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Calculus, EXT1 C1 EQ-Bank 2

75 Tasmanian Devils are placed in a Devil's Ark sanctuary that can support a maximum population of 500 devils. The increase in the devil population is proportional to the difference between the devil population and the number of devils that the sanctuary can support.

  1. Show that    is a solution to the differential equation , where is the current devil population, is time in years and is the constant of proportionality.   (1 mark)

  2. After two years, the population has grown to 245. Show that  , correct to three significant figures.   (2 marks)

  3. Hence, determine how many devils will be on the island after 5 years.  (1 mark)

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Calculus, EXT1 C1 2017 HSC 14c

The concentration of a drug in a body is  , where is the time in hours after the drug is taken.

Initially the concentration of the drug is zero. The rate of change of concentration of the drug is given by   

.

  1. By differentiating the product   show that

     

    .  (2 marks)

  2. Hence, or otherwise, show that  .  (2 marks)

  3. The concentration of the drug increases to a maximum.

     

    For what value of does this maximum occur?  (2 marks)

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i. 

 

 
 
 

 

ii.   

♦♦ Mean mark 37%.
 
 

 

 

 

 

iii.   
 

 

♦♦ Mean mark 39%.

Calculus, EXT1 C1 2016 HSC 12b

In a chemical reaction, a compound is formed from a compound . The mass in grams of and are and respectively, where is the time in seconds after the start of the chemical reaction.

Throughout the reaction the sum of the two masses is 500 g. At any time , the rate at which the mass of compound is increasing is proportional to the mass of compound .

At the start of the chemical reaction,  and  .

  1.  Show that  .  (3 marks)

  2.  Show that    satisfies the equation in part (i), and find the value of .  (2 marks)

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i.  

♦ Mean mark (part i) 43%.

 

 

 

 

ii.   
 

 

 

 

Calculus, EXT1 C1 2010 HSC 2b

The mass of a whale is modelled by

 

where    is measured in tonnes,    is the age of the whale in years and    is a positive constant.

  1. Show that the rate of growth of the mass of the whale is given by the differential equation
     
       (1 mark)

  2. When the whale is 10 years old its mass is 20 tonnes.

     

    Find the value of  ,  correct to three decimal places.    (2 marks)

  3. According to this model, what is the limiting mass of the whale?    (1 mark)

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i. 

IMPORTANT: Know this standard proof well and be able to produce it quickly.

 
 

 

ii.  

 
 

 

iii.