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4. D. Simonton claims that the "productivity levels" of people in different fields can be described as a function of their "career age" t by p(t) =e " - e" where a and b are constants which depend on the field of work, and career age is approximately 20 less than the actual age of the individual. a) Based on this model, at what ages do mathematicians (a=.03, b=.05), geologists (a=.02, b=.04), and historians (a=.02, b=.03) reach their maximum productivity? b) Simonton says "With a little calculus we can show that the curve (p(t) ) maximizes at b-a t=* In(b/a). Use calculus to show that Simonton is correct. Note: Models of this type have uses for describing the behavior of groups, but it is dangerous and usually invalid to apply group descriptions or comparisons to individuals in the group. (Scientific Genius, by Dean Simonton, Cambridge University Press, 1988, pp. 69 -73) 5. Fig. 10 shows f(x) and g(x) for 0 s x $ 5. Let h(x) = f(x) / s(x). (a) At what value(s) of x does h(x) have a root? (b) Where does h(x) have vertical asymptotes? Fig. 10