Chapter 1.4, Problem 9E

The point P(1, 0) lies on the curve $y=\sin \left(10\pi /x\right)$.

(a) If Q is the point $\left(x,\sin \left(10\pi /x\right)\right)$, find the slope of the secant line PQ (correct to four decimal places) for x=2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8 and 0.9. Do the slopes appear to be approaching a limit.

(b) Uses a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at P.

(c) By choosing appropriate secant lines, estimate the slope of the tangent line at P.