Chapter 6.1, Problem 70E

Another look at the Fundamental Theorem

70. Use Exercise 69 to prove that if two runners start and finish at the same time and place, then regardless of the velocities at which they run, their displacements are equal.

69. Suppose that f and g have continuous derivatives on an interval [a, b]. Prove that if f(a) = g(a) and f(b) = g(b), then ${\displaystyle {\int }_a^b{f}^{\prime }(x)dx={\displaystyle {\int }_a^b{g}^{\prime }(x)dx}}$.