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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, MET1 2023 VCAA SM-Bank 5

Let  ,  where  .

  1. Calculate the average rate of change of between and .  (1 mark)

  2. Calculate the average value of between and .  (2 marks)

  3. Four trapeziums of equal width are used to approximate the area between the functions    and the -axis from to .
  4. The heights of the left and right edges of each trapezium are the values of , as shown in the graph below.

  1. Find the total area of the four trapeziums.  (2 marks)

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Calculus, MET2 2022 VCAA 17 MC

A function is continuous on the domain and has the following properties:

  • The average rate of change of between and is positive.
  • The instantaneous rate of change of at is negative.

Therefore, on the interval , the function must be

  1. many-to-one.
  2. one-to-many.
  3. one-to-one.
  4. strictly decreasing.
  5. strictly increasing.
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The first property shows that the point ( , __ ) must be below ( , __ ) to give a positive rate of change.

The second property gives the midpoint of and . But this point has a negative instantaneous rate of change.

Therefore, this shows that the function has turning points and is in the shape of a cubic, so many-to-one is the correct description.

 


♦♦ Mean mark 39%.

Calculus, MET2 2021 VCAA 13 MC

The value of an investment, in dollars, after months can be modelled by the function

where  .

The average rate of change of the value of the investment over the first 12 months is closest to

  1. $10.00 per month.
  2. $10.20 per month.
  3. $10.50 per month.
  4. $125.00 per month.
  5. $127.00 per month.
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