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Calculus, 2ADV C4 2022 HSC 29

  1. The diagram shows the graph of   `y=2^{-x}`. Also shown on the diagram are the first 5 of an infinite number of rectangular strips of width 1 unit and height  `y=2^{-x}`  for non-negative integer values of  `x`. For example, the second rectangle shown has width 1 and height `(1)/(2)`. 
     

  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 `int_(0)^(4)2^(-x)\ dx=(15)/(16 ln 2)`.   (2 marks)

  4. Use parts (a) and (b) to show that  `e^(15) < 2^(32)`.   (2 marks)

Show Answers Only

a.    `text{Proof (See Worked Solutions)}`

b.    `text{Proof (See Worked Solutions)}`

c.    `text{Proof (See Worked Solutions)}`

Show Worked Solution

a.    `text{Consider the rectangle heights:}`

`2^0=1, \ 2^(-1)=1/2, \ 2^(-2)= 1/4, \ 2^(-3)= 1/8, …`

`=>\ text{Rectangle Areas}\ = 1, \ 1/2, \  1/4, \ 1/8, …`

`a=1,\ \ r=1/2`

`S_oo=a/(1-r)=1/(1-1/2)=2\ \ text{… as required}`
 

b.   `text{Show}\ \ int_0^4 2^(-x)\ dx = 15/(16ln2)`

`int_0^4 2^(-x)\ dx ` `=(-1)/ln2[2^(-x)]_0^4`
  `=(-1)/ln2(1/16-1)`
  `=1/ln2-1/(16ln2)`
  `=(16-1)/(16ln2)`
  `=15/(16ln2)\ \ text{… as required}`

 


Mean mark (b) 56%.

c.    `text{Show}\ \ e^15<2^32`

`text{Area under curve < Sum of rectangle areas}`

`15/(16ln2)` `<2`  
`15` `<32ln2`  
`15/32` `<ln2`  
`e^(15/32)` `<e^(ln2)`  
`root(32)(e^15)` `<2`  
`e^15` `<2^32\ \ text{… as required}`  

♦♦♦ Mean mark (c) 9%.

Calculus, 2ADV C4 2016 HSC 12d

  1. Differentiate  `y = xe^(3x)`.   (1 mark)

  2. Hence find the exact value of  `int_0^2 e^(3x) (3 + 9x)\ dx`.   (2 marks)

Show Answers Only

a.    `e^(3x) (1 + 3x)`

b.    `6e^6`

Show Worked Solution

a.    `y = xe^(3x)`

`text(Using product rule:)`

`(dy)/(dx)= x · 3e^(3x) + 1 · e^(3x)= e^(3x) (1 + 3x)`
 

b.    `int_0^2 e^(3x) (3 + 9x)\ dx`

`= 3 int_0^2 e^(3x) (1 + 3x)\ dx`

`= 3 [x e^(3x)]_0^2`

`= 3 (2e^6-0)= 6e^6`