MATH 210 Exam 2, Spring 2010

Math 210 W Quiz # 2, April 26, 2010 1. Let f : [a,b] — R be a continuous function. State, without proof, four properties of f. Do not supply proofs, just give clear and precise statements of those properties. 2. Let f : [a,b] > R and z € [a,b]. If f is continuous at the point x, and f(x) 4 0, prove that + is continuous at xz. Give an e — 6 proof. If you need a property of f related to those properties in problem 1, state that property clearly, but do not prove it. 3. Let f : R — R be the function defined by fe)={79rt 240}. z=0 (a) Prove that f is differentiable everywhere on R and compute f’(z) for each « ER. (b) Prove that f’ is bounded on R. (c) Prove that f’ is not continuous at 0. 4, Prove the inequality 2+ - ia) ew< 5 <2 <2. -—2 5. Let f : R > R be a differentiable function, and suppose that there is a constant A < 1 such that |f’(¢)| < A for all real t. Let zo € R, and define a sequence {z,} by 22m + 3f (In) = Inti = 5 =0,1,2,--- Prove that the sequence {xn} is convergent, and that its limit is the unique fixed point of f. 6. Suppose (a) f is continuous on [0,00); (b) differentiable on (0,00); (c) f(0) = 0; (d) f’ is monotonically increasing. Put g(x) = ie)3ve z > 0, and prove that g is monotonically increasing.