MATH 410 Midterm: MATH410_BOYLE-M_FALL2008_0101_MID_SOL

• School: University of Maryland
• Department: Mathematics
• Course: MATH 410
• Professor: All

Document Summary

Math 410 fall 2008 midterm 1 solutions: (15 points) state each of the following. (a) the completeness axiom for the real numbers. If a nonempty subset of the real numbers has an upper bound, then it has a least upper bound. (b) rolle"s theorem. Suppose a real valued function f is continuous on [a, b], di erentiable on (a, b) and satis es f (a) = f (b). Then there is a point x in (a, b) such that f (x) = 0. (c) the cauchy mean value theorem. Suppose f, g are continuous on [a, b], di erentiable on (a, b) and g is nonzero on (a, b). Then there exists a point c such that a < c < b and f (c)/g (c) = [f (b) f (a)]/[g(b) g(a)]: (15 points) suppose c is a real number and c > 1. Prove for all positive integers n that (1 + c)n 1 + nc.