Exam 1 Questions with Solutions - Analysis II | MATH 4001, Exams of Mathematical Methods for Numerical Analysis and Optimization

Download Exam 1 Questions with Solutions - Analysis II | MATH 4001 and more Exams Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! Math 4001, Spring 2012 Test 1 Your name is: Use complete sentences. Cite needed theorems completely. 1. (25 pts) Let f be a monotone function on [0, 1]. Show that f is Riemann integrable. (This entails properly recalling what it means for a function to be Riemann intergrable.) Need to show that for any  > 0, there is a partition π of [0, 1] such that U(f, π)− L(f, π) ≤  where U and L are the upper and lower Riemann sums. In particular, U = ∑ supx∈∆k f(x)|∆k| with ∆k the kth increment of the partition, and in L we replace the sup by inf. Take the case that f is monotone increasing, and pick the partition πN with xk = k N . Then U(f, πN )− L(f, πN ) = N−1∑ k=0 (f(k + 1/N)− f(k/N))(1/N) = (1/N)(f(1)− f(0)), and this can be made as small as desired by choosing N large. 1 2. (25 pts) Consider the the space X = C([0, 1]) of real valued continuous functions on [0, 1]. This may equipped with a variety of metrics, such as d∞ defined by d∞(f, g) = supx∈[0,1] |f(x) − g(x)| or d1 defined by d1(f, g) = ∫ 1 0 |f(x)− g(x)|dx. Define F : X 7→ R by F (f) = f(0). Is this a continuous function for (X, d∞)? How about for (X, d1)? It holds that |F (f)− F (g)| = |f(0)− g(0)| ≤ sup x∈[0,1] |f(x)− g(x)| = d∞(f, g), so F is continuous on (X, d∞). For the case of d1 consider the functions fn(x) = (1− x)n. Certainly F (fn) = 1 for all n. On the other hand: with O representing the function that is identically d1(fn,O) = ∫ 1 0 (1− x)ndx = 1 n+ 1 , which can be made arbitrarily small by taking n large, but of course F (O) = 0. So F is not continuous under this metric. 2