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Calculus, MET2 2020 VCAA 3 MC

 Let  `f^(′)(x)=(2)/(sqrt(2x-3))`. 

If  `f(6)=4`, then
 

  1. `f(x)=2sqrt(2x-3)`
  2. `f(x)=sqrt(2x-3)-2`
  3. `f(x)=2sqrt(2x-3)-2`
  4. `f(x)=sqrt(2x-3)+2`
  5. `f(x)=sqrt(2x-3)`
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`=>C`

Show Worked Solution
`f^(′)(x)` `=2/(sqrt(2x-3))`  
`f(x)` `=2 int(2x-3)^{- 1/2}`  
  `=2*1/2*2(2x-3)^{1/2}+c`  
  `=2sqrt(2x-3)+c`  

 
`text(When)\ \ x=6, \ f(x)=4:`

`4=2sqrt(12-3) + c \ => \ c=-2`

`:. f(x) = 2sqrt(2x-3) – 2`

`=>C`

Calculus, MET1 2011 VCAA 2a

Find an antiderivative of  `1/(3x-4)`  with respect to `x.`   (1 mark)

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`1/3 log_e | 3x-4 |`

Show Worked Solution

`int (3x-4)^-1\ dx`

♦ Mean mark 43%.
MARKER’S COMMENT: The constant `c` is required for the general derivative `int f(x)\ dx` but is not necessary for an antiderivative.

`= 1/3 log_e (| 3x-4 |)+c`

Calculus, MET1 2012 VCAA 2

Find an anti-derivative of  `1/(2x - 1)^3`  with respect to `x.`  (2 marks)

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`(-1)/(4(2x – 1)^2) + c`

Show Worked Solution
♦ Mean mark 37%.
MARKER’S COMMENT: The most common error was answering with a log equation. The low mean mark was a surprise here. Worth attention!
`int (2x – 1)^-3 dx` `= -1/(2 xx 2) (2x – 1)^-2 +c`
  `= (-1)/(4(2x – 1)^2) + c`