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Calculus, MET1 2022 VCAA 7

A tilemaker wants to make square tiles of size 20 cm × 20 cm.

The front surface of the tiles is to be painted with two different colours that meet the following conditions:

  • Condition 1 - Each colour covers half the front surface of a tile.
  • Condition 2 - The tiles can be lined up in a single horizontal row so that the colours form a continuous pattern.

An example is shown below.
 

There are two types of tiles: Type A and Type B.

For Type A, the colours on the tiles are divided using the rule , where .

The corners of each tile have the coordinates (0,0), (20,0), (20,20) and (0,20), as shown below.
 

  1.  i. Find the area of the front surface of each tile.   (1 mark)

    ii. Find the value of so that a Type A tile meets Condition 1.   (1 mark)


Type B tiles, an example of which is shown below, are divided using the rule .
 

  1. Show that a Type B tile meets Condition 1.   (3 marks)

  2. Determine the endpoints of and on each tile. Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2.   (2 marks)

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a.i.    cm²

a.ii.  

b.     cm²

c.     See worked solution

Show Worked Solution
a.i   Area  
  cm²  

  
a.ii 
 

b.   Area  
   
   
   
  cm²  

 
The area of the coloured section of the Type B tile is 200 cm² which is half the 400 cm² area of the tile.
 

c.   
 
   
   
   

 
→
 

   The endpoints for are and and for are also and .

So the tiles can be placed in any order to make the continuous pattern.


♦♦ Mean mark (c) 30%.
MARKER’S COMMENT: Students often only found and , however, and also needed to be found to verify the pattern match.

Calculus, MET2 2020 VCAA 1

Let  , where . Part of the graph of is shown below. 
 

  1. Show that  .    (1 mark)

  2. Express    in the form    where and are integers.   (1 mark)

Part of the graph of the derivative function  is shown below.
 
     
 

  1.  i. Write the rule for in terms of .   (1 mark)

  2. ii. Find the minimum value of the graph of on the interval  .   (2 marks)  

Let  . Parts of the graph of and are shown below.
 
 
       
 

  1. Write a sequence of two transformations that map the graph of onto the graph of .   (1 mark)

     
       
     

  2.   i. State the values of for which the graphs of and intersect.   (1 mark)

  3.  ii. Write down a definite integral that will give the total area of the shaded regions in the graph above.   (1 mark)

  4. iii. Find the total area of the shaded regions in the graph above. Give your answer correct to two decimal places.   (1 mark)

  5. Let be the vertical distance between the graphs of and.
  6. Find all values of for which is at most 2 units. Give your answers correct to two decimal places.   (2 marks)

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  3.  i.
  4. ii.
  6.   i.
  7.  ii.
  8. iii. ²
Show Worked Solution

a.   

 
 
 

 

b.    
   

 
c.i. 
 

c.ii. 


 

d. 


 

e.i. 


 

e.ii. 


 

e.iii.  ²

♦♦ Mean mark part (f) 26%.

 

f.   

Calculus, MET2 2021 VCAA 2

Four rectangles of equal width are drawn and used to approximate the area under the parabola    from    to  .

The heights of the rectangles are the values of the graph of  at the right endpoint of each rectangle, as shown in the graph below.
 

  1. State the width of each of the rectangles shown above.  (1 mark)
  2. Find the total area of the four rectangles shown above.  (1 mark)
  3. Find the area between the graph of  , the -axis and the line  .  (2 marks)
  4. The graph of is shown below.
     
         

    Approximate    using four rectangles of equal width and the right endpoint of each rectangle.  (1 mark)

Parts of the graphs of    and    are shown below.
 
     

  1. Find the area of the shaded region.  (1 mark)
  2. The graph of  is transformed to the graph of  , where  .
  3. Find the values of such that the area defined by region(s) bounded by the graphs of    and    and the lines    and    is equal to . Give your answer correct to two decimal places.  (4 marks)
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Show Worked Solution

a.   

 

b.    
   

 

c.    
   

 

d.

♦♦♦ Mean mark part (d) 16%.

 

 

e.  
   

 
f.   

♦♦♦ Mean mark part (f) 18%.


  

Calculus, MET1 2017 VCAA 9

The graph of    is shown below.
 

  1. Calculate the area between the graph of and the -axis.   (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of is  .   (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis, where  .

  1. Find the equation of the line through and in the form  , for  .   (2 marks)

  2. Find the coordinates of when  .   (4 marks)

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Show Worked Solution
a.  
   
   
   
   
   

 

♦♦ Mean mark part (b) 35%.
MARKER’S COMMENT: Establishing the common denominator in the working was required!
b.  
 
   
   

 

c. 

♦♦♦ Mean mark part (c) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as to solve.

 

 

   
 
   

 

 

 

d. 

♦♦♦ Mean mark part (d) 17%.