anonymous
  • anonymous
A curve is such athat {dy/dx}= {12 over (2x+1)^2} and P(1,5) is a point on the curve. (i) The normal to the curve at P crosses the x-axis at Q. Find the coordiantes of Q.
Mathematics
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SOLVED
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chestercat
  • chestercat
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Callisto
  • Callisto
slope of normal = -1/ (dy/dx) you've got the slope and point so find he equation for the normal and you can find the coordinates of Q
Callisto
  • Callisto
Hmm.. normal is perpendicular to tangent. dy/dx is the slope of tangent. Do you understand what i've written?
anonymous
  • anonymous
the slope of the normal meeting the curve = -1 so m1m2= -1 -1/dxdy= m2 (the slope of the normal) Does this equal to (2x+1)^2/12?

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Callisto
  • Callisto
you've forgotten the negative sign..
anonymous
  • anonymous
Oh, yes. - (2x+1)^2 /12
anonymous
  • anonymous
Then I have to find the equation of the line? How?
Callisto
  • Callisto
point - slope form
anonymous
  • anonymous
y=0 no?
Callisto
  • Callisto
(y-y1) = m(x-x1) (x1,y1) = (1,5)
Callisto
  • Callisto
m = - (2x+1)^2 /12 and put x=1into m=...
anonymous
  • anonymous
Hm, Ok... m= -(2x+1)^2/12 m= -9/12 m= -3/4?
Callisto
  • Callisto
should be like that
anonymous
  • anonymous
So, then y=-3/4x+b Then do I plug in the y and x values of (1,5)?
Callisto
  • Callisto
(y-y1) = m(x-x1) (x1,y1) = (1,5) Sorry if i've used the wrong words...
anonymous
  • anonymous
I mean, would the equation of the line be y= -3/4x+5 3/4?
Callisto
  • Callisto
shouldn't it be y= -3/4x+4 3/4?
anonymous
  • anonymous
Why?
anonymous
  • anonymous
the m value is -3/4, so if you move that, it's 5+3/4
Callisto
  • Callisto
Sorry my mistakes :(
anonymous
  • anonymous
That's OK... So then do you plug in 0 in the y value?
anonymous
  • anonymous
to get x, when it's Q
Callisto
  • Callisto
yup!
anonymous
  • anonymous
Can you see what you get and so I can double check my answer?
anonymous
  • anonymous
I got 7.67 = x?
Callisto
  • Callisto
something ugly, 7 2/3, 7.67 if you need a decimal number
anonymous
  • anonymous
Yes. It looks ugly... I hope it's correct! :)
anonymous
  • anonymous
The next question says find the equation of the curve?....?
Callisto
  • Callisto
use integration
Callisto
  • Callisto
\[y=\int\limits dy/dx\]
anonymous
  • anonymous
I thought we have the slope of the curve, which is 12/(2x+1)^2? So, can't we just use that? Why do we have to integrate?
Callisto
  • Callisto
Ehhh.. you need the curve right?
anonymous
  • anonymous
I need the equation of the curve...
Callisto
  • Callisto
then you need to know that dy/dx is the slope of the curve, to find the curve you need to do the reverse..
anonymous
  • anonymous
Oh... I just thought you could use that slope and do y=mx+b, where m is the slope??
Callisto
  • Callisto
nope... it's a curve, not a straight line. For a curve, the slope differs from point to point. So, there are no constant slope
Callisto
  • Callisto
*there is
anonymous
  • anonymous
Oh, Ok. So.... how do you integrate it?
Callisto
  • Callisto
I'm not good at using math latex here.. give me some time to write it..
anonymous
  • anonymous
Sure. I'll be back in a moment.
anonymous
  • anonymous
am back
Callisto
  • Callisto
1 Attachment
anonymous
  • anonymous
Thank you! So, is that the equation?
anonymous
  • anonymous
Wait, why is it timesed by 1/2?
Callisto
  • Callisto
Personally, i think it is, but not sure
anonymous
  • anonymous
Should I ask Sam?
Callisto
  • Callisto
you can d/dx(equation) and see if you understand why it is multiplied by 1/2
Callisto
  • Callisto
Sure,you should , I though he should have taught you since you asked the question.. but that falls short of my expectation :(
anonymous
  • anonymous
@.Sam. Can you check all these answers.?
anonymous
  • anonymous
Callisto, there's a last part of the question.
Callisto
  • Callisto
Honestly, i've learnt differentiation for a year and integration for half a year.. So , i'm not really good at that :S (no time to practise)
anonymous
  • anonymous
Um... the last part of the question says, A point is moving along the curve in such a way that the x=coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the y-coordinate when x=1
anonymous
  • anonymous
your integretion was correct :)
anonymous
  • anonymous
Um, I have to go very soon. Can you see if you can help me with the last question, and then it'll be good?
Callisto
  • Callisto
Sorry was away from computer. I'm not sure for this problem from the question, you'll have dx/dt = 0.03 and you can do it like that dy/dt = (dy/dx)(dx/dt) to find dy/dt..
Callisto
  • Callisto
sorry again, it should be dx/dt = 0.3
Callisto
  • Callisto
then it's like dy/dt = {12 over (2x+1)^2} * 0.3 Put x=1into dy/dt and you can find the answer, i think

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