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Calculus, 2ADV C3 2010 HSC 8d

Let  `f(x) = x^3-3x^2 + kx + 8`, where `k` is a constant.

Find the values of `k` for which `f(x)` is an increasing function.   (2 marks)

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`k>3`

Show Worked Solution
`f(x)` `= x^3-3x^2 + kx + 8`
`f^{′}(x)` `= 3x^2-6x + k`

  
`f(x)\ text(is increasing when)\ \ f^{′}(x) > 0`

`=> 3x^2-6x + k > 0`

♦♦ Mean mark 28%.
MARKER’S COMMENT: The arithmetic required to solve `36-12k<0`  proved the undoing of many students.

 

`f^{′}(x)\ text(is always positive)`

`=> f^{′}(x)\ text(is a positive definite.)`

`text(i.e. when)\ \ a > 0\ text(and)\ Delta < 0`
 

`a=3>0`

`Delta = b^2-4ac`

`:. (-6)^2-(4 xx 3 xx k)` `<0`
`36-12k` `<0`
`12k` `>36`
`k` `>3`

 

`:.\ f(x)\ text(is increasing when)\ \ k > 3.`

Calculus, 2ADV C1 2013 HSC 11b

Evaluate  `lim_(x->2) ((x-2)(x+2)^2)/(x^2-4)`.   (2 marks)

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 `4`

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`lim_(x ->2) ((x-2)(x+2)^2)/(x^2-4)`

COMMENT: This question has been simplified as students no longer need to factorise the difference between 2 cubes (`x^3-2^3`).

`=lim_(x->2) ( (x -2)(x+2)^2)/( (x-2)(x+2)`

`=lim_(x->2) (x+2)`

`=4`