smc-968-20-Cos

Calculus, 2ADV C2 SM-Bank 3

Prove that \(\cos x+\sin x\, \tan x=\sec x\) (1 mark)

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Hence, or otherwise, find \(\dfrac{d}{dx}\left(\dfrac{1}{\cos x+\sin x \,\tan x}\right)\). (2 marks)

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a. \(\text{Proof (See worked solutions)}\)

b. \(-\sin x\)

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a. \(\text{Prove}\ \ \cos x+\sin x\, \tan x=\sec x\)

\(\text{LHS}\)	\(=\cos x+\sin x \cdot \dfrac{\sin x}{\cos x}\)
	\(=\dfrac{\cos ^2 x+\sin ^2 x}{\cos x}\)
	\(=\dfrac{1}{\cos x}\)
	\(=\sec x\)

b.	\(\begin{array}{r} \dfrac{d}{d x}\end{array} \left(\dfrac{1}{\cos x+\sin x\, \tan x}\right)\)	\(=\dfrac{d}{d x}\left(\dfrac{1}{\sec x}\right)\)
		\(=\begin{array}{r}\dfrac{d}{dx}\end{array} \left(\cos x\right)\)
		\(=-\sin x\)

Calculus, 2ADV C2 2022 HSC 25

Let `f(x)=sin(2x)`.

Find the value of `x`, for `0 < x < pi`, for which `f^(′)(x)=-sqrt3` AND `f^(″)(x)=2`. (3 marks)

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`(7pi)/12`

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`f^(′)(x)=2cos(2x)`

`2cos(2x)`	`=-sqrt3`
`cos(2x)`	`=- sqrt3/2`
`2x`	`=pi-pi/6,\ \ pi+pi/6`
	`=(5pi)/6,\ \ (7pi)/6`
`x`	`=(5pi)/12,\ \ (7pi)/12`

`f^(″)(x)=-4sin(2x)`

`-4sin(2x)`	`=2`
`sin(2x)`	`=- 1/2`
`2x`	`=pi+pi/6,\ \ 2pi-pi/6`
	`=(7pi)/6,\ \ (11pi)/6`
`x`	`=(7pi)/12,\ \ (22pi)/12`

`:.x=(7pi)/12\ \ text{(satisfies both equations)}`

Mean mark 53%.

Calculus, 2ADV C2 2019 MET1 1b

Let `f(x) = x^2 cos(3x)`.

Find `f^{′}(pi/3)`. (2 marks)

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`-(2pi)/3`

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	`f(x)`	`= x^2 cos 3x`
	`f^{′}(x)`	`= x^2 ⋅ 3(-sin 3x) + 2x cos 3x`
	`f^{′}(pi/3)`	`= (pi/3)^2 ⋅ 3 (-sin pi) + 2 (pi/3) cos pi`
		`= -(2pi)/3`

Calculus, 2ADV C2 2016 HSC 5 MC

What is the derivative of `ln (cos x)?`

`– sec x`
`– tan x`
`sec x`
`tan x`

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`B`

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`y`	`= ln (cos x)`
`(dy)/(dx)`	`= (-sin x)/(cos x)`
	`= -tan x`

`=> B`

Calculus, 2ADV C2 2010 HSC 2a

Differentiate `cosx/x` with respect to `x`. (2 marks)

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`(-x sinx – cos x)/(x^2)`

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MARKER’S COMMENT: A significant number of students did not know the quotient rule and many found assistance by writing out `u,\ u prime,\ v,\ v prime\ ` before going into calculations.

`y = cos x/x`

`text(Let)`	`\ \ u = cos x`	`u prime = – sin x`
	`\ \ v = x`	`v prime = 1`

`text(Using quotient rule:)`

`dy/dx`	`= (u prime v – u v prime)/(v^2)`
	`= (-sinx x – cos x1)/x^2`
	`= (-x sin x – cos x)/(x^2)`

Calculus, 2ADV C2 2012 HSC 12aii

Differentiate with respect to `x`.

`(cos x)/(x^2)`. (2 marks)

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`(-x sin x\ – 2 cos x)/(x^3)`

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`y = cosx/(x^2)`

`u = cos x`	`\ \ \ \ \ \ u prime = – sin x`
`v = x^2`	`\ \ \ \ \ \ v prime = 2x`

`text(Using the quotient rule,)`

`dy/dx`	`= (u prime v\ – v prime u)/(v^2)`
	`= (- sin x * x^2\ – 2x * cos x)/(x^4)`
	`= (- x sin x\ – 2 cos x)/(x^3)`

Calculus, 2ADV C2 2013 HSC 4 MC

What is the derivative of `x/cosx`?

`(cosx+xsinx)/(cos^2 x)`
`(cosx-xsinx)/(cos^2 x)`
`(xsinx-cosx)/(cos^2 x)`
`(-xsinx-cosx)/(cos^2 x)`

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`A`

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`y = x/cosx`

`text(Let)\ \ \ \ `	`u = x\ \ \ \ \ \ \ `	`u prime = 1`
	`v = cosx\ \ \ \ \ \ \ `	`v prime =-sin x`

`:.\ dy/dx`	`= (vu prime\-uv prime)/v^2`
	`= (cosx 1-x (- sinx))/(cosx)^2`
	`= (cosx + xsinx)/(cos^2x)`

`=> A`