smc-725-10-Sin

Calculus, MET2 2022 VCAA 5

Consider the composite function `g(x)=f(\sin (2 x))`, where the function `f(x)` is an unknown but differentiable function for all values of `x`.

Use the following table of values for `f` and `f^{\prime}`.

Find the value of `g\left(\frac{\pi}{6}\right)`. (1 mark)

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The derivative of `g` with respect to `x` is given by `g^{\prime}(x)=2 \cdot \cos (2 x) \cdot f^{\prime}(\sin (2 x))`.

Show that `g^{\prime}\left(\frac{\pi}{6}\right)=\frac{1}{9}`. (1 mark)

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Find the equation of the tangent to `g` at `x=\frac{\pi}{6}`. (2 marks)

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Find the average value of the derivative function `g^{\prime}(x)` between `x=\frac{\pi}{8}` and `x=\frac{\pi}{6}`. (2 marks)

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Find four solutions to the equation `g^{\prime}(x)=0` for the interval `x \in[0, \pi]`. (3 marks)

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Show Answers Only

a. `3`

b. `1/9`

c. `y=1/9x+3-pi/54`

d. `-48/pi`

e. ` x = pi/8 , pi/4 , (3pi)/8 ,(3pi)/4`

Show Worked Solution

a. `g(pi/6)`	`= f(sin(pi/3))`
	`= f(sqrt3/2)`
	`= 3`

b. `g\ ^{prime}(x)`	`= 2\cdot\ cos(pi/3)\cdot\ f\ ^{prime}(sin(pi/3))`
`g\ ^{prime}(pi/6)`	`= 2 xx 1/2 xx f\ ^{prime}(sqrt3/2)`
	`= 1/9`

c. `m = 1/9` and `g(pi/6) = 3`

`y – y_1`	`= m(x – x_1)`
`y – 3`	`= 1/9(x – pi/6)`
`y`	`= 1/9x + 3 – pi/54`

♦♦ Mean mark (c) 45%.
MARKER’S COMMENT: Some students did not produce an equation as required.

d. The average value of `g^{\prime}(x)` between `x=\frac{\pi}{8}` and `x=\frac{\pi}{6}`

Average	`= \frac{1}{\frac{\pi}{6}-\frac{\pi}{8}}\cdot\int_{\frac{\pi}{8}}^{\frac{\pi}{6}} g^{\prime}(x) d x`
	`=24/pi \cdot[g(x)]_{\frac{\pi}{8}}^{\frac{\pi}{\6}}`
	`= 24/pi \cdot(f(sqrt3/2) -f(sqrt2/2))`
	`= 24/pi (3 – 5) = -48/pi`

♦♦ Mean mark (d) 30%.
MARKER’S COMMENT: Those who used the Average Value formula were generally successful.
Some students substituted `g^{\prime}(x)`, not `g(x)`.

e. `2 \cos (2 x) f^{\prime}(\sin (2 x)) = 0`

`:.\ 2 \cos (2 x) = 0\ ….(1)` or ` f^{\prime}(\sin (2 x)) = 0\ ….(2)`

(1): ` 2 \cos (2 x)`	`= 0`	`x \in[0, \pi]`
`\cos (2 x)`	`= 0`	`2 x \in[0,2 \pi]`
`2x`	`= pi/2 , (3pi)/2`
`x`	`= pi/4 , (3pi)/4`

(2): ` f^{\prime}(\sin (2 x)) `	`= sqrt2/2`
`2x`	`= pi/4 , (3pi)/4`
`x`	`= pi/8 , (3pi)/8`

`:. \ x = pi/8 , pi/4 , (3pi)/8 ,(3pi)/4`

♦♦ Mean mark (e) 30%.
MARKER’S COMMENT: Some students were able to find `pi/4, (3pi)/4`. Some solved `2 cos(2x)=0` or `f^{\prime}(sin(2x))=0` but not both.

Functions, MET1 2021 VCAA 3

Consider the function `g: R -> R, \ g(x) = 2sin(2x).`

State the range of `g`. (1 mark)
State the period of `g`. (1 mark)
Solve `2 sin(2x) = sqrt3` for `x ∈ R`. (3 marks)

Show Answers Only

`[–2,2]`
`pi`
`x= pi/6 + npi, pi/3 + npi\ \ \ (n in ZZ)`

Show Worked Solution

a. `text(S)text(ince) -1<sin(2x)<1,`

`text(Range)\ g(x) = [–2,2]`

b. `text(Period) = (2pi)/n = (2pi)/2 = pi`

c.	`2sin(2x)`	`=sqrt3`
	`sin(2x)`	`=sqrt3/2`
	`2x`	`=pi/3, (2pi)/3, pi/3 + 2pi, (2pi)/3 + 2pi, …`
	`x`	`=pi/6, pi/3, pi/6+pi, pi/3+pi, …`

`:.\ text(General solution)`

`= pi/6 + npi, pi/3 + npi\ \ \ (n in ZZ)`

Functions, MET1 2011 VCAA 3b

Solve the equation

`qquad sin (2x + pi/3) = 1/2\ \ text(for)\ \ x in [0, pi].` (2 marks)

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`x = pi/4, (11 pi)/12`

Show Worked Solution

`sin (2x + pi/3) = 1/2`

`=>\ text(Base angle is)\ \ pi/6`

`(2x + pi/3)`	`= pi/6, (5pi)/6, (13pi)/6, (17pi)/6, …`
`2x`	`= – pi/6, pi/2, (11pi)/6, (15pi)/6, …`
`x`	`= – pi/12, pi/4, (11pi)/12, (15pi)/12, …`

`:. x = pi/4, (11 pi)/12,\ \ x in [0, pi]`

Algebra, MET2 2017 VCAA 12 MC

The sum of the solutions of `sin(2x) = (sqrt3)/2` over the interval `[−pi,d]` is `−pi`.

The value of `d` could be

`0`
`pi/6`
`(3pi)/4`
`(7pi)/6`
`(3pi)/2`

Show Answers Only

`C`

Show Worked Solution

`sin(2x)`	`=sqrt3/2`
`2x`	`=pi/3, (2pi)/3, ..`
`x`	`=pi/6, pi/3, ..`

`text(Expand and test possible solutions to solve:)`

♦ Mean mark 45%.

`x=-(5pi)/6, -(2pi)/3, pi/6, pi/3, ..`

`-(5pi)/6+ -(2pi)/3+ pi/6+ pi/3=-pi`

`=> C`

Algebra, MET2 2009 VCAA 4 MC

The general solution to the equation `sin (2x) = -1` is

`x = n pi - pi/4,\ n in Z`
`x = 2n pi + pi/4 or x = 2n pi - pi/4,\ n in Z`
`x = (n pi)/2 + (-1)^n pi/2,\ n in Z`
`x = (n pi)/2 + (-1)^n pi/4,\ n in Z`
`x = n pi + pi/4 or x = 2n pi + pi/4,\ n in Z`

Show Answers Only

`A`

Show Worked Solution

`2x`	`= 2n pi – pi/2,\ \ n in Z`
`x`	`= n pi – pi/4,\ \ n in Z`

`=> A`

Functions, MET1 2007 VCAA 8

Let `f: R -> R`, `f(x) = sin((2pix)/3)`.

Solve the equation `sin((2pix)/3) = -sqrt3/2` for ` x ∈ [0,3]`. (2 marks)
Let `g: R -> R`, `g(x) = 3f(x - 1) + 2`.
Find the smallest positive value of `x` for which `g(x)` is a maximum. (2 marks)

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`x = 2, 5/2`
`7/4`

Show Worked Solution

a. `sin((2pix)/3) = -sqrt3/2`

`=>\ text(Base angle)\ = pi/3`

`(2 pi x)/3`	`=(4pi)/3, (5pi)/3, (10pi)/3, …`
`:.x`	`=2 or 5/2, \ \ \ text(for)\ x ∈ [0,3]`

b. `g(x) = 3sin[(2pi)/3 (x – 1)] + 2`

♦♦ Mean mark 29%.
STRATEGY: Max/min questions involving trig functions can often use the powerful identity, `-1 <= sin theta“<=1` to solve efficiently and reduce errors.

`text(Maximum occurs when:)`

`sin[(2pi)/3 (x – 1)]`	`= 1`
`(2pi)/3 (x – 1)`	`= pi/2`
`x-1`	`= pi/2 xx 3/(2pi)`
`:. x`	`=7/4`

Functions, MET1 2013 VCAA 4

Solve the equation `sin (x/2) = -1/2` for `x in [2 pi, 4 pi].` (2 marks)

Show Answers Only

`x = (7 pi)/3, (11 pi)/3`

Show Worked Solution

`x/2`	`=pi/6 + pi, 2pi – pi/6, 2pi + (pi/6 +pi), …`
	`=(7pi)/6, (11pi)/6, (19pi)/6, …`
`:. x`	`=(7pi)/3, (11pi)/3, (19pi)/3, …`

`text(Given)\ \ x in [2 pi, 4 pi],`

`:. x = (7 pi)/3, (11 pi)/3`

Functions, MET1 2015 VCAA 5

On any given day, the depth of water in a river is modelled by the function

`h(t) = 14 + 8sin((pit)/12),\ \ 0 <= t <= 24`

where `h` is the depth of water, in metres, and `t` is the time, in hours, after 6 am.

Find the minimum depth of the water in the river. (1 mark)
Find the values of `t` for which `h(t) = 10`. (2 marks)

Show Answers Only

`6\ text(m)`
`14quadtext(or)quad22`

Show Worked Solution

a. `h_(text(min))\ text(occurs when)\ \ sin((pit)/12)=-1`

MARKER’S COMMENT: Students who used calculus to find the minimum were less successful.

`:. h_(text(min))`	`= 14 – 8`
	`= 6\ text(m)`

b.	`14 + 8sin(pi/12t)`	`= 10`
	`sin(pi/12t)`	`= – 1/2`

`text(Solve in general:)`

`pi/12t`	`=(7pi)/6 + 2pi n\ \ \ \ text(or)\ \ \ `	`pi/12t`	`= (11t)/6 + 2pi n,`
`t`	`= 14 + 24n`	`t`	`=22 + 24n`

`text(Substitute integer values for)\ n,`

`:. t = 14quadtext(or)quad22,\ \ \ (t ∈ [0,24])`