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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.   

Graphs, SPEC1 2013 VCAA 4

  1. State the maximal domain and the range of     (2 marks)

  2. Sketch the graph of    over its maximal domain. Label the endpoints with their coordinates.   (2 marks)

     

     
              VCAA 2013 spec 4b
     

  3. Find the gradient of the tangent to the graph of    at    (2 marks)

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  2.  

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a. 

 

 

b.   

 

c.   

 

 

 
 
 

Calculus, SPEC1 2015 VCAA 9

Consider the curve represented by 

  1. Find the gradient of the curve at any point    (2 marks)
  2. Find the equation of the tangent to the curve at the point    and find the equation of the tangent to the curve at the point

     

    Write each equation in the form    (2 marks)

  3. Find the acute angle between the tangent to the curve at the point    and the tangent to the curve at the point 

     

    Give your answer in the form  , where is a real constant  (2 marks)

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a.   
 
 
 

 

b.   

 

 

 
 

 

c.   

 
 
 
 

 

Calculus, SPEC1-NHT 2018 VCAA 9

  1.    i. Given that  , show that  .   (2 marks)

  2.  ii. Show that  .  (1 mark)

  3. iii. Hence, show that  , given that  , where  .   (1 mark)

  4.  Find the gradient of the tangent to the curve    at  .   (2 marks)

  5.  A solid of revolution is formed by rotating the region between the graph of  , the horizontal axis, and the lines    and    about the horizontal axis.
  6. Find the volume of the solid of revolution.   (3 marks)

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a.i.  
 
 
 
 
 

 

a.ii.  
 
 
 

 

a.iii.   

 
 
 
   
   

 

 

b.  
 
   

 

 
 
 

 

c.  
   
   
   
   
   
   
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