MATH 223 Final, Fall 1996

Prof. F.Abi-Khuzam Math 223 January 29,1997 Final 2 hrs. and 30 min. 1. Let f be a real -valued continuous function defined on a closed and bounded interval [a,b]. State four important properties of f and prove one of them. 2.Suppose f is a real-valued function continuous on 0, «© ) and differentiable on (0,c¢ ) .Suppose also that (0) = 0 and f is monotonically decreasing. Prove that the function g defined by g(x) = f(x)/x is monotonically decreasing. What can you say about the function Gr cos) 2 —x 2 when x is in (0, m /2)? Justify your answer. 3. Let f be a twice - differentiable function on (c, ee ) and let M,N, and P be the least upper bounds of | f(xJ, | fA, AS Od respectively on (c,c). Prove that the square of N can never exceed 4MP. 4. (a) Suppose f and g are real differentiable functions on (- 0 , © ).If f(t) is not equal to g(t) for every real ¢ how many points x satisfying f(x) = e(x) can there be? Justify your answer. (b) Suppose that there is a constant A </ such that |f(t]< A for all reals, prove that there is a point x such that f(x) = x. 5. Prove the inequalities 242x+1) < Inf (x+1 xu] < (2x+1)/2x(x+1) for x>0. 6. Let f(x) = (cosx)M1+x) and g(x) = (sin x1 +x) . Prove that the integral over [0, e ) of f equals that of g and that one of these integrals converges absolutely but not the other. 7.(a) Give an example of a bounded differentiable function whose derivative is not . bounded on (0, °). (b) If f is a continuous function between two metric spaces X and Y, does f map cauchy sequences in X to cauchy sequences in Y.Justify. (c) Prove that, in a metric space, closed subsets of compact sets are compact. (d) If & is monotonically increasing on [0,b] what is the value of the Stieltjes integral Jdcr over [0,x] for 0 <x <b.Justify. (e) State and prove the fundamental theorem of Calculus.