Your Practice Set – Applications and Interpretation for IBDP Mathematics
Exercise 58
1 . Let x and y be the populations , in thousands , of shark and whale in an ocean respectively . The changes in the populations can be modelled by the coupled differential �dx
�( 3�2 yx )
� dt equations �� . dy
�( x�6 ) y
�� dt
( a ) State the equilibrium points for the two populations .
The initial populations of shark and whale are 2000 and 2500 respectively .
[ 2 ]
( b ) ( i ) Express d y dx in terms of x and y .
( ii ) Hence , find the value of d y dx when t � 0.
( iii ) State , at the beginning , whether the population of shark is increasing or decreasing .
[ 4 ]
2 . Let x and y be the number of soldiers , in thousands , of army A and army B respectively . The changes in the number of soldiers in a battle between them can be
modelled by the coupled differential equations
�dx
�( 2 �5 yx )
� dt �� . dy
�( x�8 ) y
�� dt
( a ) State the equilibrium points for the two populations .
[ 2 ]
The initial number of soldiers of army A and army B are 4000 and 3000 respectively .
( b ) ( i ) Express d y dx in terms of x and y .
( ii ) Hence , find the value of d y dx when t � 0.
( iii ) When x � 0 , state the range of values of y such that d x
0 dt � . [ 4 ]
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