Cubic Difference Parabola
What is the first derivative of y = x3?
This can be derived algebraically, but it is always good to discover this by an empirical approach.
Personally I always like to make sure that the relationship between y = x2 and y = x3 is well understood by getting both of these graphs sketched by hand on the same axes, paying particular attention to the range -1 ≤ × ≤ 1.
On your graphing software, draw y = x3, with a point attached at x = 1, and a tangent drawn.
You can show the gradient (slope) of the tangent by placing a vector [1, 0] on the point, a vertical line on the end of it, and find the intersection with the tangent.
Clearly when x = 1, the first derivative has a value 3.
Move on round the cubic and see where all the other points are.
Graphical analysis video using Autograph
Douglas Butler illustrates using Autograph to model this parabola.
Play with this Autograph file here...
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Douglas Butler
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