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at what points does the curve intersect the paraboloid

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How many Tangent lines attend a certain taper.

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Homework Statement

How many tangent lines to the curve y = x / (x + 1) transit the point (1,2)? At which points do these tan lines touch the curve?

Homework Equations

n/a

The Set about at a Solution

I'm completely draftsmanship a blank on this ane. Facilitate.

Answers and Replies

Let's first simply imagine a tangent furrow to the curve passing through the point (1,2).
What would the equation of this tangent be?

Well we know a point, and we don't roll in the hay what the gradient is but we do know it is tangent to the curve so this would point towards the derivative equation, yes?

So we have, [tex]y-2=m(x-1)[/tex] where [tex]m=\frac{dy}{dx}=\frac{d}{dx}\left(\frac{x}{x+1}\right)[/tex]

And this line equation touches the veer, so what can we do here?

Indeed we have, [tex]y-2=m(x-1)[/tex] where [tex]m=\frac{Dy}{dx}=\frac{d}{dx}\left(\frac{x}{x+1}\right)[/tex]

And this line equation touches the curve, so what can we do here?


Find the pitch of the line, and and then follow it outward to realise at what other points it would signature the chart?
Retrieve the slope of the line, and so follow IT outward to see at what other points it would relate the chart?

Trial and mistake is one style, just a very bad way. Let's coiffure this more rigorously :smile:

You already have the equation of the tangent short letter(s), [tex]y=m(x-1)+2[/tex] and the equation of the curve [tex]y=\frac{x}{x+1}[/tex], and we know that the line touches the curve. How can we find the point where they partake past mathematical way?

If you backside't quite figure it out, it's probably that you're just being overwhelmed by the problem at hand down. As a simpler example, it's same to finding the points of intersection between the equations [tex]y=x+1[/tex] and [tex]y=2x[/tex]

you could make them equal one another. Since they both have y = (something) then puzzle out for the variable. Or that's what i'd try in your example there. Simply i don't know how to ramp that up to pass process a bigger scale, corresponding the example i have.
That's it.

Thusly we sporty have to solve [tex]m(x-1)+2=\frac{x}{x+1}[/tex] for x, where [tex]m=\frac{dy}{dx}[/tex]. This will give you the points where the describe and curve cross, which means you'll sustain the x values of where the tan(s) to the curve passing through (1,2) touch the curve.

OK, so then i would have to first get the slope of the line though wouldn't i? how stool i do that if i sole have one point?
Alright, then and then i would have to first produce the slope of the pipeline though wouldn't i? how can i do that if i only have one peak?

Healed we know the gradient is going to be the first derivative at the point x that we solve for, thus we give the sack just find the derivative and replace the variable m for IT.
So i got a rate of x to be -2 - 1√3 and -2 + 1√3. I'm intellection something has gone wrong, cause that would cost impossible to try and work out.
What do you contemptible it would be impossible to try and figure?

Soh the x values are [tex]x=-2\pm \sqrt{3}[/tex], at present ascertain the corresponding y values to complete the answer. If you deprivation to hinderance without doubt you rich person the right answer, find the derivative at those points, and so test to see if the line going through and through that sharpen with such a slope passes through (1,2).

But I'll save you the deed and tell you now that it's right hand.

haha thanks. I just had a moment where i forgot these can likewise be decimal fraction values. As i read that too i thought, this job doesn't look worth all this effort. Merely thanks for checking them for me. I appreciate information technology mankin. Thanks for all your help over again.

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at what points does the curve intersect the paraboloid

Source: https://www.physicsforums.com/threads/how-many-tangent-lines-go-to-a-certain-point.469774/

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