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Calculus, SPEC2 2016 VCAA 7 MC

Given that  `x = sin(t) - cos(t)`  and  `y = 1/2 sin(2t)`, then  `(dy)/(dx)`  in terms of  `t`  is

A.  `cos(t) - sin(t)`

B.  `cos(t) + sin(t)`

C.  `sec(t) + text(cosec)(t)`

D.  `sec(t) - text(cosec)(t)`

E.  `(cos (2t))/(cos(t) - sin(t))` 

Show Answers Only

`A`

Show Worked Solution

`(dx)/(dt)= cos(t) + sin(t),\ \ dy/(dt)=cos(2t)`

♦ Mean mark 37%.

`(dy)/(dx)` `= (dy)/(dt) ⋅ (dt)/(dx)`
  ` = cos(2t) /(cos(t) + sin(t))`
  `= (2 cos^2(t) – 1)/(cos(t) + sin(t))`
  ` = (cos^2(t) – sin^2(t))/(cos(t) + sin(t))`
  `= {(cos(t) – sin(t))(cos(t) + sin(t))}/(cos(t) + sin(t))`
  `= cos(t) – sin(t)`

 
`=>  A`