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Calculus, MET2 2023 VCE SM-Bank 1

The function is defined as follows.

  1. Sketch the graph of on the axes below. Label the vertical asymptote with its equation, and label any axial intercepts, stationary points and endpoints in coordinate form, correct to three decimal places.   (3 marks)

     

     

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

  3. ii. Sketch the graph of the tangent to the graph of at on the axes in part a.   (1 mark)

Newton's method is used to find an approximate -intercept of , with an initial estimate of .

  1. Find the value of .   (1 mark)

  2. Find the horizontal distance between and the closest -intercept of , correct to four decimal places.   (1 mark)

  3.  i. Find the value of , where , such that an initial estimate of    gives the same value of    as found in part . Give your answer correct to three decimal places.   (2 marks)

  4. ii. Using this value of , sketch the tangent to the graph of at the point where    on the axes in part a.   (1 mark)

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a.    
  
b.i
  
b.ii
  
c.
  
d.
  
e.i
  
e.ii
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a.

b.i   
 
 
 

  

b.ii

 
c.   

 
 


  
 

d.    

 

e.i. 

 
e.ii

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.

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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 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.
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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.
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a.   


 

 


 
 

b.     i.   


 

ii.   


  

c.    


 

d.  


 
 

e. 

 


 


 

f.    


 

 

Algebra, MET2 2017 VCAA 4

Let  . Part of the graph of  is shown below.
 

  1. The transformation    maps the graph of    onto the graph of  .

     

    State the values of and .   (2 marks)

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

  3. Find the area bounded by the graphs of  and  .   (3 marks)

  4. Part of the graphs of  and  are shown below.
     

         
     
    Find the gradient of  and the gradient of    at  .   (2 marks)

The functions of  , where  , are defined with domain such that  .

  1. Find the value of such that  .  (1 mark)

  2. Find the rule for the inverse functions  of  , where  .   (1 mark)

  3. i. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

    ii. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

  4. The lines and are the tangents at the origin to the graphs of    and    respectively.
  5. Find the value(s) of for which the angle between and is 30°.   (2 marks)

  6. Let be the value of for which    has only one solution.
  7.  i. Find .   (2 marks)

  8. ii. Let    be the area bounded by the graphs of    and    for all  .
  9.     State the smallest value of such that  .   (1 mark)

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a. 

b. 

c.  ²

d.

e. 

f. 

g.i. 

g.ii. 

h. 

i.i. 

i.ii. 

Show Worked Solution

a.   

   

 

 

b.   

 

 

c.   

MARKER’S COMMENT: Specifically recommends using    in this type of integral to avoid errors in stating the integral.

 

  ²

 

d.   
 

 

e.   

 

f.   

 

♦♦ Mean mark part (g)(i) 31%.

g.i.   
 

 

 

♦♦ Mean mark part (g)(ii) 30%.

g.ii.   
 

 

 

h.   

♦♦♦ Mean mark part (h) 13%.

 

 

 

i.i   

♦♦♦ Mean mark part (i)(i) 4%.

 

 

i.ii 

♦♦♦ Mean mark part (i)(ii) 2%.

 

Calculus, MET1 2007 VCAA 9

The graph of  ,    is shown. The normal to the graph of where it crosses the -axis is also shown.
 

MET1 2007 VCAA Q9
 

  1. Find the equation of the normal to the graph of where it crosses the -axis.   (2 marks)

  2. Find the exact area of the shaded region.   (3 marks)

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  2. ²
Show Worked Solution

a.  

MARKER’S COMMENT: A common error was made in calculating the point of tangency. Be careful!

  

  

 

♦ Mean mark 44%.
b.   
   
   
   
    ²

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  .