W.W. Sawyer's book on calculus begins with some simple graphs which unfortunately missed their mark with me, at first. There is such a thing as being too visual and too literal when studying mathematics, and this is one of my problems. The first two graphs attempt to show "snapshots" of an object moving vertically (on the plane of the page, perpendicular to the top and moving towards the top of the page) at regularly sequential times. The two quantities being measured are movement in space, and time elapsed. However, the sequence of dots at different times looks like a diagonal line if you join them, which I was tempted to do. If I joined the dots, it made me think that the object was moving in a diagonal line, when it was not. It took me quite a while to figure this out.

If the hypothetical object (represented only by a dot) was indeed moving in a diagonal line, it wouldn't have made much difference in the graph, since it was still moving at a steady pace. But if the graph were depicting the steady, unchanging pace rather than the movement in space, then it wouldn't look the same. The line would be flat, or horizontal.

Remember that a couple of entries ago, I realized that the graph of a linear equation, a straight line, is also the graph of an arithmetic progression in which a quantity increases by the same amount over and over again. The slope of the diagonal line represents the amount of the arithmetic increase. If the slope is 3, for instance, then the graph depicts a progression which is always increasing by adding 3. It's counting up by threes. But the slanted line that results from y = 3x also looks like the trajectory of something moving. When I first encountered the acceleration of gravity, I just could not get it through my head what "thirty-two feet per second per second" meant. Why did they repeat the "per second" bit? Finally I was able to understand (after working with progressions) that the peculiar repeated phrase really meant "thirty-two feet per second FASTER EACH second." If it's free-falling in the ideal abstract Newtonian universe, it's going 32 feet (or 9.8 meters) faster each second.

So that straight line is the line of something which is accelerating at a steady pace, adding the same amount of speed each second as it goes faster. But the steep graph of y = 3x looks like it's going up, not down. Well, I suppose it could look like something going down the slope if I looked at it another way. But in this graph, nothing is going either up or down. The only thing that it's measuring is how fast its rate of whatever is changing.

It gets even more confusing with graphs that depict things that have varying rates of change. If something starts off slow, then adds speed as it accelerates at a faster rate each second, the curve will start wide and then get vertically steep quickly. To graphic-visual me, it looks like a rocket-plane taking off, and could depict that if you used the curve as a visual depiction of motion in space rather than rate of change. But the same curve could just as easily depict some other varying quantity which had nothing to do with moving in space. I have to stop making pictures out of graphs, which is hard to do because if I am in artist mode, everything I see is a picture.

Posted at 1:41 am | link