f(alpha) exercises

First, alpha(q) is the negative of the slope of the tau(q) curve.

From this we can derive a formula for alpha(q). In the special case where all scaling factors ri take on a common value r, the formula simplifies.

Exercise 1. Is alpha only positive, only negative, or both? Answer

Exercise 2. Estimating some values of alpha from the slope of the tau(q) curve.

For each alpha(q), f(alpha) is the y-intercept of the tangent line through the point (q, tau(q)). From this we see

f(alpha) = q*alpha + tau(q)

Exercise 3 Use the estimates of alpha in Exercise 2 to estimate the corresponding f(alpha).

The oblique asymptotes of the tau(q) curve give the maximum and minimum values of the slope of the tangent line, hence the maximum and minimum values of alpha. If these asymptotes also pass through the origin, then f(alphamax) = 0 and f(alphamin) = 0.

Exercise 4. Add this information and the point corresponding to q = 0 to that of Exercise 3 to obtain a very crude approximation of the f(alpha) curve. Answer

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