Chapter 12.5, Problem 54ES

These exercises are concerned with the problem of creating a single smooth curve by piecing together two separate smooth curves. If two smooth curves $C_1\text{ and }{C}_2$ are joined at a point P to form a curve C, then we will say that $C_1\text{ and }{C}_2$ make a smooth transition at P if the curvature of C is continuous at P.

(a) Sketch the graph of the curve defined piecewise by $y=x^2\text{ for }x<0,y=x^4\text{ for }x\ge 0.$

(b) Show that for the curve in part (a) the transition at $x=0$ is not smooth.