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Calculus, EXT2 C1 2022 HSC 14b

Let  , where "n" is a non-negative integer.

  1. Show that  .  (1  mark)

  2. Show that  .  (2 marks)

  3. Show that  .  (2 marks)

  4. Using parts (i) and (iii), show by mathematical induction, or otherwise, that for all ,
  5.            (2 marks)

  6. Using parts (ii) and (iv) prove that  .  (1  mark)

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i.   
   
   
   

 


Mean mark (i) 93%.

ii. 

 
   
   
   
   

 


♦♦ Mean mark (ii) 28%.
 

iii. 

 
   
   

 
iv.   


 


   


♦ Mean mark (iv) 50%.

 
   
   
   
   

 


 

v.   

  

 
 
 
   
   
   

♦♦ Mean mark (v) 34%.

Calculus, EXT2 C1 2021 HSC 13c

  1. The integral    is defined by    for integers  .
    Show that    for   .  (2 marks)

  2. The diagram shows two regions.

Region is bounded by    and    between    and  .

Region is bounded by    and    between    and  .
 
   
 
The volume of the solid created when the region between the curve    and the  -axis, between    and  , is rotated about the -axis is given by  .

The volume of the solid of revolution formed when region is rotated about the -axis is .

The volume of the solid of revolution formed when region is rotated about the -axis is .

Using part (i), or otherwise, show that the ratio  is .  (4 marks)

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i.   
 
 

 

ii.