BC 2
Cubics, Tangents, and Area Extra Credit
Name:
Due Date: 8 Dec 06
Graph a general cubic polynomial of the form y  ax3  bx2  cx  d . Draw a tangent line to the graph
of your cubic at some point x = p. Be sure this tangent line intersects the graph of the cubic at another
point. Determine the second point (say x = q) where this tangent intersects the cubic. Determine the
area of the region bounded by the tangent and the cubic. Next, draw the tangent line to the cubic at x = q
and determine second point (say x = r) where this tangent intersects the cubic. Determine the area of the
region bounded by this second tangent and the cubic. What is the ratio of the larger area to the smaller
area? Prove this result will hold for all cubic polynomials.
x=p
x=q
Note: The proof of this result may
involve quite a bit of not-too-nice
algebra. You are welcome to use the TI89 or Mathematica to do the “grunt”
work (like locating points of intersection
of the tangent line and the cubic or
calculating the values of integrals).
What you must be sure to do to receive
credit is explain the steps you are taking
and why you are taking them.
x=r
IMSA
F06