Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number. t (hours) 3 8 P (t) (passengers) 35 204 600 728 The number of passengers who have boarded a ship is modeled by the differentiable function P, where t is the number of hours since boarding began. Values of P (t) for selected values of t are given in the table above.

(b)      Write P′(4.5) as the limit of a difference quotient. Use the data in the table to approximate P′(4.5). Show the computations that lead to your 

(c)      Must there be a time  t, for 3≤t≤6, at which P(t)=500 ? Justify your answer. 

d)      The total number of gallons of water used by the passengers on the ship is modeled by the function  G defined by G(t)=120tt√ for 0≤t≤8, where t is the number of hours since boarding began. Find G′(4), the rate at which passengers use water, in gallons per hour, at time t=4hours. 

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