Real analysis: final 1

Grading

This is the final exam for Real analysis: course.

1. Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
2. Prove that an open ball in a metric space is an open set.
3. Prove that a compact set in a metric space is bounded and closed.
4. Suppose $S,T$ are metric spaces and $f,g:S\rightarrow T$ are continuous functions. Prove that the set $A=\{x\in S:f(x)=g(x)\}$ is closed in $S.$ What can you say about $B=\{x\in S:f(x)\neq g(x)\}?$
5. State and prove the fundamental lemma of differentiation for $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$.
6. State the definition of a differentiable function $f:\mathbf{R}^{N}\rightarrow\mathbf{R.}$ Give an example of a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ such that both partial derivatives of $f$ exist at $x=a,$ but $f$ is not differentiable.
7. State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$
8. Give example of such a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ that $f$ is not continuous at $(0,0)$ but both partial derivatives exist at $(0,0).$
9. State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
10. Describe Newton's method. Give an example of a function for which the method does not apply.
11. Let $S$ be a complete metric space. Then every subset $A$ of $S$ ia also a metric space. Whan is and when is not $A$ a \textit{complete} metric space?
12. Give examples of functions $f:\mathbf{R}\rightarrow\mathbf{R}$ that satisfy and don't satisfy the Lipschitz condition.
13. Find an parametric equation of an ascending spiral in space. Define the arc-length of a parametric curve and provide its basic properties. Provide the integral formula.
14. Define the curvature of a curve. Find the curvature of the curve $<t^{2},t,5>$ as a function of $t>0.$ Under what circumstances is the acceleration perpendicular to the velocity?

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1. (a) State the definition of the limit of a function and three most important theorems about it. (b) Prove one of these theorems.
2. Suppose $\lim_{x\rightarrow\infty}f(x)=\infty$ and $\lim_{x\rightarrow\infty}g(x)=-\infty$. Give examples of such $f$ and $g$ for each possible $\lim_{x\rightarrow\infty}[f(x)+g(x)].$
3. Show that if a function is differentiable at a point, then it is continuous at that point.
4. Evaluate $\lim_{x\rightarrow0}(1+x)^{1/x}.$
5. Prove that if $f$ is continuous on $[a,b]$ then it is integrable on $[a,b].$
6. Discuss the continuity, differentiability, and integrability of the function: $f(x)=1$ if $x$ is rational and $f(x)=0$ if $x$ is irrational.
7. (a) State the Root Test and the Ratio Test. Give examples of their application. (b) Prove one of them.
8. State and prove the theorem about the interval of convergence of power series. Give examples of specific series for each case of the theorem, i.e., for each different type of interval.
9. Prove that the uniform limit of a sequence of continuous functions is continuous.
10. Show that the sequence $f_{n}(x)=x^{n}$ converges for each $x\in \lbrack0,1]$ but the convergence is not uniform. What happens to the sequences of the derivatives and the antiderivatives of $f_{n}?$