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Calculus, 2ADV C3 2024 HSC 17

In a particular electrical circuit, the voltage (volts) across a capacitor is given by

,

where is a positive constant and is the number of seconds after the circuit is switched on.

  1. Draw a sketch of the graph of , showing its behaviour as increases.   (2 marks)
     

  1. When , the voltage across the capacitor is 2.6 volts.
  2. Find the value of , correct to 3 decimal places.   (2 marks)
  1. Find the rate at which the voltage is increasing when , correct to 3 decimal places.   (2 marks)

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a.   
       

b.  

c. 

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a.   
       
♦ Mean mark (a) 49%.

b.   

 
 
 
 
 
   

 
c.   

 
   
   

Calculus, 2ADV C3 2022 HSC 20

A scientist is studying the growth of bacteria. The scientist models the number of bacteria, , by the equation

,

where is the number of hours after starting the experiment.

  1. What is the initial number of bacteria in the experiment?  (1 mark)

  2. What is the number of bacteria 24 hours after starting the experiment?  (1 mark)

  3. What is the rate of increase in the number of bacteria 24 hours after starting the experiment?  (2 marks)

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a.   
   

 

b.   

 
   
   

 

c.   
 
   

 

 
   
   

Calculus, 2ADV C3 2021 HSC 23

A population, , which is initially 5000, varies according to the formula

where is a positive constant and is time in years, .

The population is 1250 after 20 years.

Find the value of , correct to one decimal place, for which the instantaneous rate of decrease is 30 people per year.  (4 marks)

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♦ Mean mark 42%.

 

 

 

 
 

Calculus, 2ADV C3 2020 HSC 21

Hot tea is poured into a cup. The temperature of tea can be modelled by  , where is the temperature of the tea, in degrees Celsius, minutes after it is poured.

  1. What is the temperature of the tea 4 minutes after it has been poured?  (1 mark)

  2. At what rate is the tea cooling 4 minutes after it has been poured?  (2 marks)

  3. How long after the tea is poured will it take for its temperature to reach 55°C?  (3 marks)

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a.   
   
   

 

b.   
   

 

 
 

 

c.   

♦ Mean mark part (c) 44%.
 
 

Calculus, 2ADV C3 2016 HSC 16b

Some yabbies are introduced into a small dam. The size of the population, , of yabbies can be modelled by the function

where is the time in months after the yabbies are introduced into the dam.

  1. Show that the rate of growth of the size of the population is
  2. .  (2 marks)

  3. Find the range of the function , justifying your answer.  (2 marks)

  4. Show that the rate of growth of the size of the population can be written as
  5. .  (1 mark)

  6. Hence, find the size of the population when it is growing at its fastest rate.  (2 marks)

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i.  
 
   
   

 

ii.  

♦♦♦ Mean mark (ii) 21%.

 

iii. 

♦♦♦ Mean mark (iii) 18%.

 
 

 

iv.  

♦♦♦ Mean mark (iv) 14%.

 

hsc-2016-16bi

Calculus, 2ADV C4 2009 HSC 7a

The acceleration of a particle is given by

,

where    is the displacement in metres and    is the time in seconds.

Initially its velocity is  and its displacement is 5 m.

  1. Show that the displacement of the particle is given by
  2. .   (2 marks) 

  3. Find the time when the particle comes to rest.    (3 marks)

  4. Find the displacement  when the particle comes to rest.    (1 mark)

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i.   

 

 

 
 

 

 

 

ii.  

 
 

 


 


  

iii. 

ALGEBRA TIP: Helpful identity  . Easily provable as follows:



.

Calculus, 2ADV C3 2011 HSC 7b

The velocity of a particle moving along the -axis is given by

,

where is the time in seconds and is the displacement in metres.

  1. Show that the particle is initially at rest.     (1 mark)

  2. Show that the acceleration of the particle is always positive.     (1 mark)

  3. Explain why the particle is moving in the positive direction for all  .     (2 marks)

  4. As  , the velocity of the particle approaches a constant.

     

    Find the value of this constant.     (1 mark) 

  5. Sketch the graph of the particle's velocity as a function of time.     (2 marks)

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i.   

 
 

 

 

MARKER’S COMMENT: Students whose working showed as , tended to score highly in this question.

ii.   

.

.
 

.
 

iii. 

♦♦♦ Mean mark 22%
COMMENT: Students found part (iii) the most challenging part of this question by far.


 

.
 

iv.   ,  


 

 

IMPORTANT: Use previous parts to inform this diagram. i.e. clearly show velocity was zero at    and the asymptote at  . 
v.   

Calculus in the Physical World, 2UA 2011 HSC 7b Answer