Chapter 7.8, Problem 86E

The Eiffel Tower property Let R be the region between the curves y = e^−cx and y = −e^−cx on the interval [a, ∞), where a ≥ 0 and c > 0. The center of mass of R is located at $(\bar{x},0)$, where $\bar{x}=\frac{{\displaystyle {\int }_a^{\infty }x{e}^{-cx}dx}}{{\displaystyle {\int }_a^{\infty }{e}^{-cx}dx}}$. (The profile of the Eiffel Tower is modeled by the two exponential curves; see the Guided Project The exponential Eiffel Tower.)

1. a. For a = 0 and c = 2, sketch the curves that define R and find the center of mass of R. Indicate the location of the center of mass.
2. b. With a = 0 and c = 2, find equations of the lines tangent to the curves at the points corresponding to x = 0.
3. c. Show that the tangent lines intersect at the center of mass.
4. d. Show that this same property holds for any a ≥ 0 and any c > 0; that is, the tangent lines to the curves y = ± e^−cx at x = a intersect at the center of mass of R.