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Functions, MET2 2023 VCAA 9 MC

The function \(f\) is given by
 

\(f(x) = \begin {cases}
\tan\Bigg(\dfrac{x}{2}\Bigg)         &\ \ 4 \leq x \leq 2\pi \\
\sin(ax) &\ \ \ 2\pi\leq x\leq 8
\end{cases}\)

 
The value of \(a\) for which \(f\) is continuous and smooth at  \( x\) = \(2\pi\)  is

  1. \(-2\)
  2. \(-\dfrac{\pi}{2}\)
  3. \(-\dfrac{1}{2}\)
  4. \(\dfrac{1}{2}\)
  5. \(2\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Solve for}\ a\ \text{given}\ \ x=2\pi:\)

\(\tan\Bigg(\dfrac{2\pi}{2}\Bigg)=\sin(a2\pi)=0\)

\(a=\pm \dfrac{1}{2}\)
 

\(\text{For smoothness, solve for}\ a\ \text{given}\ \ x=2\pi:\)

\(\dfrac{d}{dx}\tan\Bigg(\dfrac{x}{2}\Bigg)=\dfrac{d}{dx}\sin(ax)\)

\(\therefore a=-\dfrac{1}{2}\)

 
\(\Rightarrow C\)


♦ Mean mark 46%.

Graphs, MET2 2019 VCAA 10 MC

Which one of the following statements is true for  `f: R -> R, \ f(x) = x + sin(x)`?

  1. The graph of  `f` has a horizontal asymptote
  2. There are infinitely many solutions to  `f(x) = 4`
  3. `f` has a period of  `2 pi`
  4. `f^{\prime}(x) >= 0`  for  `x in R`
  5. `f^{\prime}(x) = cos(x)`
Show Answers Only

`D`

Show Worked Solution

`text(By CAS, sketch)\ \ f(x) = x + sin(x):`

`text(By inspection, eliminate A, B, C)`

`text(By CAS, sketch)\ \ d/(dx)\ f(x):`

` text(Graph range) >= 0\ \ text(for)\ \ x in R`
 

`=>   D`

Graphs, MET2 2016 VCAA 8 MC

The UV index, `y`, for a summer day in Melbourne is illustrated in the graph below, where `t` is the number of hours after 6 am.
 

     
 

The graph is most likely to be the graph of

A.   `y = 5 + 5 cos ((pi t)/7)`

B.   `y = 5 - 5 cos ((pi t)/7)`

C.   `y = 5 + 5 cos ((pi t)/14)`

D.   `y = 5 - 5 cos ((pi t)/14)`

E.   `y = 5 + 5 sin ((pi t)/14)`

Show Answers Only

`B`

Show Worked Solution

`text(Median) = (0 + 10)/2 = 5`

`text(Amplitude) = 5`

`text(Period:)\ \ 14` `= (2 pi)/n`
`n` `= pi/7`

 

`:.\ text(Graph:)\ \ y = 5 – 5 cos ((pi t)/7)`

`=>   B`

Graphs, MET2 2011 VCAA 15 MC

met2-2011-vcaa-15-mc 
 

The graph shown could have equation

  1. `y = 2cos(x + pi/6) + 1`
  2. `y = 2cos4(x - pi/6) + 1`
  3. `y = 4sin2(x - pi/12) - 1`
  4. `y = 3cos(2x + pi/6) - 1`
  5. `y = 2sin(4x + (2pi)/3) - 1`
Show Answers Only

`=> B`

Show Worked Solution

`text{Amplitude = 2  (range from – 1 to 3)}`

`text(Median) = 1`

`:.\ text(Solution is)\ A\ text(or)\ B.`

`text(From graph:)`

`text(Period) = (2pi)/3 – pi/6 = pi/2`

`text(Consider option)\ B,`

`text(Period)= (2pi)/n= (2pi)/4 = pi/2`

`=> B`

Graphs, MET2 2012 VCAA 6 MC

A section of the graph of  `f` is shown below.

VCAA 2012 6mc

The rule of `f` could be

  1. `f (x) = tan (x)`
  2. `f (x) = tan (x - pi/4)`
  3. `f (x) = tan(2 (x - pi/4))`
  4. `f (x) = tan(2 (x - pi/2))`
  5. `f (x) = tan(1/2 (x - pi/2))`
Show Answers Only

`C`

Show Worked Solution

`text(Period) = pi/2`

`=>\ text(must be)\ C\ text(or)\ D`

`text(Shift)\ \ y = tan(x)\ \ text(right)\ \ pi/4.`

`=>   C`