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Calculus, MET2 2023 VCAA 1

Let . Part of the graph of is shown below.

  1. State the coordinates of all axial intercepts of .   (1 mark)
  2. Find the coordinates of the stationary points of .   (2 marks)
    1. Let .
    2. Find the values of for which .   (1 mark)
    1. Write down an expression using definite integrals that gives the area of the regions bound by and .  (2 marks)
    2. Hence, find the total area of the regions bound by and , correct to two decimal places.   (1 mark)
  1. Let , where and .
  2. Find the possible values of and .   (4 marks)
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a.   

b.   

c.i. 

c.ii.

 

c.iii.

d.  

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a.   
  

b.   


  

ci   

  

cii  


  

ciii 
 

  

d.   

  

 

Calculus, MET2 2019 VCAA 5

Let  . The tangent to the graph of at  , where  , intersects the graph of again at and intersects the horizontal axis at . The shaded regions shown in the diagram below are bounded by the graph of , its tangent at    and the horizontal axis.
 

  1. Find the equation of the tangent to the graph of at  , in terms of .   (1 mark)

  2. Find the -coordinate of , in terms of .   (1 mark)

  3. Find the -coordinate of , in terms of .   (2 marks)

Let    be the function that determines the total area of the shaded regions.

  1. Find the rule of , in terms of .   (3 marks)

  2. Find the value of for which is a minimum.   (2 marks)

Consider the regions bounded by the graph of , the tangent to the graph of at  , where  , and the vertical axis.

  1. Find the value of for which the total area of these regions is a minimum.   (2 marks)

  2. Find the value of the acute angle between the tangent to the graph of and the tangent to the graph of at  .   (1 mark)

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b.   


 

c.   


 

d.   

 
 

 

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Calculus, MET2 2017 VCAA 1

Let  . Part of the graph of is shown below.
 

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

  2.   and   are two points on the graph of .

     

    1. Find the equation of the straight line through and .   (2 marks)

    2. Find the distance .   (1 mark)

Let  .

  1. Let  and be two points on the graph of .

     

    1. Find the distance in terms of .   (2 marks)

    2. Find the values of such that the distance is equal to  .   (1 mark)

  2. The diagram below shows part of the graphs of and  . These graphs intersect at the points with the coordinates and .
  3.  
       
  4.  
    1. Find the value of in terms of .   (1 mark)

    2. Find the area of the shaded region in terms of .   (2 marks)

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a.   

 


 

b.i.  

 

MARKER’S COMMENT: Students skilled in the use of technology will be much more efficient and minimise errors here.
b.ii.   
   
   

 

c.i.   
   
   

 

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Calculus, MET1 2011 VCAA 9

Parts of the graphs of the functions

are shown in the diagram below.

The graphs intersect when    and when  

vcaa-2011-meth-9a

The area of the shaded region is 64.

Find the value of and the value of   (4 marks)

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♦ Mean mark 41%.
MARKER’S COMMENT: Few students correctly used the intersection to achieve the final answer.

 

 

 

Calculus, MET1 2014 VCAA 5

Consider the function  ,  .

  1. Find the coordinates of the stationary points of the function.   (2 marks)

  2. On the axes  below, sketch the graph of .

     

    Label any end points with their coordinates.   (2 marks)

     

     
        met1-2014-vcaa-q5

     

  3. Find the area enclosed by the graph of the function and the horizontal line given by  .   (3 marks)

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  2.  
    met1-2014-vcaa-q5-answer3
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a.   
   
 


 

 

b.    met1-2014-vcaa-q5-answer3

 

♦ Mean mark (c) 48%.
c.    met1-2014-vcaa-q5-answer4

 
 
 
   
²

 

 
 
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Calculus, MET2 2015 VCAA 1

Let  . The point    is on the graph of  , as shown below.

The tangent at cuts the y-axis at and the x-axis at

VCAA 2015 1ai

  1. Write down the derivative   of .   (1 mark)

  2.  i. Find the equation of the tangent to the graph of   at the point  .   (1 mark)

    ii. Find the coordinates of points and .   (2 marks)

  3. Find the distance and express it in the form  , where and are positive integers.  (2 marks)

VCAA 2015 1di

  1. Find the area of the shaded region in the graph above.   (3 marks)

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  2.  i.
    ii.
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a.   
 
   

 

b.i.   


  

  
b.ii.   


  

 

  

c.  

 
 

 

d.  

MARKER’S COMMENT: Many students complicated their answer by splitting up the area.

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