b336a53425 exp(t) and sinh(t), are supported and whitespace is allowed. Its shortcomings, discussed in detail in the last lab, nameley its inaccuracy and its slowness, are just too great. And remember, we don't even have any guarantee that the concavity of the curve remains consistent. and . In order to get this "ideal point" we still need to ride along an "ideal prediction line," but what should we use as this line's slope? If you've really been following the foregoing discussion you can probably guess by now what slope we'll use. sloperight=f(xn h,yn hf(xn,yn)) . One last substitution, and our formula is complete. This program will allow you to obtain the numerical solution to the first order initial value problem:.
and that . We don't use the right tangent line itself to make our prediction, since, in a sense, it's "already there" at the right end of the interval which we are spending so much time trying to predict. Obviously its slope is too steep to be used as the slope of our "ideal prediction line," and results in an overestimate if it is used. 31.13 Asymptotic Approximations. sloperight=f(xn h,yn hf(xn,yn)) . 31.11 Expansions in Series of Hypergeometric Functions. In reality, this turns out to be asking too much. We wish to predict the right hand end-point's coordinates. Consider the following illustration.
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