Wikipedia:Reference desk/Archives/Mathematics/2024 December 15
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December 15
[edit]What is the cause of this paradox?
[edit]I recently completed a calculus term, in which one of the last units involving how much one aspect of an object was changing in relation to time at a certain point, given the rate of change of another aspect. Many specific questions could be analyzed as a right triangle with one leg (the x) remaining constant and the other leg (the y) growing at a specified rate. When it came time to solve for the value of the dz/dt (the rate of the hypotenuse’s growth with respect to time) at a certain point, it ended up as less than the provided dy/dt. Here’s an illustration:
The x is the distance from me to a tower. This remains constant.
The y is the distance from the tower to a flying bird.
The dy/dt is the speed at which the bird is flying from the tower.
The z is my distance from the bird.
In this illustration, the distance between me and the bird is increasing at a slower rate than the speed at which the bird itself is flying. What is the cause of this paradox? Primal Groudon (talk) 19:43, 15 December 2024 (UTC)
- I do not see any paradox here. Ruslik_Zero 20:30, 15 December 2024 (UTC)
- If the bird is between you and the tower (0 ≤ y < x), the distance between you and the bird is even decreasing: dz/dt < 0. By the time it flies right overhead (y = x), the distance is momentarily stationary: dz/dt = 0. After that, it increases: dz/dt > 0. This rate of increase will asymptotically approach dy/dt from below as the bird flies off into an infinite distance. --Lambiam 00:34, 16 December 2024 (UTC)
- I think the issue here is that even though the rate of change of z is less than the rate of change of y, z never actually becomes less than y. You can see this graphically, for example, by comparing the graphs of y=x and y=√(x2+1). The second graph is always above the first graph, but the slope of the first graph is x/√(x2+1), which is always less than 1, the slope of the first graph. But this is typical behavior when a graph has an asymptote. As a simpler example, the slope of 1/x is negative, but the value never goes below 0 (at least for x>0). Similarly, the slope of x+1/x is always less than 1, but the value of x+1/x is always greater than x (again, for x>0). The graph of y=√(x2+1) is one branch of a hyperbola having y=x as an asymptote, and this looks very much like the x>0 part of y=x+1/x. In general the difference in rates of change can imply that that two quantities get closer and closer to each other, but this does not mean they ever become equal. This phenomenon is, perhaps, counterintuitive for many people, but the math says it can happen anyway. I don't know if this rises to the level of a paradox, but I can see that it might be concerning for some. --RDBury (talk) 09:39, 16 December 2024 (UTC)
- For x > 0, the graph of y=√(x2+1) looks even more like that of y=x+1/(2x). For example, when x = 5, √(x2+1) = √26 ≈ 5.09902 is approximated much more closely by 5.1 than by 5.2. --Lambiam 18:35, 16 December 2024 (UTC)
- I think the issue here is that even though the rate of change of z is less than the rate of change of y, z never actually becomes less than y. You can see this graphically, for example, by comparing the graphs of y=x and y=√(x2+1). The second graph is always above the first graph, but the slope of the first graph is x/√(x2+1), which is always less than 1, the slope of the first graph. But this is typical behavior when a graph has an asymptote. As a simpler example, the slope of 1/x is negative, but the value never goes below 0 (at least for x>0). Similarly, the slope of x+1/x is always less than 1, but the value of x+1/x is always greater than x (again, for x>0). The graph of y=√(x2+1) is one branch of a hyperbola having y=x as an asymptote, and this looks very much like the x>0 part of y=x+1/x. In general the difference in rates of change can imply that that two quantities get closer and closer to each other, but this does not mean they ever become equal. This phenomenon is, perhaps, counterintuitive for many people, but the math says it can happen anyway. I don't know if this rises to the level of a paradox, but I can see that it might be concerning for some. --RDBury (talk) 09:39, 16 December 2024 (UTC)