APSC 172 Lecture 1: 172 cubic-tangent solution

• School: Queen's University
• Department: Applied Science
• Course: APSC 172
• Professor: Ping Li

Document Summary

At the right is the graph of the equation y = x (x 1) (x+1). There are two lines passing through ( 1,0) that are tangent to the curve. Think of the family of lines passing through ( 1, 0). So we can find these by counting intersections. Using the point-slope form of a straight line, the family of lines has equation yl = m(x+1) where the slope m serves as the parameter identifying different lines in the family. Writing the curve as: yc = x (x 1) (x+1) they intersect when: yc = yl x (x 1) (x+1) = m(x+1) x (x 1) = m x2 x m = 0 and this solves to give. The story is told by the sign of the discriminant d = 1 + 4m: 1 + 4m > 0 m m < m = m > . Pay attention to that step when we divided both sides by (x+1).