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?
The slope of the tangent line at a point (x,y) on the graph of y=x^3 is given by the derivative of the function, which is equal to 3x^2. So, the formula for finding the slope of the tangent line is m = 3x^2.
To find the intersection point, we need to equate the tangent line equation with the function y=x^3. This will give us a point where both the tangent line and the graph intersect. We can then solve for the x-coordinate of the intersection point by setting y=x^3 equal to the tangent line equation and solving for x. The y-coordinate can be found by plugging in the x-coordinate into the original function.
The intersection point of the tangent line and the graph of y=x^3 represents the point where the slope of the tangent line is equal to the slope of the graph at that point. This means that the tangent line is a perfect approximation of the graph at that point, making it useful in understanding the behavior of the function at that specific point.
Yes, the slope of the tangent line can be negative for y=x^3. It depends on the x-coordinate of the point at which the tangent line is drawn. If the x-coordinate is negative, then the slope will be negative, and if the x-coordinate is positive, then the slope will be positive.
The slope of the tangent line gives us information about the rate of change of the graph at a specific point. If the slope of the tangent line is increasing, then the graph is concave up, and if the slope of the tangent line is decreasing, then the graph is concave down. This relationship is due to the fact that the slope of the tangent line is equal to the derivative of the function, which also gives us information about the concavity of the graph.