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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)

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  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.   

Calculus, SPEC1-NHT 2017 VCAA 3

Find the gradient of the curve with equation  `x = sin (y/15)`  when  `x = 1/4`. Give your answer in the form  `a sqrt b`, where  `a, b \ in Z^+`.  (3 marks)

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 `m = 4 sqrt 15`

Show Worked Solution
`x` `= sin (y/15)`
`sin^(-1)x` `= y/15`
`y` `= 15 sin^(-1)x`
`dy/dx` `= 15/(sqrt(1-x^2))`

 

`text(When)\ \ x=1/4:`

`dy/dx` `= 15/(sqrt(1-(1/4)^2))`
  `= 15/(sqrt(15/16))`
  `= 60/sqrt15`
  `=4sqrt15`

 
`:. m = 4 sqrt 15`

Calculus, SPEC2 2017 VCAA 6 MC

Given that  `(dy)/(dx) = e^x\ text(arctan)(y)`, the value of  `(d^2y)/(dx^2)`  at the point  `(0,1)`  is

  1.      `1/2`
  2.     `(3pi)/8`
  3.  `−1/2`
  4.     `pi/4`
  5.  `−pi/8`
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`B`

Show Worked Solution
`(d^2y)/(dx^2)` `= d/(dx)(e^x) xx text(arctan)(y) + d/(dx)(text(arctan)(y)) xx e^x`
  `= e^xtext(arctan)(y) + e^x (1/(1 + y^2))(dy)/(dx)`

♦ Mean mark 46%.

`(d^2y)/(dx^2)|_(text{(0, 1)})` `= e^0text(arctan)(1) + e^0(1/(1 + 1^2)) xxe^0text(arctan)(1)`
  `= 1 xx pi/4 + 1 xx 1/2 xx pi/4`
  `= (3pi)/8`

`=>B`