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12 Optimisation - 12 OPTIMISATION 1 2 Fora>0theline a(a 1 y...

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12.OPTIMISATION 1. For  a  > 0 the line  x a y a a - = + ) 1 ( 2  forms a triangle in the first quadrant with the  x-  and  y -axes. (a) [2 marks] Find the  x-  and  y- intercepts of the line in terms of  a . __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (b) [2 marks] Find the area of the triangle as a function of  a . __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (c) [6 marks] Find analytically the value of  a  which maximises the area. Fully justify your answer.  What is the value of the maximum area? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET Calculus 70 12. Optimisation
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2. Suppose that ) 1 2 )( 4 ( ) ( 2 - - = x x x f  and  4 ) ( 2 - = x x g The line  p x = , where 0 <  p  < 2, intersects the graph of  f  at point A and the graph of  g  at point B.  A  triangle has vertices A, B and the origin O. (a) [2 marks] Draw a sketch showing the graphs of the functions  f  and  g , the line  p x =  and  points A, B and O.  (b) [5 marks] Write the area of triangle OAB as a function of  p . __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (c) [2 marks] What value of  p  maximizes the area of the triangle?   __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET Calculus 71 12. Optimisation
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__________________________________________________________________________________ VET Calculus 72 12. Optimisation
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3. The voltage  V  of an electrical outlet and the time  t  seconds are related by the formula V  = 30 sin (60 π t ). (a) What is the rate of change of voltage at any time  t  ? [2] ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ (b)When is the rate of change in voltage a maximum and what is the maximum rate of change? [4] ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ (c) When is the rate of change equal to zero and what does this mean in terms of voltage? [3] ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET Calculus 73 12. Optimisation
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____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ ____________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET Calculus 74 12. Optimisation
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4. Consider the functions  f ( x ) =  2 1 ( x 2  – 5 x ) x  and  g ( x ) = -3 x . S is the point of intersection of the graph of the function  f  with the positive x-axis. P is a point of the graph of the function  g .
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