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Calculus, MET1 SM-Bank 21

The rule for function   is  .  The diagram shows the graph  .

 Inverse Functions, EXT1 2010 HSC 3b

The graph has two points of inflection. 

  1. Find the coordinates of these points.   (2 marks)

  2. Explain why the domain of must be restricted if is to have an inverse function.    (1 mark)

  3. Find the rule for the inverse function if the domain of is restricted to     (2 marks)

  4. Find the domain for .    (1 mark)

  5. Sketch the curve  .   (1 mark)

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  5. Inverse Functions, EXT1 2010 HSC 3b Answer

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a.   
 
 
   
   

 

 
   
 
   
COMMENT: It is also valid to show that is an even function and if a P.I. exists at , there must be another P.I. at .
 

 


 

 

 

 

b.  
 
 
 

 

c.  

 

 

 

 

d.  

 

e. 

Inverse Functions, EXT1 2010 HSC 3b Answer

Functions, MET1 2008 VCAA 10

Let  .

  1. Find the rule and domain of the inverse function  .   (2 marks)

  2. On the axes provided, sketch the graph of    for its maximal domain.   (1 mark)


     

    met1-2008-vcaa-q2
     

  3. Find    in the form    where , and are real constants.   (2 marks)

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  2.  
    met1-2008-vcaa-q10-answer
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a.  

 

 


  

b.  

♦ Mean mark part (b) 19%.
MARKER’S COMMENT: Few students were aware that a function of its own inverse function is the line    over the appropriate domain.

met1-2008-vcaa-q10-answer

 

♦ Mean mark 34%.
c.   
 
   
   
   

Calculus, MET2 2010 VCAA 1

  1. Part of the graph of the function    is shown on the axes below

     

         

    1. Find the rule and domain of  , the inverse function of  .   (3 marks)

    2. On the set of axes above sketch the graph of  . Label the axes intercepts with their exact values.   (3 marks)

    3. Find the values of , correct to three decimal places, for which  .   (2 marks)

    4. Calculate the area enclosed by the graphs of    and  . Give your answer correct to two decimal places.   (2 marks)

  2. The diagram below shows part of the graph of the function with rule
  3.         , where , and are real constants.
     

    • The graph has a vertical asymptote with equation  .
    • The graph has a y-axis intercept at 1.
    • The point on the graph has coordinates  , where is another real constant.
       

      VCAA 2010 1b

    1. State the value of .   (1 mark)

    2. Find the value of .   (1 mark)

    3. Show that  .   (2 marks)

    4. Show that the gradient of the tangent to the graph of at the point is  .   (1 mark)

    5. If the point    lies on the tangent referred to in part b.iv., find the exact value of .   (2 marks)

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  1.   i.
  2.  ii. 
     
  3. iii.
  4. iv. ²
  5.   i.
  6.  ii.
  7. iii.
  8.  iv.
  9.   v.
Show Worked Solution

a.i.  

 

ii.  

 

 iii. 

 

 

  iv.  
    ²

 

b.i.  

 

  ii.  

 

iii.  

 

  iv.  
 
   
   

 

  v.  

♦♦ Mean mark 33%.
MARKER’S COMMENT: Many students worked out the equation of the tangent which was unnecessary and time consuming.