smc-744-30-tan

Calculus, MET1 2006 VCAA 3a

Let `y = x tan(x)`. Evaluate `(dy)/(dx)` when `x = pi/6`. (3 mark)

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`(3sqrt3 + 2pi)/9`

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`text(Using the product rule:)`

MARKER’S COMMENT: A common error was not knowing the exact trig function values.

`d/(dx)(uv)`	`= u^{prime}v + uv^{prime}`
`(dy)/(dx)`	`= tan(x) + x/(cos^2(x))`

`text(When)\ x = pi/6,`

`(dy)/(dx)`	`= tan(pi/6) + (pi/6)/((cos(pi/6))^2)`
	`= 1/sqrt3 + pi/6 xx (2/sqrt3)^2`
	`= sqrt3/3 + (4pi)/18`
	`= (3sqrt3 + 2pi)/9`

Calculus, MET1 2007 ADV 2ai

Let `f(x)=(1 + tan x)^10.` Find `f^{\prime}(x)`. (2 marks)

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`10 sec^2 x \ (1 + tan x)^9`

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`f(x) = (1 + tan x)^10`

`f^{\prime}(x)`	`= 10 (1 + tan x)^9 xx d/(dx) (tan x)`
	`= 10 sec^2 x \ (1 + tan x)^9`

Calculus, MET1 2006 ADV 2ai

Differentiate with respect to `x`:

Let `f(x)=x tan x`. Find `f^{prime}(x)`. (2 marks)

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`f^{prime}(x) = x sec^2 x + tan x `

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`y = x tan x`

`text(Using product rule)`

`f^{prime} (uv)`	`= u^{prime}v + uv ^{prime}`
`:.f^{prime}(x)`	`= tan x + x xx sec^2 x`
	`= x sec^2 x + tan x`

Calculus, MET1 SM-Bank 20

If `f(x)= 2 sin 3x - 3 tan x`, find `f^{prime}(0)`. (2 marks)

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`3`

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`y`	`= 2 sin 3x-3 tan x`
`(dy)/(dx)`	`= 6 cos 3x-3 sec^2 x`

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

`(dy)/(dx)`	`= 6 cos 0-3 sec^2 0`
	`= 6 (1)-3/(cos^2 0)`
	`= 6-3`
	`= 3`

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`