Probem42+Hannah+W.,+Ari

42. For a fish swimming at speed v relative to the water, the energy expenditure per unit time is proportional to v^3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u (ubut I believe you are trying to MINIMIZE energy! 

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 * E(v) = a(v^3)L/(v-u)
 * find critical values
 * E'(v) = 3a(v^2)L/(v-u) - La(v^3)/(v-u)^2 = 0
 * 3a(v^2)L/(v-u) = La(v^3)/(v-u)^2
 * 3 = v/(v-u)
 * 3v - 3u = v
 * 2v = 3u
 * v = 3u/2
 * next you have to sub this value back into E(v) to determine if it creates the largest value. You have to compare E(3u/2) to lim as v approaches 0 ( //ms sweeney -- no, the smallest value that velocity can be would be u - in which case the fish is not making any progress. You are right; if the velocity is 0 then the fish really IS going backwards, so we can't consider that!) // of E(v) and lim as v approaches infinity of E(v). The velocity cant be negative because then the fish would be going the wrong way which would definitely not maximize E And I don't know how fast these fish are physically able to swim but theoretically they can swim at an infinate velocity. show this! What is the limit as v approaches u? What is the limit as v approaches infinity (you will need to use l'hopital's here!)

b) sketch the graph of E