smc-1073-10-Trig

Functions, EXT1 F1 2021 HSC 7 MC

Which curve best represents the graph of the function `f(x) = -a sin x + b cos x` given that the constants `a` and `b` are both positive?

A.		B.
C.		D.

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`D`

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`text(By elimination:)`

`text(At)\ \ x = 0, -asinx = 0\ \ text(and)\ \ bcosx = b`

COMMENT: When told constants are “positive”, pay close attention!

`:. f(0) = b > 0`

`->\ text(Eliminate A and C)`

`f′(x) = -acosx – bsinx`

`text(At)\ \ x = 0, -acos x = -a\ \ text(and)\ \ bsinx = 0`

`:. f′(0) = -a < 0`

`->\ text(Eliminate B)`

`=>\ D`

Functions, EXT1 F1 SM-Bank 16

Given `f(x) = 2 sinx` and `g(x) = x`, sketch `y = f(x) + g(x)` for `0 <= x <= 2pi`.

Clearly label endpoints and all local maximum and minimums. (3 marks)

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`y`	`= 2sinx + x`
`(dy)/(dx)`	`= 2cosx + 1`
`(dy)/(dx)`	`= 0 \ => \ cosx = −1/2 => x = (2pi)/3, (4pi)/3`

`text(MAX at)\ ((2pi)/3, 3.83)`

`text(MIN at)\ ((4pi)/3, 2.46)`

Functions, EXT1 F1 EQ-Bank 15

Given `f(x) = sin x` and `g(x) = cos x`, sketch `y = f(x) + g(x)` for `−pi <= x <= pi`. (2 marks)

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State the range of `y`. (2 marks)

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`text(Range)\ {−sqrt2 <= y <= sqrt2}`

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i.	`y`	`= sinx + cosx`
	`(dy)/(dx)`	`= cosx – sinx`
	`(dy)/(dx)`	`= 0\ \ text(when)\ \ sinx = cosx`

COMMENT: Question style consistent with NESA exemplar questions in official Topic Guidance.

`:. text(SP’s when)\ \ x = pi/4, −(3pi)/4`

ii. `text(Max/min when)\ \ (dy)/(dx) = 0`

`x = pi/4, −(3pi)/4`

`y_(text(max))`	`= sin\ pi/4 + cos\ pi/4`
	`= 2 xx 1/sqrt2`
	`= sqrt2`

`:.\ text(By symmetry, range)\ {−sqrt2 <= y <= sqrt2}`

Functions, EXT1′ F1 2013 HSC 9 MC

Which diagram best represents the graph `y = (sin x)/x?`

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`B`

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`f(x)`	`=(sin x)/x`
`f(−x)`	`=(sin (−x))/-x`
	`=(sin (x))/x`
	`=f(x)`

`=>f(x)\ \ text(is even)`

`lim_(x -> 0) (sin(x))/x =1`

`=> B`

Functions, EXT1′ F1 2011 HSC 3a

Draw a sketch of the graph

`quad y = sin\ pi/2 x` for `0 < x < 4.` (1 mark)

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Find `lim_(x -> 0) x/(sin\ pi/2 x).` (1 mark)

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Draw a sketch of the graph

`quad y = x/(sin\ pi/2 x)` for `0 < x < 4.` (2 marks)
(Do NOT calculate the coordinates of any turning points.)

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ii. `2/pi`

iii.

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ii. `lim_(x->0) x/(sin\ pi/2 x)`	`= 2/pi lim_(x->0) (pi/2 x)/(sin\ pi/2 x)`
	`= 2/pi xx 1`
	`= 2/pi`

iii.