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Trigonometry, 2ADV T3 2020 HSC 31

The population of mice on an isolated island can be modelled by the function.

,

where    is the time in weeks and  . The population of mice reaches a maximum of 35 000 when    and a minimum of 5000 when  . The graph of    is shown.
 

  1. What are the values of and ?  (2 marks)

  2. On the same island, the population of cats can be modelled by the function
     

     
    Consider the graph of    and the graph of  .

     

    Find the values of  , for which both populations are increasing.  (3 marks)

  3. Find the rate of change of the mice population when the cat population reaches a maximum.  (2 marks)

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a.   
   

 

 
 

 

b.   

♦♦ Mean mark part (b) 30%.

 

c.   

♦♦♦ Mean mark part (c) 27%.
 

 

Calculus, 2ADV C3 2004 HSC 5b

A particle moves along a straight line so that its displacement, metres, from a fixed point is given by  , where    is measured in seconds.

  1. What is the initial displacement of the particle?  (1 mark)

  2. Sketch the graph of    as a function of    for  .  (2 marks)

  3. Hence, or otherwise, find when AND where the particle first comes to rest after  .  (2 marks)

  4. Find a time when the particle reaches its greatest magnitude of velocity. What is this velocity?  (2 marks)

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i.   

 
 

 

 

 ii. 

 

 Calculus in the Physical World, 2UA 2004 HSC 5b

 

iii.  
 

 

 

 
 
 

 

 

iv. 

   

 

 

 
 

 

Calculus, 2ADV C4 2015 HSC 15c

Water is flowing in and out of a rock pool. The volume of water in the pool at time hours is litres. The rate of change of the volume is given by

At time  , the volume of water in the pool is 1200 litres and is increasing.

  1. After what time does the volume of water first start to decrease?  (2 marks)

  2. Find the volume of water in the pool when  .  (2 marks)

  3. What is the greatest volume of water in the pool?  (1 mark)

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i.   

♦ Mean mark (i) 37%.

 

ii.   
 
 

 

 

 
 

 

iii. 

♦♦ Mean mark (iii) 27%.