Chapter 2, Problem 2PS

Area Let P(x, y) be a point on the parabola $y=x^2$ in the first quadrant. Consider the triangle $\Delta PAO$ formed by P, A (0, 1), and the origin O (0, 0), and the triangle $\Delta PBO$ formed by P, B (1, 0), and the origin (see figure).

(a) Write the area of each triangle in terms of x.

(b) Let a (x) be the ratio of the areas of the two triangles,

$a\left(x\right)=\frac{\Delta PBO}{\Delta PAO}\cdot$

Complete the table. Calculate $\underset{x\to 0^{+}}{\lim }a\left(x\right)$.

x 4210.1 0.01

Area $\Delta PAO$

Area $\Delta PBO$

a(x)