, then 8, then 12 rectangles before we let the calculators tell us what is going on. Very cool.
#10 Isis the Scientist
March 8, 2009

Matt, this was hot. The only thing that would have made me lose complete bladder control was seeing a little “dx” action thrown in there. I do love a sexy equation.
#11 J Meckley
March 8, 2009

What program did you use to draw the graphs?
#12 Winter Toad
March 8, 2009

Hank Roberts: the inner and outer surfaces have the same (infinite) area. So, you could fill the volume, but you could not paint the inside surface. It can take a while to wrap one’s mind around it.
#13 Matt Springer
March 8, 2009

#11, I drew the graphs with Mathematica 6 and filled in the shaded area and text in Photoshop.

Good call on the Gabriel’s Horn, several of you. I smell another Sunday Function idea!
#14 Bala Narayanaswamy
March 10, 2009

This is one superb post. I love such connections – they are lurking somewhere but have to be pulled out like a rabbit out of a hat!

Nary ( New Delhi / India)