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Calculus, 2ADV C3 2019 HSC 14b

The derivative of a function    is given by  .

  1. Find the -values of the two stationary points of  , and determine the nature of the stationary points.  (2 marks)

  2. The curve passes through the point  .

     

    Find an expression for  .  (2 marks)

  3. Hence sketch the curve, clearly indicating the stationary points.  (2 marks)

  4. For what values of is the curve concave down?  (1 mark)

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a.   

 


 

 

b.   
   
   

 

 

c.   
 

 

 

d.   

♦ Mean mark 36%.

Calculus, 2ADV C3 2018 HSC 13a

Consider the curve  .

  1. Find the stationary points and determine their nature.  (3 marks)

  2. Given that the point  (2,16)  lies on the curve, show that it is a point of inflection.  (2 marks)

  3. Sketch the curve, showing the stationary points, the point of inflection and the and intercepts.  (2 marks)

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i.   


 


 


 

 

ii. 

 

 

 

iii.  

Calculus, 2ADV C3 2017 HSC 13b

Consider the curve  .

  1. Find the stationary points of the curve and determine their nature.  (4 marks)

  2. Sketch the curve, labelling the stationary points.  (2 marks)

  3. Hence, or otherwise, find the values of for which is positive.  (1 mark)

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  1.  

  2.    
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i.  
 
 

 

 

 


 

 

ii.  

 

iii. 

 

 
 

Calculus, 2ADV C3 2015 HSC 13c

Consider the curve  .

  1. Find the stationary points and determine their nature.   (4 marks)

  2. Given that the point    lies on the curve, prove that there is a point of inflection at  .  (2 marks)

  3. Sketch the curve, labelling the stationary points, point of inflection and -intercept.  (2 marks)

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  3.  
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i.
 
 

 

 
 

 

 

ii. 

 

 

iii. 

Calculus, 2ADV C3 2004 HSC 4b

Consider the function  .

  1. Find the coordinates of the stationary points of the curve    and determine their nature.   (3 marks)

  2. Sketch the curve showing where it meets the axes.   (2 marks)

  3. Find the values of    for which the curve    is concave up.   (2 marks)

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  2.  
    1. Geometry and Calculus, 2UA 2004 HSC 4b Answer
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(i)   
 
 

 

 

 

 

(ii)  

 Geometry and Calculus, 2UA 2004 HSC 4b Answer

(iii)  

 

Calculus, 2ADV C3 2005 HSC 4b

A function    is defined by  .

  1. Find all solutions of   (2 marks)

  2. Find the coordinates of the turning points of the graph of  , and determine their nature.  (3 marks)

  3. Hence sketch the graph of  , showing the turning points and the points where the curve meets the -axis.  (2 marks)

  4. For what values of is the graph of    concave down?  (1 mark)

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i.   
   
 

 

ii.  
   
   
 
 

 

 

 
 

 

 

iii.   Geometry and Calculus, 2UA 2005 HSC 4b Answer

 

iv. 

Calculus, 2ADV C3 2006 HSC 5a

A function    is defined by  .

  1. Find the coordinates of the turning points of    and determine their nature.  ( 3 marks)

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

  3. Hence sketch the graph of  , showing the turning points, the point of inflection and the points where the curve meets the -axis.  (3 marks)

  4. What is the minimum value of    for  ?  (1 mark)

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  3.  
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i.       
 

 

 

 

ii. 

 

 

 

iii. 

2UA HSC 2006 5a

 

(iv) 

 

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)

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  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

Calculus, 2ADV C3 2010 HSC 6a

Let  .

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

  2. Find the values of    for which the graph    is concave down, and the values for which it is concave up.    (2 marks)

  3. Sketch the graph  ,  indicating the values of the    and   intercepts.   (2 marks)

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  2.  

     

  3.  
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i. 

 

 

 
 
 

 

 

ii. 

MARKER’S COMMENT: The significance of the sign of the second derivative was not well understood by most students.

 

♦♦ Mean mark 33%.
MARKER’S COMMENT: Students are reminded to bring a ruler to the exam and use it to draw the axes for graphing and to help with an appropriate scale.

iii. 

 

 

Calculus, 2ADV C3 2011 HSC 7a

Let  . 

  1. Find the coordinates of the stationary points of  , and determine their nature.   (3 marks)

  2. Hence, sketch the graph    showing all stationary points and the  -intercept.   (2 marks)

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  2.  
    Geometry and Calculus, 2UA 2011 HSC 7a
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i.   
 
 

 

 

 

 
MARKER’S COMMENT: Graphs should be large (around ½ page), axes drawn with a ruler, with intercepts and turning points clearly shown. The scale can be different on each axe for clarity, as shown in the Worked Solution.

 

ii.   
 

Geometry and Calculus, 2UA 2011 HSC 7a