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Calculus, SPEC1 2020 VCAA 6

Let  `f(x) = arctan (3x - 6) + pi`.

  1. Show that  `f^{\prime}(x) = 3/(9x^2 - 36x + 37)`.  (1 mark)

  2. Hence, show that the graph of  `f`  has a point of inflection at  `x = 2`.  (2 marks)

  3. Sketch the graph of  `y = f(x)`  on the axes provided below. Label any asymptotes with their equations and the point of inflection with its coordinates.   (2 marks)

Show Answers Only

  1. `text(Proof)\ text{(See Worked Solutions)}`
  2. `text(Proof)\ text{(See Worked Solutions)}`
  3. `text(See Worked Solutions)`

Show Worked Solution

a.    `f^{\prime}(x)` `= (d/(dx) (3x – 6))/(1 + (3x – 6)^2)`
    `= 3/(9x^2 – 36x + 37)`

 

b.   `f^{\prime\prime}(x) = (3(18x – 36))/(9x^2 – 36x + 37)^2`

♦ Mean mark part (b) 42%.

`f^{\prime\prime}(x) = 0\ \ text(when)\ \ 18x – 36 = 0 \ => \ x = 2`

`text(If)\ \ x < 2, 18x – 36 < 0 \ => \ f^{\prime\prime}(x) < 0`

`text(If)\ \ x > 2, 18x – 36 > 0 \ => \ f^{\prime\prime}(x) > 0`

`text(S) text(ince)\ \ f^{\prime\prime}(x)\ \ text(changes sign about)\ \ x = 2,`

`text(a POI exists at)\ \ x = 2`

 

c.   

Graphs, SPEC2 2013 VCAA 4 MC

The graphs of  `y = ax`  and  `y = arctan(bx)`  intersect exactly three times if

A.   `0 < b < a`

B.   `a < b < 0`

C.   `a = b`

D.   `b < a < 0`

E.   `0 < b^2 < a^2`

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`D`

Show Worked Solution

`text{Graphs intersect at (0,0).}`

♦ Mean mark 43%.

`y = tan^(−1)(bx)\ \ =>\ \ dy/dx = b/(1 + (bx)^2)`

`text(At)\ \ x=0,\ \ m=b`

`y=ax\ \ =>\ \ dy/dx = a`
 

`text(To intersect exactly 3 times:)`

`text(Gradient of)\ \ y=ax\ \ text(must be less than)\ b\ text(for positive gradients.)`

`0<a<b`

`text(Gradient of)\ \ y=ax\ \ text(must be greater than)\ b\ text(for negative gradients.)`

`b<a<0`

`=> D`

Algebra, SPEC2-NHT 2017 VCAA 3 MC

For the function  `f: R -> R, \ f(x) = k arctan (ax - b) + c`, where  `k > 0, \ c > 0`  and  `a, b in R, \ f(x) > 0`  if

A.   `c < (k pi)/2`

B.   `c >= (k pi)/2`

C.   `x > b/a`

D.   `c + k > pi/2`

E.   `c >= pi/2`

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`B`

Show Worked Solution
`tan^(-1)(ax -b)` `in ((-pi)/2, pi/2)`
`k tan^(-1) (ax – b)` `in ((- k pi)/2, (k pi)/2),\ text(as)\ k > 0`

 
`f(x) = k tan^(-1) (ax – b) + c in (c – (k pi)/2, (k pi)/2 + c)`

`text(When)\ \ f(x)>0:`

`c – (k pi)/2` `>= 0\ \ \ text{(domain doesn’t include end limits)}`
`:. c` `>= (k pi)/2`

 
`=>   B`

Graphs, SPEC2 2018 VCAA 1MC

Part of the graph of  `y = 1/2 tan^(-1)(x)`  is shown below.
 

 
The equations of its asymptotes are

A.  `y = +- 1/2`

B.  `y = +- 3/4`

C.  `y = +- 1`

D.  `y = +- pi/2`

E.  `y = +- pi/4` 

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`E`

Show Worked Solution

`y = tan(x): qquad x in (-pi/2, pi/2)`

`-> y = tan^(-1)(x): qquad y in (-pi/2, pi/2)`

`-> y = 1/2 tan^(-1)(x): qquad y in (-pi/4, pi/4)`

 
`=>  E`