Mathematics Problem Of the Week

Spring 2006

POW #6

Follow the Intercept

The graph below shows the function y=f(x)=xⁿ for n>1 and x>0. At any point (c,f(c)), draw the tangent line to the curve, then draw another line, perpendicular to that tangent line through the same point. Let Y(c) denote the value at which that line hits the vertical y-axis (as in the diagram below).

For n=1, it should be obvious that . Is this always the case?

Are there values for n>1 for which the limit exists but is not zero?

Are there cases where the limit does not exist?

Give a complete description of the behavior of for all n>1 and justify your reasoning.

Due Friday, March 10^th Noon

Solutions should be submitted to Dr. R. Lock's mailbox in the Math office or sent via e-mail to rlock@stlawu.edu.