smc-739-80-Trig overlap

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

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

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

Let `f(x) = (e^x)/(cos(x))`.

Evaluate `f′(pi)`. (2 marks)

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`text(See Worked Solutions)`

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`f′(x) = (e^x)/(cos(x))`

`u`	`= e^x`	`v`	`= cos(x)`
`u′`	`= e^x`	`v′`	`= −sin(x)`

`f′(x)`	`= (u′v – uv′)/(v^2)`
	`= (e^x · cos(x) + e^x sin(x))/(cos^2(x))`

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

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`B`

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`text(When)\ \ x = 0,`

`dy/dx= 2`

`=> B`

Let `g(x) = log_e(tan(x))`. Evaluate `g′(pi/4)`. (2 marks)

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`g′(pi/4) = 2`

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

`text(Using Chain Rule:)`

`g′(x) = (sec^2(x))/(tan(x))`

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