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Calculus, MET2 2024 VCAA 2

A model for the temperature in a room, in degrees Celsius, is given by

where represents time in hours after a heater is switched on.

  1. Express the derivative as a hybrid function.   (2 marks)
  1. Find the average rate of change in temperature predicted by the model between and .
  2. Give your answer in degrees Celsius per hour.   (1 mark)
  1. Another model for the temperature in the room is given by .
  2.  i. Find the derivative .   (1 mark)
  3. ii. Find the value of for which .
  4.     Give your answer correct to three decimal places.   (1 mark)
  1. Find the time when the temperatures predicted by the models and are equal.
  2. Give your answer correct to two decimal places.   (1 mark)
  1. Find the time when the difference between the temperatures predicted by the two models is the greatest.
  2. Give your answer correct to two decimal places.   (1 mark)
  1. The amount of power, in kilowatts, used by the heater hours after it is switched on, can be modelled by the continuous function , whose graph is shown below.

The amount of energy used by the heater, in kilowatt hours, can be estimated by evaluating the area between the graph of and the -axis.
 

  1.   i. Given that is continuous for , show that .   (1 mark)
  2.  ii. Find how long it takes, after the heater is switched on, until the heater has used 0.5 kilowatt hours of energy.
  3.     Give your answer in hours.   (1 mark)
  4. iii. Find how long it takes, after the heater is switched on, until the heater has used 1 kilowatt hour of energy.
  5.     Give your answer in hours, correct to two decimal places.   (2 marks)

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Calculus, MET2 2023 VCAA 3

Consider the function .

  1. State the value of .   (1 mark)
  2. The derivative, , can be expressed in the form .
  3. Find the real number .   (1 mark)
  4.  i. Let be a real number. Find, in terms of , the equation of the tangent to at the point .   (1 mark)

    ii. Hence, or otherwise, find the equation of the tangent to that passes through the origin, correct to three decimal places.   (2 marks)

  1.  

Let .

  1. Find the coordinates of the point of inflection for , correct to two decimal places.   (1 mark)
  2. Find the largest interval of values for which is strictly decreasing.
  3. Give your answer correct to two decimal places.   (1 mark)
  4. Apply Newton's method, with an initial estimate of , to find an approximate -intercept of .
  5. Write the estimates and in the table below, correct to three decimal places.   (2 marks)
      


  6. For the function , explain why a solution to the equation should not be used as an initial estimate in Newton's method.   (1 mark)
  7. There is a positive real number for which the function has a local minimum on the -axis.
  8. Find this value of .   (2 marks)
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b.   

ci. 

cii.

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e.

f. 

g.

h.

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a.   


  

b.    
   
   
 
     
 
   
ci. 


♦♦ Mean mark (c)(i) 50%.

cii. 

  

♦♦♦ Mean mark (c)(ii) 30%.
MARKER’S COMMENT: Some students did not substitute (0, 0) into the correct equation or did not find the value of and used .
d.    
 
 

e.   


♦♦ Mean mark (e) 40%.
MARKER’S COMMENT: Round brackets were often used which were incorrect as endpoints were included. Some responses showed the interval where the function was strictly increasing.

f.     

g.   

♦♦♦ Mean mark (g) 20%.

h.   

 


♦♦♦ Mean mark (h) 10%.
MARKER’S COMMENT: Many students showed that but failed to couple it with . Ensure exact values are given where indicated not approximations.

Calculus, MET1 2018 VCAA 1b

Let  .

Evaluate  .   (2 marks)

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