SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C3 2023 HSC 30

Let  .

  1. Find the coordinates of the stationary points of for  . You do NOT need to check the nature of the stationary points.  (3 marks)

  2. Without using any further calculus, sketch the graph of  , for  , showing stationary points and intercepts.  (2 marks)
     


Show Answers Only

a.   

b.    
         

Show Worked Solution

a.   

 
 
 
Mean mark (a) 54%.


 


 

 
b.   
 

 

♦♦ Mean mark (b) 34%.

Calculus, 2ADV C3 2022 HSC 27

Let  .

It is given that  .

  1. Show that  .  (2 marks)

  2. Find any stationary points of    and determine their nature.  (2 marks)

  3. Sketch the curve  , showing any stationary points, points of inflection and intercepts with the axes.  (3 marks)

Show Answers Only
  3.  
Show Worked Solution

a.   

 
   
   
   

 

b.   

 
 

 

 
   

 

 

c.   


 


Mean mark part (c) 54%.

Calculus, 2ADV C3 2019 MET1 4

Given the function  ,

  1. State the domain and range of .  (1 mark)

  2. i.  Find the equation of the tangent to the graph of  at  .  (2 marks)

     

    ii. On the axes below, sketch the graph of the function  , labelling any asymptote with its equation.

     

        Also draw the tangent to the graph of    at  .  (4 marks)
     

Show Answers Only

  1.  

  2. i. 

     

    ii.

Show Worked Solution
a.  
 

 

b.i.  
 
 

 

 

b.ii.  

Calculus, 2ADV C3 2004 HSC 9c

Consider the function  , for  .

  1. Show that the graph of    has a stationary point at  .  (2 marks)

  2. By considering the gradient on either side of  , or otherwise, show that the stationary point is a maximum.  (1 mark)

  3. Use the fact that the maximum value of    occurs at    to deduce that    for all  .  (2 marks)

Show Answers Only
Show Worked Solution

i.     

 
 

 

 

 

ii.   

Geometry and Calculus, 2UA 2004 HSC 9c Answer

 

 

 

iii.  
   

MARKER’S COMMENT: Very challenging for most students. Successful students recognised the link to part (ii).

 

Calculus, 2ADV C3 2009 HSC 10

  1. Show that the graph of    has no turning points.   (2 marks)

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

  3. i. Show that  for  .   (1 mark)

     

    ii. Let  .

     

        Use the result in part c.i. to show that    for all  .   (2 marks)

  1. Sketch the graphs of    and    for  .    (2 marks)

  2. Show that  .   (2 marks)

  3. Find the area enclosed by the graphs of    and  , and the straight line  .   (2 marks)

Show Answers Only
  3. i. 

     

    ii.  

  4.  
        Geometry and Calculus, 2UA 2009 HSC 10 Answer
  6. ²
Show Worked Solution
a.   
♦♦ Mean mark 28% for all of Q10 (note that data for each question part is not available).
 

 
 

 

 

b.   

 

 
 

 

c.i.     
 
 

 

c.ii.  
 
 

MARKER’S COMMENT: When 2 graphs are drawn on the same set of axes, you must label them. 
 

d. 

Geometry and Calculus, 2UA 2009 HSC 10 Answer
e.   
 
 
 
 

 

f.   
   
   
   
   
   
    ²