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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 3

Consider the function .

  1. State the value of .   (1 mark)
  2. The derivative, , can be expressed in the form .
  3. Find the real number .   (1 mark)
  4.  i. Let be a real number. Find, in terms of , the equation of the tangent to at the point .   (1 mark)

    ii. Hence, or otherwise, find the equation of the tangent to that passes through the origin, correct to three decimal places.   (2 marks)

  1.  

Let .

  1. Find the coordinates of the point of inflection for , correct to two decimal places.   (1 mark)
  2. Find the largest interval of values for which is strictly decreasing.
  3. Give your answer correct to two decimal places.   (1 mark)
  4. Apply Newton's method, with an initial estimate of , to find an approximate -intercept of .
  5. Write the estimates and in the table below, correct to three decimal places.   (2 marks)
      


  6. For the function , explain why a solution to the equation should not be used as an initial estimate in Newton's method.   (1 mark)
  7. There is a positive real number for which the function has a local minimum on the -axis.
  8. Find this value of .   (2 marks)
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a.   

b.   

ci. 

cii.

d.

e.

f. 

g.

h.

Show Worked Solution

a.   


  

b.    
   
   
 
     
 
   
ci. 


♦♦ Mean mark (c)(i) 50%.

cii. 

  

♦♦♦ Mean mark (c)(ii) 30%.
MARKER’S COMMENT: Some students did not substitute (0, 0) into the correct equation or did not find the value of and used .
d.    
 
 

e.   


♦♦ Mean mark (e) 40%.
MARKER’S COMMENT: Round brackets were often used which were incorrect as endpoints were included. Some responses showed the interval where the function was strictly increasing.

f.     

g.   

♦♦♦ Mean mark (g) 20%.

h.   

 


♦♦♦ Mean mark (h) 10%.
MARKER’S COMMENT: Many students showed that but failed to couple it with . Ensure exact values are given where indicated not approximations.

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 2020 VCAA 4

The graph of the function  , where  , is shown below.
 

  1. Find the slope of the tangent to at  .   (1 mark)

  2. Find the obtuse angle that the tangent to at    makes with the positive direction of the horizontal axis. Give your answer correct to the nearest degree.   (1 mark)

  3. Find the slope of the tangent to at a point  . Give your answer in terms of  .   (1 mark)

  4.  i. Find the value of for which the tangent to at  and the tangent to at    are perpendicular to each other. Give your answer correct to three decimal places.   (2 marks)

  5. ii. Hence, find the coordinates of the point where the tangents to the graph of at    and    intersect when they are perpendicular. Give your answer correct to two decimal places.   (3 marks)

Two line segments connect the points   and    to a single point  , where  , as shown in the graph below.
 
         
 

  1.   i. The first line segment connects the point and the point , where .
  2.      Find the equation of this line segment in terms of  .   (1 mark)

  3.  ii. The second line segment connects the point and the point  , where  .
  4.      Find the equation of this line segment in terms of .   (1 mark)

  5. iii. Find the value of , where  , if there are equal areas between the function and each line segment.
  6.      Give your answer correct to three decimal places.   (3 marks)

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  4.  i.
  5. ii.
  6.   i.
  7.  ii.
  8. iii.
Show Worked Solution

a.   

♦ Mean mark part (b) 37%.

 

b.   


 

c.  


 

d.i.   


 

d.ii. 

♦ Mean mark part (d)(ii) 41%.

♦ Mean mark part (e)(i) 44%.

 
e.i. 

♦♦ Mean mark part (e)(ii) 28%.
 

e.ii. 

 
 
♦♦ Mean mark part (e)(iii) 28%.

 

e.iii. 

Calculus, MET2 2021 VCAA 3

Let  .

  1. State the maximal domain and the range of .   (2 marks)

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

  3. ii. Find the equation of the line that is perpendicular to the graph of when    and passes through the point  (-2, 0).   (1 mark)

Let 

  1. Explain why is not a one-to one function.   (1 mark)

  2. Find the gradient of the tangent to the graph of at  .   (1 mark)

The diagram below shows parts of the graph of and the line  .
 
 
                     
 
The line    and the tangent to the graph of at    intersect with an acute angle of between them.

  1. Find the value(s) of for which  . Give your answer(s) correct to two decimal places.   (3 marks)

  2. Find the -coordinate of the point of intersection between the line  and the graph of , and hence find the area bounded by  , the graph of and the -axis, both correct to three decimal places.   (3 marks)

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  2. i. 
    ii.
Show Worked Solution

a.   


 

 


 
 

b.     i.   


 

ii.   


  

c.    


 

d.  


 
 

e. 

 


 


 

f.    


 

 

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, MET1 2018 VCAA 9

Consider a part of the graph of  , as shown below.

  1.  i. Given that  , evaluate    when is a positive even integer or 0.
      
    Give your answer in simplest form.   (2 marks)

    ii. Given that  , evaluate    when is a positive odd integer.
    Give your answer in simplest form.   (1 mark)

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

  4. The translation maps the graph of    onto the graph of  , where
  6. and is a real constant.
  7. State the value of .   (1 mark)

  8. Let    and  .
    The line is the tangent to the graph of at the point and the line is the tangent to the graph of at , as shown in the diagram below.
     
         
    Find the total area of the shaded regions shown in the diagram above.   (2 marks)

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  1.  i.
    ii.
Show Worked Solution

a.i. 

♦♦ Mean mark 31%.

 

a.ii.

♦♦♦ Mean mark 19%.

 

b.   
 

 

♦ Mean mark part (b) 49%.

 
 

 

 

 

♦♦ Mean mark part (c) 34%.

c.   
   

 

 

d.   

♦♦♦ Mean mark 7%.

 

π

 


 

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)

Show Answers Only
  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 2016 VCAA 1

Let  .

  1. Find the period and range of .   (2 marks)

  2. State the rule for the derivative function .   (1 mark)

  3. Find the equation of the tangent to the graph of at  .   (1 mark)

  4. Find the equations of the tangents to the graph of    that have a gradient of 1.   (2 marks)

  5. The rule of   can be obtained from the rule of  under a transformation , such that
      

     

     

    Find the value of and the value of .   (3 marks)

  6. Find the values of  , such that  .   (2 marks)

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Show Worked Solution

a.   

MARKER’S COMMENT: Including round brackets rather than square ones was a common mistake.


  

b.   
 

c.  


 

d.   

♦ Mean mark part (d) 50%.

 

e.  

♦♦ Mean mark part (e) 27%.
   

 
   

 

f.   

♦ Mean mark part (f) 50%.

Calculus, MET1 2012 VCAA 10

Let  ,  where is a positive rational number.

  1.  i. Find, in terms of , the -coordinate of the stationary point of the graph of     (2 marks)

  2. ii. State the values of such that the -coordinate of this stationary point is a positive number.   (1 mark)

  3. For a particular value of , the tangent to the graph of    at    passes through the origin.
  4. Find this value of .   (3 marks)

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  1.  i.
  2. ii.
Show Worked Solution

a.i.  

 

 

♦♦♦ Part (a.ii.) mean mark 18%.
  ii.   
 
 
 

 

b.  

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: Many confused with the more common use of for gradient in  .

 

 

 

Calculus, MET2 2014 VCAA 5

Let  

  1. Express    in the form  .   (2 marks)

  2. Describe the translation that maps the graph of    onto the graph of  .   (1 mark)

  3. Find the values of such that the graph of   has
    1. one positive -axis intercept.   (1 mark)

    2. two positive -axis intercepts.   (1 mark)

  4. Find the value of for which the equation    has one solution.   (1 mark)

  5. At the point  , the gradient of    is and at the point , the gradient is  , where is a positive real number.
    1. Find the value of  .   (2 marks)

    2. Find and if  .   (1 mark)

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

    2. Find the equations of the tangents to the graph of    that pass through the point with coordinates  .   (3 marks)

Show Answers Only
  3.  i.
  4. ii.
  6.  i.
  7. ii.
  8. i.
  9. ii.
Show Worked Solution

a. 


 

 
 
 

 

b.   
   
   

 

♦ Mean mark (b) 37%.


 

c.i. 

♦♦♦ Mean mark 7%.

 
met2-2014-vcaa-sec5-answer
 


 

c.ii.  

♦♦♦ Mean mark part (c)(ii) 19%.


 

d.   

♦♦♦ Mean mark 17%.

 

met2-2014-vcaa-sec5-answer1
 

 
 

 

e.i.  

♦♦ Mean mark (e.i.) 28%.

 

e.ii.  

♦♦♦ Mean mark (e.ii.) 10%.


 

f.i.   

 

♦♦ Mean mark (f.i.) 22%.

 

f.ii.  

♦♦♦ Mean mark (f.ii.) 14%.

   
   
   
   

 

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%.
 
 

Graphs, MET2 2013 VCAA 1

Trigg the gardener is working in a temperature-controlled greenhouse. During a particular 24-hour time interval, the temperature   is given by  , where is the time in hours from the beginning of the 24-hour time interval.

  1. State the maximum temperature in the greenhouse and the values of when this occurs.   (2 marks)

  2. State the period of the function    (1 mark)

  3. Find the smallest value of for which     (2 marks)

  4. For how many hours during the 24-hour time interval is     (2 marks)

Trigg is designing a garden that is to be built on flat ground. In his initial plans, he draws the graph of    for    and decides that the garden beds will have the shape of the shaded regions shown in the diagram below. He includes a garden path, which is shown as line segment

  1. The line through points    and    is a tangent to the graph of    at point

    1. Find    when     (1 mark)

    2. Show that the value of is     (1 mark)

In further planning for the garden, Trigg uses a transformation of the plane defined as a dilation of factor from the -axis and a dilation of factor from the -axis, where and are positive real numbers.

  1. Let and be the image, under this transformation, of the points and respectively. 

     

    1. Find the values of and  if    and     (2 marks)

    2. Find the coordinates of the point    (1 mark)

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  5.  i. 
    ii. 
  6.  i. 
    ii. 
Show Worked Solution

a.  

 

b.   
   

 

c.  

 

 

d.  

met2-2013-vcaa-sec1-answer1

 

 

e.i.  

 

e.ii. 

 

 

 

f.i.  

 
♦♦♦ Mean mark part (f)(i) 14%.

 

 
♦♦♦ Mean mark part (f)(ii) 12%.

 

f.ii.   
   

Calculus, MET2 2012 VCAA 2

Let  

  1. Sketch the graph of   on the set of axes below. Label the axes intercepts with their coordinates and label each of the asymptotes with its equation.   (3 marks)

           VCAA 2012 2a
     

  2.   i. Find .   (1 mark)

  3.  ii. State the range of  .   (1 mark)

  4. iii. Using the result of part ii. explain why has no stationary points.   (1 mark)

  5. If    is any point on the graph of  , show that the equation of the tangent to    at this point can be written as     (2 marks)

  6. Find the coordinates of the points on the graph of    such that the tangents to the graph at these points intersect at     (4 marks)

  7. A transformation    that maps the graph of   to the graph of the function    has rule
  8.      , where , and are non-zero real numbers.
  9. Find the values of and .   (2 marks)

Show Answers Only
  1. met2-2012-vcaa-sec2-answer
  2.   i.
     ii.
    iii.
Show Worked Solution

a.  

met2-2012-vcaa-sec2-answer

 

b.i.  

 

b.ii.  

MARKER’S COMMENT: Incorrect notation in part (b)(ii) was common, including .

 

b.iii.   
   

 

 

c.  

♦♦ Mean mark 29%.
 

 

 

d.  

♦♦♦ Mean mark 19%.


 

e.