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Calculus, SPEC2 2023 VCAA 1

Viewed from above, a scenic walking track from point to point is shown below. Its shape is given by

The minimum turning point of section occurs at point . Point is a point of inflection and the curves meet at point . Distances are measured in kilometres.
 

  1. Show that    and  .  (1 mark)

  2. Verify that the two curves meet smoothly at point .  (2 marks)

  3.  i. Find the coordinates of point .  (1 mark)

  4. ii. Find the coordinates of point .  (1 mark)

The return track from point to point follows an elliptical path given by

, where .

  1. Find the Cartesian equation of the elliptical path.  (2 marks)

  2. Sketch the elliptical path from to on the diagram above.  (1 mark)

  3.  i. Write down a definite integral in terms of that gives the length of the elliptical path from to .  (1 mark)

  4. ii. Find the length of the elliptical path from to .
  5.     Give your answer in kilometres correct to three decimal places.   (1 mark)

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a.   

   


 

b.   


 

c.i. 
 

c.ii. 
 

d.   
 

e.    
          
 

f.i.   
 

f.ii. 

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a.   

   


 

b.   


 

c.i. 


 

c.ii. 


 

d.   


 

e.    
          
 

f.i.   


 

f.ii.   

Calculus, SPEC1 2020 VCAA 9

Consider the curve defined parametrically by

where .

  1. can be written in the form  where    and    are real numbers.
  2. Show that    and  .  (2 marks)

  3. Find the arc length, , of the curve from    to  . Give your answer in the form  , where  .  (3 marks)

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Show Worked Solution

a.  
 
 
   

 

♦♦ Mean mark part (b) 31%.

 

b.  
   
   
   
   
   
   

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

Show Worked Solution

a.   


 

b.   

♦ Mean mark 49%.

 

c.   
    ²

 

d.   


 


  

e. 
 

♦♦ Mean mark 27%.