SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C4 2019 MET1 7

The shaded region in the diagram below is bounded by the vertical axis, the graph of the function with rule    and the horizontal line segment that meets the graph at  , where  .
 


 

Let    be the area of the shaded region.

Show that  .  (3 marks)

Show Answers Only

Show Worked Solution

   

 
 
 

Calculus, 2ADV C4 2007 HSC 7b

2ua 2007 7b
 

The diagram shows the graphs of    and  . The first two points of intersection to the right of the -axis are labelled    and  .

  1. Solve the equation    to find the -coordinates of    and  .  (2 marks)

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

Show Answers Only
  2. ²
Show Worked Solution
i. 

 

 

 

 

ii. 

²

Calculus, 2ADV C4 2006 HSC 5b

  1. Show that   (1 mark)


  2.   
     
    2006 5b
     
    The shaded region in the diagram is bounded by the curve    and the lines    and 

     

    Using the result of part (i), or otherwise, find the area of the shaded region.  (3 marks)

Show Answers Only
  2. ²
Show Worked Solution
i.   
 

 

ii. 

­

­

²

Trigonometry, 2ADV T3 2006 HSC 7b

A function    is defined by  .

  1. Show that the graph of    cuts the -axis at  .   (1 mark)

  2. Sketch the graph of   for    showing where the graph cuts each of the axes.   (3 marks)

  3. Find the area under the curve   between   and  .   (3 marks)

Show Answers Only
  2.   
  3. ²
Show Worked Solution

i.   

 

 

ii.   2UA HSC 2006 7b

 

iii. 
 
  ­­
 
 
  ²

Calculus, 2ADV C3 2010 HSC 9b

Let    be a function defined for  , with  . 

The diagram shows the graph of the derivative of  ,  . 

The shaded region    has area    square units. The shaded region    has area    square units. 

  1. For which values of    is    increasing?  (1 mark)

  2. What is the maximum value of  ?     (1 mark)

  3. Find the value of  .   (1 mark)

  4. Draw a graph of    for  .   (2 marks)

Show Answers Only
  4.  
  5.  
Show Worked Solution
i.   
 
 

 

ii.  
 
 
 
 
 

 

♦♦♦ Parts (ii) and (iii) proved particularly difficult for students with mean marks of 12% and 11% respectively.

 

 

iii.  
   

 
 
 

 

 

  

♦♦ Mean mark 28%
EXAM TIP: Clearly identify THE EXTREMES when given a defined domain. In this case, the origin is obvious graphically, and the other extreme at , is CLEARLY LABELLED! 
iv.  2UA HSC 2010 9bi

Calculus, 2ADV C4 2011 HSC 6c

The diagram shows the graph   . 
  

2011 6c 
  

  1. State the coordinates of  .   (1 mark)

  2. Evaluate the integral  .    (2 marks)

  3. Indicate which area in the diagram,  ,  ,   or  , is represented by the integral
     
           .   (1 mark)

  4. Using parts (ii) and (iii), or otherwise, find the area of the region bounded by the curve    and the  -axis, between    and   .   (1 mark)

  5. Using the parts above, write down the value of
     
         .   (1 mark)

Show Answers Only
  4. ²
Show Worked Solution

i. 

 

ii. 


 

iii. 
 

iv.   
 

 

  ²

 

MARKER’S COMMENT: “Using the parts above” in part (v) was ignored by many students. Important to know when finding an area and evaluating an integral may differ.

v.