When a fish nothing river up at a speed v. against a stream vw , the energy that expends to travel to a river point above is given by a function of the form: Cyk E(v) = v – Vw where C>0 and k >2 is a number that depends on the species of fish involved. a) Show that E(v) has exactly one critical point and explain whether this point is a relative minimum or maximum. b) The critical number in part a) depends on k. Let F (k) be the function of the critical number. Graph F(k). What can be said about F(k) if k is very large

At what value of x, f(x) = Answer: z(1-x) attains its absolute minimum value over the interval [2,3]?

The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…

4527 44 -) Determine the values of constants a and b so that f(x): has an absolute maximum at the point (1,2). = = ax² + bx