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Calculus, MET2 2019 VCAA 1

Let  .

  1. Find .  (1 mark)

  2. i.  State the nature of the stationary point on the graph of    at the origin.  (1 mark)

  3. ii.  Find the maximum value of the function    and the values of    for which the maximum occurs.  (2 marks)

  4. iii. Find the values of    for which    is always negative.  (1 mark)

  5. i.  Find the equation of the tangent to the graph of    at  .  (1 mark)

  6. ii. Find the area enclosed by the graph of    and the tangent to the graph of    at  , correct to four decimal places.  (2 marks)

  7. Let    be a point on the graph of  , where  .
  8. Find the minimum distance between    and the point  , and the value of    for which this occurs, correct to three decimal places.  (3 marks)

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  2. i. 
  3. ii.
  4. iii.
  5. i.   
  6. ii.

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a.   
   

​

b.i.   
 

 

b.ii.   
 

 

b.iii.   
 

 

c.i.   
 

 

c.ii.   
   

 

d.   

Calculus, SPEC1 2011 VCAA 3

  1. Show that    can be written in the form    (1 mark)

  2. Sketch the graph of the relation    on the axes below.
  3. Label any asymptotes with their equations and label any intercepts with the axes, writing them as coordinates.  (3 marks)

     
    VCAA 2011 spec 3b
     

  4. Find the area enclosed by the graph of the relation  , the -axis, and the lines    and    (3 marks)

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  2. “
    VCAA 2011 spec 3bi

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a.   
   
   

 

b.   


 

VCAA 2011 spec 3bi

 

c.   

 
   
   
   

Calculus, SPEC2 2017 VCAA 3

A brooch is designed using inverse circular functions to make the shape shown in the diagram below.
 


 

The edges of the brooch in the first quadrant are described by the piecewise function

  1. Write down the coordinates of the corner point of the brooch in the first quadrant.  (1 mark)

  2. Specify the piecewise function that describes the edges in the third quadrant.  (1 mark)

  3. Given that each unit in the diagram represents one centimetre, find the area of the brooch.
  4. Give your answer in square centimetres, correct to one decimal place.  (3 marks)

  5. Find the acute angle between the edges of the brooch at the origin. Give your answer in degrees, correct to one decimal place.  (3 marks)

  6. The perimeter of the brooch has a border of gold.
    Show that the length of the gold border needed is given by a definite integral of the form  , where  . Find the values of and .  (2 marks)

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  3. ²

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a.   


 

b.   

♦ Mean mark 49%.

 

c.   
    ²

 

d.   


 


  

e. 
 

♦♦ Mean mark 27%.

  

Calculus, SPEC1 2015 VCAA 8

  1.  Show that    (2 marks)


  2.  The graph of    is shown below
     
         
     
  3. i.  Write down the equations of the asymptotes.  (1 mark)

  4. ii. On the axes above, sketch the graph of  , labelling any asymptotes with their equations.  (1 mark)

  5.  Find    (1 mark)

  6.  Find the area enclosed by the graph of  , the -axis and the line    (2 marks)

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a.   

b. (i)  
  (ii)  

c.   

d.   

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a.   

 
 
 
 

 

b.i. 

 

b.ii.

 

c.  
   
   

 

d.   
 
 

 

 

♦ Mean mark part (d) 49%.

 
 
 
 
 

Calculus, SPEC1 2014 VCAA 7

Consider  .

  1.  Write down the range of  .  (1 mark)

  2.  Show that  .  (1 mark)

  3.  Hence evaluate the area enclosed by the graph of  , the -axis and the lines    and  .  (3 marks)

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a. 

♦♦♦ Mean mark 4%.


 


 
 

 

b.  

 

 

c.