Question

Question of utmost urgency, 4 hours for my exam!!!!!

Finding all the points on the curve at which the tangent passes through the origin

y = 4x^3 - 2x^5

How do I approach and solve this?

Answer 1

@phi @zepdrix

Answer 2

@WordGEEK

Answer 3

Sorry I do not know this :(

Answer 4

Do you know anyone who does? I really need to learn this :(

Answer 5

@Kainui

Answer 6

@ganeshie8

Answer 7

Not gonna lie, this is a tough one. It seems like graphing this would be one of the easiest ways to see, then take a line through the origin and just rotate it. When it first hits the graph, that's exactly where the tangent line would fall.

Interesting question, I'm sure there's a better way to do this, let me think about it a little.

Answer 8

So the general lines we're looking to compare this to are of the form:

y=mx

since the y-intercept is 0. We also know that because m=rise/run and that this line goes through the origin: \[\Large m= \frac{y_2-y_1}{x_2-x_1}=\frac{y_2}{x_2}\]because (x1,y1)=0.

That's actually not so bad now that we notice this. Notice all I did here wasn't any kind of magic genius, all I did was begin to start looking at the definitions of things and plugging in numbers and it's getting us closer to the answer.

Answer 9

(x1, y1)=(0,0) is what i meant to write haha. Ok, let's take this a little further. The slope of the line we're making is just the y value divided by the x value at that point. This means that for a line to have the same slope as our line, we just compare their slopes! \[\Large \frac{y}{x}=\frac{dy}{dx}\] Now we just solve this formula by plugging in. Does that make sense?

y/x is just the slope of our linear line through the origin and dy/dx is the slope of the curve we were given. When these are equal, we will have a tangent line at the same place on our curve because we set them equal to each other.

Answer 10

so we're actually using y - y_1 = m(x - x_1)

y - 0 = m(x - 0) so its y=mx

What we do next is find f'(x)?

f'(x) = 12x^2 - 10x^4?

Answer 11

Yes exactly, now we're just setting the derivative equal to y/x since that's the slope of our line.

Answer 12

Okay our next step is to put our f'(x) into y = mx?

Answer 13

It actually solves very nicely! To phrase it in another notation, but identical to what I have above:\[\Large f'(x)=\frac{f(x)}{x}\]

I have to go eat dinner right now I'm afraid. I'll be back very soon to help you out though, I want to help make sure you understand so you can do well! =)

Answer 14

I'll have to go by then.. I'm leaving right now as my exam center is over 10 km away :( I'm nervous as hell..

Answer 15

Good luck! Try not to freak out too much and breathe.

Answer 16

Thanks bud :) I'll try my best! bye! :) I'll be back this afternoon