Wednesday, July 31, 2013
On first blush, it might seem that if you add an infinite number of positive numbers together, the sum must be unbounded. But if the numbers are getting small enough fast enough, the sum of an infinite series can be both finite and exactly defined.
Probably the simplest infinite sum is the powers of ½. Let's say that half of a room is painted in the first hour, and half of what remains is painted in the second hour, so now 3/4 of the room is painted. For reasons unknown, you decide to let this increasingly lazy person continue their plan of painting less and less each hour, one eighth in the third hour, one sixteenth in the fourth hour, on and on infinitely. I say infinitely because mathematically the room is never finished. (In reality, we will get to a sliver so tiny that it must be painted completely, if only because a molecule of paint once split isn't a molecule of paint anymore.)
In math, this is well understood now to be an infinite sum that adds up to 1. You might fairly say "we never get there", which is true. This was the point of one of the famous paradoxes of the ancient Greek mathematician and philosopher Zeno. But in modern math we have the concept of a limit as n approaches infinity. We can agree that the sum never gets to be more than 1, and the idea of a limit is that you get to choose how close you want the sum to be to 1 and I have to give a number of steps which will guarantee a sum that is closer than your given definition of "close enough". For example, if we want the sum to be within one millionth of 1, which is to say larger than .999999 but still less than 1, I can guarantee that after twenty steps, the sum will be that large, because ½ raised to the twentieth power is slightly less than one millionth.
Tomorrow, one of the tricks used for the general power series of positive numbers less than 1.
Monday, July 29, 2013
This is the bell shaped curve from x ranging from -3 to 3. Note that this is NOT the normal curve, the famous workhorse of statistics. In math, "normal" usually means there is something about object whose measure is 1. This curve has a highest point at (0,1) but the normal curve has an area between the curve and the x-axis of 1. (In math, we would talk about this as the value of the definite integral from negative infinity to infinity.)
While this is not the normal curve, one calculus related attribute it shares with the normal curve is that the points of inflection are at x = -1 and x = 1.
Here is how the curve compares to the top of the unit circle from -1 to 1 and the parabola y = 1 - x² over the same range.
The curve that is very close to the same shape when the bottom points (-1, 0) and (1, 0) and top point (0, 1) are lined up the cosine function. The bell shaped curve is shown in blue dots and the cosine function in purple dashes. The most they disagree by is about .00647, a little more than six parts in one thousand. Still, they do disagree and are not the same function as far as mathematicians are concerned.
Sunday, July 28, 2013
If the two ends of a chain are held fast at the same height, what is the shape of the hanging section? This question has been around for a long time. Galileo speculated it was a parabola, but some experimentation showed him that wasn't accurate.
The name of the shape is the catenary. The simplest equation for the shape is the hyperbolic cosine function, y = ½(e^x + e^-x). Unlike sine and cosine that oscillate us and down, hyperbolic cosine has just the one lowest point, like a parabola, and hyperbolic sine looks something like the cubic. The thing that makes them sort of like the trig functions is that each is the derivative and anti-derivative of the other. (I write "sort of" because the derivative of sine is cosine, but the derivative of cosine is negative sine.)
As pretty as the tangent line representations can be, looking at the actual functions is clearer in this case.
Red curve is the catenary from x = -3 to 3.
The blue shape is y = x² + 1.
The 1 added so they coincide more closely. A catenary grows slightly more slowly than the parabola at the vertex, but it is an exponential function, so when it starts to grow faster, it will overwhelm a quadratic function very quickly.
This drawing is the catenary in red and the parabola in blue yet again, but this time from -5 to 5. The shape of the hanging chain depends on the ratio of the length of the chain to the distance apart of the points from which the shape hangs. To get the exact comparison of the catenary and parabola with those values given is a harder thing to compute and describe, as it involves the arc length formula for the parabola and the catenary, which turn into messy integrals.
Tomorrow we will look at the top of the bell shaped curve before the second derivative goes to zero and compare it to the top of a circle.
Saturday, July 27, 2013
The bottom half of a circle has tangents that don't look anything like the tangents of a parabola or the trough of a sine wave. Most notably, the last two tangent lines, the one on the left and the one on the right, are both vertical.
Both the parabola and the circle have tangents that are neatly spaced out, but the parabola seen here only has slopes that range from -2 to 2, while the circle has slopes from negative infinity to infinity.
Tomorrow, a somewhat lesser known curve sometimes mistaken for a parabola, the catenary.
Friday, July 26, 2013
These are the tangent lines defined by the trough of a sine wave, with x between -pi and 0. Over this particular stretch, the curve might be hard to distinguish from the parabola from x varying from -1 to 1, but a close look at the tangent patterns should show how the two curves are different. Notice where the first downward sloping red tangent line meets the last upward sloping purple line.
Here in the sine wave curve, the first few tangents on the left are almost on top of each other, as are the last few tangents on the right.
Here are the parabola tangents again. The slopes of these tangents are increasing steadily, while the sine curve tangents max out at a slope of 1 and begin to get smaller as we leave the trough.
Tomorrow, another famous "round valley" curve, the bottom half of a circle.
Thursday, July 25, 2013
Over the next few days, I'm going to play with describing a function using just the tangent lines. The first function is a parabola y = x² with x between -1 and 1 in increments of 0.1. Since this uses a total of 21 lines, I decided to use a seven colored rainbow sequence (ROY G. BIV for all you spectrum nerds out there) where three consecutive lines are red, then three orange, etc.
I think these are pretty and I'm going to try some other functions as well over the next few days.