Finding Tangent Line Slope for y=x^3 and its Intersection Point

[Willowz]

Homework Statement

show that the tangent line to the curve y=x^3 at any point (a,a^3) meets the curve again at a point where the slope is four times the slop at (a,a^3).

Homework Equations

y=3x^2, point slope formula

The Attempt at a Solution

Not really sure where to begin. A hint should be enough.

 

[micromass]

Can you find the equation of the tangent line at a?

 

[Willowz]

Can you find the equation of the tangent line at a?

That would be, y=3a^2?

 

[micromass]

That would be, y=3a^2?

No, where in Earth did you get that?? A tangent line is a line. Its equation must contain an x and a y variable.

 

[Willowz]

No, where in Earth did you get that?? A tangent line is a line. Its equation must contain an x and a y variable.

Not sure what you are asking for. Could you clarify?

 

[micromass]

Not sure what you are asking for. Could you clarify?

I'm asking for the equation of the tangent line at the curve. How do you find the tangent line at a curve?? What is the slope of such a tangent line??

 

[HallsofIvy]

Do you know what the equation of a line, perhaps with slope m and passing through point (x_0, y_0) looks like?

 

[Willowz]

I'm asking for the equation of the tangent line at the curve. How do you find the tangent line at a curve?? What is the slope of such a tangent line??

I got y=3a^2*x -2a^3 for the tangent line. Is that alright?

 

[chrisbaird]

The equation of a line is y = m x + b. You need to find the right slope m and intercept b that will make the line be the right one that is tangent and touches a certain point.

 

[Willowz]

Do you know what the equation of a line, perhaps with slope m and passing through point (x_0, y_0) looks like?

Yes, I do have an idea how it may look like. A straight line touching the curve and then intersecting the function at some other point where the slope is four times the value of the tangent line.

 

[HallsofIvy]

Do you know what the equation of a line, perhaps with slope m and passing through point (x_0, y_0) looks like?

Yes, I do have an idea how it may look like. A straight line touching the curve and then intersecting the function at some other point where the slope is four times the value of the tangent line.

The scary thing is that you think that answers my question. I asked about what the equation looks like and you said nothing about an equation.

 

[Willowz]

The scary thing is that you think that answers my question. I asked about what the equation looks like and you said nothing about an equation.

Well, that means that I misunderstood your question. I think this thread has had a bad start. Thank you.

 

[eumyang]

I got y=3a^2*x -2a^3 for the tangent line. Is that alright?

This is correct. Now find out where else does this line intersect with y = x³. Set the two right hand sides equal to each other and solve for x (treating a as a constant).