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Calculus, MET1 2022 VCAA 1b

Find and simplify the rule of `f^{\prime}(x)`, where `f:R \rightarrow R, f(x)=\frac{\cos (x)}{e^x}`.   (2 marks)

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`\frac{-(\sin x+\cos x)}{e^x}`

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 Using the quotient rule

`f(x)` `=\frac{\cos (x)}{e^x}`  
`f^{\prime}(x)` `=\frac{-e^x \sin x-e^x \cos x}{e^{2 x}}`  
  `= \frac{-e^x(\sin x+\cos x)}{e^{2 x}}`  
  `=\frac{-(\sin x+\cos x)}{e^x}`  

Calculus, MET1 2023 VCAA 1b

Let  \(f(x)=\sin(x)e^{2x}\).

Find  \(f^{'}\Big(\dfrac{\pi}{4}\Big)\).   (2 marks)

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\(\dfrac{3\sqrt{2}}{2}e^{\frac{\pi}{2}}\ \text{or}\ \dfrac{3e^{\frac{\pi}{2}}}{\sqrt{2}}\)

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\(\text{Using the product rule}\)

\(f'(x)\) \(=e^{2x}\cos(x)+2e^{2x}\sin(x)\)
  \(=e^{2x}\Big(\cos(x)+2\ \sin(x)\Big)\)
\(\therefore\ f’\Big(\dfrac{\pi}{4}\Big)\) \(=e^{2(\frac{\pi}{4})}\Bigg(\cos(\dfrac{\pi}{4})+2\ \sin(\dfrac{\pi}{4})\Bigg)\)
  \(=e^{\frac{\pi}{2}}\Bigg(\dfrac{1}{\sqrt{2}}+\sqrt{2}\Bigg)\)
  \(=e^{\frac{\pi}{2}}\Bigg(\dfrac{1+\sqrt2 \times \sqrt2}{\sqrt2} \Bigg) \)
  \(=\dfrac{3\sqrt{2}}{2}e^{\frac{\pi}{2}}\ \ \text{or}\ \ \dfrac{3e^{\frac{\pi}{2}}}{\sqrt{2}}\)

Calculus, MET1 2018 VCAA 1b

Let  `f(x) = (e^x)/(cos(x))`.

Evaluate  `f^{prime}(pi)`.   (2 marks)

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`text(See Worked Solutions)`

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`f^{prime}(x) = (e^x)/(cos(x))`

`u` `= e^x` `v` `= cos(x)`
`u^{prime}` `= e^x` `v^{prime}` `=-sin(x)`
`f^{prime}(x)` `= (u^{prime}v-uv^{prime})/(v^2)`
  `= (e^x · cos(x) + e^x sin(x))/(cos^2(x))`

 

`f^{prime}(pi)` `= (e^pi · cospi + e^pi sinpi)/(cos^2 pi)`
  `= (e^pi(-1) + e^pi · 0)/((-1)^2)`
  `= -e^pi`

Calculus, MET2 2009 VCAA 7 MC

For  `y = e^(2x) cos (3x)`  the rate of change of `y` with respect to `x` when  `x = 0`  is

  1. `0`
  2. `2`
  3. `3`
  4. `– 6`
  5. `– 1`
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`B`

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`y` `= e^(2x) cos (3x)`
`dy/dx` `=e^(2x) xx -3sin(3x) + 2e^(2x) xx cos (3x)`
  `=e^(2x)(-3sin(3x) + 2cos(3x))`

 
`text(When)\ \ x = 0,`

`dy/dx= 2`

`=>   B`

Calculus, MET1 2007 VCAA 2b

Let  `g(x) = log_e(tan(x))`.  Evaluate `g^{prime}(pi/4)`.   (2 marks)

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`g^{prime}(pi/4) = 2`

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`g(x) = log_e (tan(x))`

`text(Using Chain Rule:)`

`g^{prime}(x) = (sec^2(x))/(tan(x))`

`text(When)\ \ x = pi/4,`

`g^{prime}(pi/4)` `= (sec^2(pi/4))/(tan (pi/4))`
  `=1/(1/sqrt2)^2`
  `=1/(1/2)`
  `=2`