The discussion revolves around finding the tangent line to the curve y=x^3 at a given point (a, a^3) and determining where this tangent line intersects the curve again, specifically at a point where the slope is four times the slope at (a, a^3).
Some participants have provided insights into the form of the tangent line and the necessary components such as slope and intercept. There is an ongoing exploration of how to equate the tangent line with the original curve to find additional intersection points.
There is some confusion regarding the formulation of the tangent line equation, with participants questioning assumptions about its structure and the process of finding it. The original poster has requested hints rather than complete solutions.
That would be, y=3a^2?micromass said:Can you find the equation of the tangent line at a?
Willowz said:That would be, y=3a^2?
Not sure what you are asking for. Could you clarify?micromass said: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 said:Not sure what you are asking for. Could you clarify?
I got y=3a^2*x -2a^3 for the tangent line. Is that alright?micromass said: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??
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 said:Do you know what the equation of a line, perhaps with slope m and passing through point [itex](x_0, y_0)[/itex] looks like?
HallsofIvy said:Do you know what the equation of a line, perhaps with slope m and passing through point [itex](x_0, y_0)[/itex] looks like?
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 said: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.
Well, that means that I misunderstood your question. I think this thread has had a bad start. Thank you.HallsofIvy said: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.
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).Willowz said:I got y=3a^2*x -2a^3 for the tangent line. Is that alright?