Exercise 3 3 27 Suppose that the size of a population at time t is given by N t

# Exercise 3 3 27 suppose that the size of a population

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Exercise 3 . 3 : 27 Suppose that the size of a population at time t is given by N ( t ) = 500 t 3 + t , t 0 . ( a ) Use a graphing calculator to sketch the graph of N ( t ). ( b ) Determine the size of the population as t → ∞ . We call this the limiting population size . ( c ) Show that, at time t = 3, the size of the population is half its limiting size. Exercise 3 . 3 : 29 Suppose that the size of a population at time t is given by N ( t ) = 100 1 + 3 e - t , t 0 . ( a ) Use a graphing calculator to sketch the graph of N ( t ). ( b ) Determine the size of the population as t → ∞ , using the basic rules for limits compare your answer with the graph that you sketched in ( a ). For Section 3 . 4 Exercise 3 . 4 : 1 , 3 , 5 , 7 , 9 In the following problems, evaluate the trigonometric limits, 1) lim x 0 sin(3 x ) 3 x , 2) lim x 0 sin(5 x ) x , 3) lim x 0 sin( πx ) x , 4) lim x 0 sin( πx ) x , 5) lim x 0 sin( x ) cos( x ) x (1 - x ) . Exercise 3 . 4 : 19 Let f ( x ) = ln( x ) x , x > 0, 1

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( a ) Use graphing calculator to graph y = f ( x ). ( b ) Use a graphing calculator to investigate the values of x for which 1 x ln( x ) x 1 x holds. ( c ) Use your result in ( b ) explain why: lim x →∞ ln( x ) x = 0.
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