What is the purpose of the paragraph/explanation on page 2, right BEFORE the heading "3 Cautionary Tales"? (The paragraph beginning with "First y=x-1..."
MIT 18.03SC Differential Equations
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I think this paragraph is giving mathematical evidence that the estimate for y(1.5) is too large.
The "First," "Second," and "Third" refer to lines in the preceding table (n= 0, 1, and 2 respectively).
Each of the paragraphs shows the same thing: y' along the line of estimation is less than or equal to the y' estimate we found at the start of the interval.
For example, the paragraph beginning with "First" gives the equation used to estimate the solution to the ODE in the first step: y=x-1. The writer explains what y' would be on that line by substituting (x-1) for y (the writer says y'=(x-1)^2 -x^2=-2x+1 on the line y=x-1). But in the interval of the first step [0,0.5], \[-2x+1\le1 \]. So on the estimated line, y' is less than or equal to 1, our estimated slope. Thus our slope is probably too large.
(Some sort of additional evidence was helpful. After all, it is not sufficient to examine only y'' at x=0, finding y''<0, because the concavity may quickly change to y''>0, causing an estimate we thought was too large to actually be too small.)
Hope that makes sense.