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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. 

Show Worked Solution

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.