PEI Math 521K case studies
PEI Math 521K · Put your knowledge to work in context.
Math 521K is a University of Prince Edward Island course in real analysis, and this page prepares you for its final assessment. It is aimed at mathematics majors and others who need a rigorous, proof-based grounding in single-variable analysis with a first look at metric spaces and multivariable differentiability. The exam tests definitions, theorem statements and, above all, your ability to build valid proofs: epsilon-delta arguments, compactness, uniform convergence, and the inverse and implicit function theorems. Expect to justify every step rather than compute answers. Topics on this exam: Real Number System and Elementary Logic, Sequences, Series, and Limits, Continuity and Uniform Continuity, Differentiation and Mean Value Theorems, Riemann and Improper Integration, Sequences and Series of Functions; Uniform Convergence, Metric Spaces, Topology of R^n, and Compactness, Multivariable Calculus: Derivatives, Inverse and Implicit Function Theorems, and Proof Technique and Rigorous Argumentation. The PEI Math 521K case studies are 5 cases and about 2,405 words.
1Supremum, Accumulation Points, and a Misused Existential Quantifier
In a real-analysis review session, a student presents the following argument. Let A={1}∪{1/n:n≥2}.
480 words, about 2 min2Uniform Continuity of Power Functions on (0,1)
An exam problem asks: for which α∈R is f α(x)=x^α uniformly continuous on (0,1)? A student writes: 'For α 0, f α is continuous and 0<f α(x)<1, so it is bounded.
450 words, about 2 min3A Derivative That Cannot Jump: Darboux Versus a Piecewise Construction
A student is asked whether there exists a differentiable function f:(-1,1)→R with f′(x)=0 for x<0 and f′(x)=1 for x 0. The student proposes f(x)=0 for x≤0 and f(x)=x for x 0, and writes: 'On each side the derivative is correct.
517 words, about 3 min4Uniform Convergence of a Parameter Series on [0,1)
Consider the series S(x)=∑ {n=1}∞ x^n/(1+x^n), defined for x∈[0,1). A homework asks two questions: (i) is the series uniformly convergent on [0,a] for each fixed 0<a<1?
493 words, about 2 min5Compactness of a Convergent Sequence and Its Terms
In R with the usual metric, let S={0}∪{1/n:n∈N} and T={1/n:n∈N}. A student writes: 'Both sets are bounded.
465 words, about 2 min