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Calculus, MET1 2023 VCAA 7

Consider . Part of the graph of    is shown below.
 

  1. State the range of .   (1 mark)
  2. Sketch the graph of the inverse function  on the axes above. Label any endpoints and axial intercepts with their coordinates.   (2 marks)
  3. Determine the equation of the domain for the inverse function  .   (2 marks)
  4. Calculate the area of the regions enclosed by the curves of   and  .   (2 marks)
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a.   

b.   

c.   

d.   

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a.   

b.   

c.   


 

 


♦ Mean mark (c) 48%.
MARKER’S COMMENT: Common error → writing the function as .

d.     

 
   
   
   
   

♦♦♦ Mean mark (d) 24%.
MARKER’S COMMENT: Using the symmetry properties of the graph and its inverse helped answer this question efficiently.

Calculus, MET1 2020 VCAA 6

Let  , where  .

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

The graph of  , where  , is shown on the axes below.
 

     
 

  1. On the axes above, sketch the graph of    over its domain. Label the endpoints and point(s) of intersection with the function  , giving their coordinates.   (2 marks)

  2. Find the total area of the two regions: one region bounded by the functions  and , and the other region bounded by    and the line  . Give your answer in the form  , where  .   (4 marks)

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  1.  

  2.  
       
  3.  ²
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a.   

 

 

 

b.     

 

c.     
 
 
 
 
 
  ²

Calculus, MET1 2017 VCAA 9

The graph of    is shown below.
 

  1. Calculate the area between the graph of and the -axis.   (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of is  .   (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis, where  .

  1. Find the equation of the line through and in the form  , for  .   (2 marks)

  2. Find the coordinates of when  .   (4 marks)

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a.  
   
   
   
   
   

 

♦♦ Mean mark part (b) 35%.
MARKER’S COMMENT: Establishing the common denominator in the working was required!
b.  
 
   
   

 

c. 

♦♦♦ Mean mark part (c) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as to solve.

 

 

   
 
   

 

 

 

d. 

♦♦♦ Mean mark part (d) 17%.

   
 
   

 

 

 

 

Calculus, MET2 2009 VCAA 1

Let  

The graph of    is shown below.

VCAA 2009 1a

  1. State the interval for which the graph of is strictly decreasing.   (2 marks)

  2. Points and are the points of intersection of    with the -axis. Point has coordinates  and point has coordinates .

     

    Find the length of such that the area of rectangle is equal to the area of the shaded region.   (2 marks)
     
    VCAA 2009 1c

  3. The points and are labelled on the diagram.

      
     

          VCAA 2009 1d

     

    1. Find , the gradient of the chord .   (Exact value to be given.)   (1 mark)

    2. Find    such that  .  (Exact value to be given.)   (2 marks)

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a. 

b. 

c.i.   

c.ii. 

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a.   vcaa-2009-1ai

 

 

b.   vcaa-2009-1ci
 
 

 

c.i.  
 

 

 c.ii.  
 

Calculus, MET2 2014 VCAA 19 MC

Jake and Anita are calculating the area between the graph of    and the -axis between    and  

Jake uses a partitioning, shown in the diagram below, while Anita uses a definite integral to find the exact area.
 

VCAA 2014 19mc
 

The difference between the results obtained by Jake and Anita is

A.  

B.  

C.  

D.  

E.  

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