Exam2022Proposed solution included

MAT 302 ANALYSIS II Exam 2022

MAT 302: End of Semester Examination for MAT302 Analysis II, Department of Mathematics, Faculty of Science, University of Buea, Instructor Dr. Ndambomve P., June 2022. Exercise 1: definitions of ten metric-space notions (including Lipschitz function and compact set) and statements of the Contraction Mapping Principle, Weierstrass M-Test and Uniform Cauchy Criterion. Exercise 2: proofs that d(x,K)=0 iff x is in the closure of K, the reverse-triangle-type bound on d(x,K), uniform continuity of f(x)=d(x,K), continuity preserving compactness (f(K) compact), and that a closed subset of a compact set is compact. Exercise 3: proof that a uniform limit of continuous functions is integrable and that limit/integral may be interchanged; pointwise/uniform convergence analysis of f_n(x)=(2n+sin(nx))/(3n+sin^2(nx)) including a limit-interchange integral computation; and a Weierstrass M-Test proof of uniform convergence for a series g_n(x)=x^(3n)/(n+2x)^2 on [0,1]. This paper is a near-duplicate pair with ID 2843 (same course, July 2021 sitting, same instructor), sharing Exercises 1-2 word-for-word and Exercise 3's structure with renamed/reparametrised function families; each is transcribed and solved independently from its own scan.

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University of Buea

Faculty of Science - FS

Mathematics - MAT

MAT3022022

ANALYSIS II

Exam · Second Semester · Level 300

From this paper

1.
Carefully define the following notions in the framework of metric spaces: a) Lipschitz function, b) fixed point of a map, c) convergent sequence, d) Cauchy sequence, e) complete metric space, f) continuous functions, g) uniformly continuous function, h) pointwise convergence of sequences of functions, i) uniform convergence of sequences of functions, j) compact set. [8 marks]
2.
Carefully state, without proof: a) the Contraction Mapping Principle, b) the Weierstrass MTest Theorem for uniform convergence of series of functions, c) the Uniform Cauchy Criterion. [9 marks]