Yes - i thought you had to show it by doing dy/dx < 0 (or is it the other way round)
That is the way to show it, what would be the "other way round"? You do NOT have to solve f′(x)<0 if that's what you're asking, it is a transcendental equation so you wouldn't get any exact answers anyway. You simply have to factor out a -1 then show how the inside of the bracket is always positive therefore f′(x)<0 in the given domain of x∈R,x>0
That is the way to show it, what would be the "other way round"? You do NOT have to solve f′(x)<0 if that's what you're asking, it is a transcendental equation so you wouldn't get any exact answers anyway. You simply have to factor out a -1 then show how the inside of the bracket is always positive therefore f′(x)<0 in the given domain of x∈R,x>0
How could i show inside the bracket is always +ve - just input some values?
How could i show inside the bracket is always +ve - just input some values?
Not really, that wouldn't prove anything for ALL x>0. Since the numerator's are already positive, consider how e2x and x2 are always positive for all real x
If this was a question in an exam do you know what the mark scheme would say? e.g. certain phrases you must include?
It would say ex>0⇒e2x>0 and x2>0⇒x23>0 where x>0 therefore f′(x)<0 hence f is decreasing over the interval but mark schemes never really mention all of it. Would touch up on the fact that the two terms are greater than 0, though.
More inequalities than 'certain phrases' to be honest. This would be like 2 or 3 marks max.
Finally, I have a normal A Level question which I'm stuck on (I probably have annoyed many here by posting stuff which should go in the STEP prep thread):
Find the volume of revolution of the shape modelled by the equation x=sint,y=sin2t,0<t<2π.
V=π∫02πy2dtdxdt so I used the fact that y2=sin22t=4sin2tcos2t and dtdx=cost to get me to 4π∫02πsin2tcos3tdt=4π∫02πcost3−cost5dt. What do I do next?
Finally, I have a normal A Level question which I'm stuck on (I probably have annoyed many here by posting stuff which should go in the STEP prep thread):
Find the volume of revolution of the shape modelled by the equation x=sint,y=sin2t,0<t<2π.
V=π∫02πy2dtdxdt so I used the fact that y2=sin22t=4sin2tcos2t and dtdx=cost to get me to 4π∫02πsin2tcos3tdt=4π∫02πcost3−cost5dt. What do I do next?
Am I bizarrely overcomplicating this?
You can use a reduction formula to integrate cos3t and cos5t but there's an easier way of doing this. If you notice that sin2tcos3t≡sin2tcost(1−sin2t), it should be much easier to integrate.