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Functions, MET2 2024 VCAA 1

Consider the function 

  1. State, in terms of where required, the values of for which .  (1 mark
  1. Find the values of for which the graph of has
      
     i. exactly three -intercepts.   (2 marks)

    ii. exactly four -intercepts.   (1 mark)

  1. Let be the function , which is the function where .
      
      i. Find    (1 mark)

     ii. Find the coordinates of the local maximum of .   (1 mark)

    iii. Find the values of for which .   (1 mark)

     iv. Consider the two tangent lines to the graph of at the points where
     and . Determine the coordinates of the point of intersection of these two tangent lines.   (2 marks)

  1. Let remain as the function , which is the function where .

    Let be the function , which is the function where .
      
     i. Using translations only, describe a sequence of transformations of , for which its image would have a local maximum at the same coordinates as that of .   (1 mark)

    ii. Using a dilation and translations, describe a different sequence of transformations of , for which its image would have both local minimums at the same coordinates as that of .   (2 marks)

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a.    

bi. 

bii.

ci. 

cii.

ciii.

civ.

di.   

dii. 

Show Worked Solution

a.    

bi. 

bii.

ci.   
   
   

  
cii.  

 

  

ciii.

civ.  

di. 

dii.

 

Calculus, MET2 2022 VCAA 1

The diagram below shows part of the graph of , where .
 

  1. State the equation of the axis of symmetry of the graph of .   (1 mark)

  2. State the derivative of with respect to .   (1 mark)

The tangent to at point has gradient .

  1. Find the equation of the tangent to at point .   (2 marks)

The diagram below shows part of the graph of , the tangent to at point and the line perpendicular to the tangent at point .
 

 

  1.  i. Find the equation of the line perpendicular to the tangent passing through point .   (1 mark)

  2. ii. The line perpendicular to the tangent at point also cuts at point , as shown in the diagram above.
  3.     Find the area enclosed by this line and the curve .   (2 marks)

  4. Another parabola is defined by the rule , where .
  5. A tangent to and the line perpendicular to the tangent at , where , are shown below.

  1. Find the value of , in terms of , such that the shaded area is a minimum.   (4 marks)

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a.   

b.   

c.    

di.   

dii.   Area units²

e.   

Show Worked Solution

a.   Axis of symmetry: 
  

b.       
   

  
c.   At gradient

 
 

 
When 
 


 
Equation of tangent at :

 
 
 


d.i   Gradient of tangent

  gradient of normal  

Equation at

 
 
 


d.ii  Points of intersection of and normal are at  and .

So equate and  to find

 
 
 

   
or

Area  
   
   
  units²  

 

e.     

At   

Gradient of tangent

Gradient of normal

Equation of normal at

 
 
 
 

 
Points of intersection of normal and parabola (Using CAS)

solve

  or 

 
Calculate area using CAS


 
Using CAS Solve derivative of   with respect to to find

solve

 
   and 

Given


♦♦ Mean mark (e) 33%.
MARKER’S COMMENT: Common errors: Students found and failed to subtract or had incorrect terminals.

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.  

Show Worked Solution

a.   
  

b.   


  

ci   

  

cii  


  

ciii 
 

  

d.   

  

 

Calculus, MET2 2020 VCAA 5

Let  .

Let    be the function representing the tangent to the graph of at  , where  .

Let be the -intercept of the graph of .

  1. Show that  .   (3 marks)

  2. State the values of for which does not exist.    (1 mark)

  3. State the nature of the graph of when does not exist.   (1 mark)

  4. i.  State all values of for which  . Give your answer correct to four decimal places.   (1 mark)

  5. ii. The graph of has an -intercept at (1, 0).
  6.      State the values of    for which  .
  7.      Give your answers correct to three decimal places.   (1 mark)

The coordinate is the horizontal axis intercept of .

Let be the function representing the tangent to the graph of at  , as shown in the graph below.
 
 
     
 

  1. Find the values of for which the graphs of and , where exists, are parallel and where  .   (3 marks)

Let  , where  .

  1. Show that    for all  .   (1 mark)

A property of the graphs of is that two distinct parallel tangents will always occur at and for all  .

  1. Find all values of such that a tangent to the graph of at , for some  , will have an -intercept at .   (1 mark)

  2. Let  , where    and  .
     
    State any restrictions on the values of , , , and , given that the image of under the transformation always has the property that parallel tangents occur at    and    for all  .   (1 mark)

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  4.  i.
  5. ii.
Show Worked Solution

a.   

 
 

  

 
 
 
 

 
b.   

♦ Mean mark part (b) 46%.

♦♦ Mean mark part (c) 23%.
 

c.   


 

d.i. 


  

d.ii. 

♦♦♦ Mean mark part (d)(ii) 13%.


 

e.   

 
 

 

♦♦♦ Mean mark part (e) 13%.
 
   

 


 

f.   
   
   
   

 
g. 

♦♦♦ Mean mark part (g) 3%.

 
 

 


 

h.   

♦♦♦ Mean mark part (h) 2%.

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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Show Worked Solution

a.   


 

b.   


 

c.   


 

d.   

 
 

 

e.   


 

f.   


 

g.   

 

Calculus, MET2 2016 VCAA 2

Consider the function  

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

  3. ii. Find the values of for which the graph of    has a stationary point.   (1 mark)

The diagram below shows part of the graph of  , the tangent to the graph at    and a straight line drawn perpendicular to the tangent to the graph at  . The equation of the tangent at the point with coordinates    is  .

The tangent cuts the -axis at . The line perpendicular to the tangent cuts the -axis at .
 


 

  1.   i. Find the coordinates of .   (1 mark)

  2.  ii. Find the equation of the line that passes through and and, hence, find the coordinates of .   (2 marks)

  3. iii. Find the area of triangle .   (2 marks)

  4. The tangent at is parallel to the tangent at . It intersects the line passing through and at .

     


     
     i. Find the coordinates of .   (2 marks)

  5. ii. Find the length of .   (3 marks)

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  1.  i.
  2. ii.
  3.   i.
  4. ii.
  5.      
  6. iii. ²
Show Worked Solution
a.i.   
   
   
 

 

  

 

a.ii.   


 

b.i.   


 

b.ii.   

   

   


 

b.iii.  
   
    ²

 

♦ Mean mark part (c)(i) 43%.
MARKER’S COMMENT: Many students gave an incorrect -value here. Be careful!
c.i.   
 
 
   

  

 

c.ii.  

 

♦♦ Mean mark 32%.

 

 

 

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.
  4. ²
Show Worked Solution
a.   
 
   

 

b.i.   


  

  
b.ii.   


  

 

  

c.  

 
 

 

d.  

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

  ²

Calculus, MET2 2013 VCAA 4

Part of the graph of a function is shown below.

VCAA 2013 4a

  1. Points and are the positive -intercept and -intercept of the graph , respectively, as shown in the diagram above. The tangent to the graph of at the point is parallel to the line segment
    1. Find the equation of the tangent to the graph of at the point    (2 marks)

    2. The shaded region shown in the diagram above is bounded by the graph of , the tangent at the point , and the -axis and -axis.
    3. Evaluate the area of this shaded region.   (3 marks)
  2. Let be a point on the graph of  .
  3. Find the positive value of the -coordinate of , for which the distance is a minimum and find the minimum distance.   (3 marks)

The tangent to the graph of at a point has a negative gradient and intersects the -axis at point  , where  
 

VCAA 2013 4c
 

  1. Find the gradient of the tangent in terms of    (2 marks)

  2.   i. Find the rule for the function of that gives the area of the shaded region.   (2 marks)

  3.  ii. Find the maximum area of the shaded region and the value of for which this occurs.   (2 marks)

  4. iii. Find the minimum area of the shaded region and the value of for which this occurs.  (2 marks)

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Show Worked Solution

a.i.  

 

 


 

a.ii.   

♦ Mean mark 37%.
MARKER’S COMMENT: A common incorrect answer:
.  Know why it’s wrong!

 

 
  ²

 

  ²

 

b.  

♦♦♦ Mean mark 20%.
 

 


 

c.  

♦♦♦ Mean mark 8%.


 


 

 


 

d.i.  

♦♦♦ Mean mark 8%.


 

 

 

d.ii.   

♦♦♦ Mean mark 5%.

 

 

 met2-2013-vcaa-sec4-answer


 

d.iii.   

♦♦♦ Mean mark 3%.