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Calculus, SPEC2 2024 VCAA 1

Consider the function with rule  .

  1. Sketch the graph of    on the set of axes below. Label the vertical asymptotes with their equations and label the stationary points with their coordinates.   (3 marks)
     


  1. The region bounded by the graph of    and the lines    and    is rotated about the -axis to form a solid of revolution.
    1. Write down a definite integral involving only the variable , that when evaluated, will give the volume of the solid.   (2 marks)

    2. Find the volume of the solid, correct to one decimal place.   (1 mark)

  1. Now consider the function with rule  , where .  
  2. For what value of will the graph of have no asymptotes?    (1 mark)

  3. The gradient function of is given by  .
  4. For what values of will the graph of have exactly
    1. one stationary point?   (1 mark)

    2. three stationary points?   (1 mark)

    3. five stationary points?   (1 mark)

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a.   

b.i.   

b.ii. 

c.   

d.i.   

d.ii. 

d.iii.

Show Worked Solution

a.   


 

b.i    
   

 
b.ii. 
 

c.    


  

d.i.

♦♦♦ Mean mark (d.i.) 27%.
♦♦♦ Mean mark (d.ii.) 27%.
♦♦♦ Mean mark (d.iii.) 25%.
 

d.ii. 

  

 


 

d.iii 

   
 

 

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, SPEC2 2021 VCAA 1

Let  .

  1. Express in the form  , where , an are real constants.   (1 mark)

  2. State the equation of the asymptotes of the graph of .   (2 marks)

  3. Sketch the graph of on the set of axes below. Label the asymptotes with their equations, and label the maximum turning point and the point of inflection with their coordinates, correct to two decimal places. Label the intercepts with the coordinate axes.   (3 marks)

     

     

  4. Let  , where is a real constant.
  5.  i. For what values of will the graph of , have two asymptotes?   (2 marks)

  6. ii. Given that the graph of has more than two asymptotes, for what values of will the graph of have no stationary points?   (2 marks)

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  3.  
  4.  i.
  5. ii.
Show Worked Solution

a.   


 

b.   

♦ Mean mark part (c) 48%.

 
c.   

 

d.i.   

♦♦ Mean mark part (d)(i) 24%.

 

d.ii.   

♦♦♦ Mean mark part (d)(ii) 16%.


 

Calculus, SPEC2 2020 VCAA 3

Let  .

  1. Find an expression for    and state the coordinates of the stationary points of  .  (2 marks)
  2. State the equation(s) of any asymptotes of  .  (1 mark)
  3. Sketch the graph of    on the axes provided below, labelling the local maximum stationary point and all points of inflection with their coordinates, correct to two decimal places.  (3 marks)
     
         

Let  , where  .

  1. Write down an expression for  .  (1 mark)
  2.  i. Find the non-zero values of for which  .  (1 mark)
  3. ii. Complete the following table by stating the value(s) of for which the graph of    has the given number of points of inflection.  (2 marks)
     
       
         
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  3.   
  5.  i.
  6. ii.
       
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a.   

 

 

b.   

 

c.   

 

d.   

 

e.i.   

♦♦♦ Mean mark (e)(ii) 9%.

 

e.ii.   

Calculus, SPEC2 2012 VCAA 11 MC

If    and    at  , then the graph of will have

  1. a local minimum at 
  2. a local maximum at    and a local minimum at 
  3. stationary points of inflection at    and  , and a local minimum at 
  4. a stationary points of inflection at  , no other points of inflection and a local minimum at 
  5. a stationary point of inflection at  , a non-stationary point of inflection at    and a local minimum at   
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Show Worked Solution

♦ Mean mark 51%.