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Calculus, 2ADV C4 2022 HSC 29

  1. The diagram shows the graph of  . Also shown on the diagram are the first 5 of an infinite number of rectangular strips of width 1 unit and height    for non-negative integer values of . For example, the second rectangle shown has width 1 and height . 
     


 

  1. The sum of the areas of the rectangles forms a geometric series.
  2. Show that the limiting sum of this series is 2. (1 mark)

  3. Show that .  (2 marks)

  4. Use parts (a) and (b) to show that  .  (2 marks)

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a.   


 

b.   

 
   
   
   
   

 


Mean mark (b) 56%.

c.   

 
 
 
 
 
 

♦♦♦ Mean mark (c) 9%.

Calculus, 2ADV C4 2013* HSC 15a

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.

2013 15a

The front of the tent has area  ². 

  1. Use the trapezoidal rule to estimate  .    (1 mark)

  2. Does the Trapezoidal rule give a higher or lower estimate of the actual area? Justify your answer.  (1 mark)

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  1. ²
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i.   
   
   
    ²

 

ii.       

Calculus, 2ADV C4 2004* HSC 10a

  1. Use the Trapezoidal rule with 3 function values to find an approximation to the area under the curve   between   and  , where    is positive.  (2 marks)

  2. Using the result in part (i), show that  .  (1 mark)

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i.   
 
 

 

ii. 


 

Calculus, 2ADV C4 2017* HSC 14b

  1. Find the exact value of  .  (1 mark)

  2. Using the Trapezoidal rule with three function values, find an approximation to the integral  leaving your answer in terms of and .  (2 marks)

  3. Using parts (i) and (ii), show that  .  (1 mark)

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i.  
   
   

 

ii.  

 
 

 

♦ Mean mark part (iii) 49%.

(iii)   
 
   

Calculus, 2ADV C4 2016* HSC 14a

The diagram shows the cross-section of a tunnel and a proposed enlargement.

hsc-2016-14a

The heights, in metres, of the existing section at 1 metre intervals are shown in Table

hsc-2016-14ai

The heights, in metres, of the proposed enlargement are shown in Table

hsc-2016-14aii

Use the Trapezoidal rule with the measurements given to calculate the approximate increase in area.   (3 marks)

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²

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  ²

Calculus, 2ADV C4 2010 HSC 3b

  1. Sketch the curve  .   (1 mark)

  2. Use the trapezoidal rule with 3 function values to find an approximation to   (2 marks) 

  3. State whether the approximation found in part (ii) is greater than or less than the exact value of  . Justify your answer.   (1 mark)

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  2. ²
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i. 2010 3b image - Simpsons
MARKER’S COMMENT: Many students failed to illustrate important features in their graph such as the concavity, -axis intercept and -axis asymptote (this can be explicitly stated or made graphically clear).

 

ii.   
 
 
 
  ²   

 

iii. 2010 13b image 2 - Simpsons

 

♦♦♦ Mean mark 12%.
MARKER’S COMMENT: Best responses commented on concavity and that the trapezia lay beneath the curve. Diagrams featured in the best responses.